Improvement of Traveling Salesman Problem Solution Using Hybrid Algorithm Based on Best-Worst Ant System and Particle Swarm Optimization

: This work presents a novel Best-Worst Ant System (BWAS) based algorithm to settle the Traveling Salesman Problem (TSP). The researchers has been involved in ordinary Ant Colony Optimization (ACO) technique for TSP due to its versatile and easily adaptable nature. However, additional potential improvement in the arrangement way decrease is yet possible in this approach. In this paper BWAS based incorporated arrangement as a high level type of ACO to upgrade the exhibition of the TSP arrangement is proposed. In addition, a novel approach, based on hybrid Particle Swarm Optimization (PSO) and ACO (BWAS) has also been introduced in this work. The presentation measurements of arrangement quality and assembly time have been utilized in this work and proposed algorithm is tried against various standard test sets to examine the upgrade in search capacity. The outcomes for TSP arrangement show that initial trail setup for the best particle can result in shortening the accumulated process of the optimization by a considerable amount. The exhibition of the mathematical test shows the viability of the proposed calculation over regular ACO and PSO-ACO based strategies.


Introduction
The solution of Traveling Salesman Problem (TSP) [1] and its variants aids in finding the minimum travel distance as a cost function for a variety of computing problems. The extraordinary theoretical importance of TSP solution has a variety of practical applications, for example, design of circuit board layout [2] and transportation [3]. With the advent of intelligent transport, multi-machine industrial applications and swift delivery services, there has been an increased interest in optimizing the solution for TSP. The shortest travel distance, with the condition of traversing each node at least once, can be found by using different approaches with each having some advantage and associated weakness. Finding solution through an exact algorithm such as branch and bound [4] can take a very long run-time, particularly for a large number of nodes. Given a build-up of nodes y = y 1 , y 2 , y 3 , . . . , y n , as the objective of finding nearest travel path solution, including as much as decreased length with one time node visit condition. Vowing to practical applications and theoretical features, it is considered NP hard problem in conjunctional optimization. The other problems associated with direct methods like branch and bound [5] is the ultra-long run-time because the size of TSP iteration increases with the characters as combination explosion, for which even dynamic programming methods can hardly provide a solution within an adequate amount of time. The other type to solve TSP are the heuristic approaches such as bat algorithm [6], fruit fly algorithm [7][8][9], particle swarm algorithm [10,11], cuttlefish optimization algorithm [12] and artificial bee colony algorithm [13], which are result of the continued development of artificial intelligence. These algorithms work with a goal to find a satisfactory result, regardless of the concern to find an ideal one, because of time constraint. Ant Colony Optimization (ACO) [14], which is proposed by Italian Dorigo, is a swarm intelligent optimization algorithm with wide range of application like multi-objective optimization problems [15], vehicle routing problem [16], resource constrained job scheduling [17], dynamic railway junction rescheduling [18]. The authors in [19] integrate a local search (LS) method with an existing PSO algorithm with strong global search ability, named comprehensive learning PSO (CLPSO). In [20], the authors have explained a new multiobjective programming model for the target disassembly sequencing. It proposes an improved multiobjective ant colony algorithm to derive optimal target disassembly sequences. ACO is based on the social solution path which is build by a group of ants to travel in food search and then return back with an optimized travel path [21,22]. The basic principle is the exchange of useful information about the path by the ants, and thus improving the solution quality [23]. Better paths are chosen by ants as indicated by the amount of pheromone, where then the pheromone is updated. So, at this stage through different iterations the global optimal solution is induced [24]. The improved ACO like parallel and direction guided algorithms are presented in order to resolve premature and algorithm performance of ACO impairments [25]. From the analysis of predatory behavior of birds practical swarm optimization (PSO) intelligent algorithm has been introduced, which function is to modify the location and speed of elements to gain optimal outcomes in terms of population information and experience [26,27]. Moreover, PSO can be easily implemented due to its simple structure, therefore, it is majorly applied on problems like resource allocation, wind power prediction, etc., [28][29][30].
The above discussed PSO and ACO algorithms present efficient performance against problems. However, the complex structure and probability of ACO has decreased its services. Therefore, low cost and simple optimization procedures are necessary to overcome on these issues. Thus, the hybrid algorithm of combined ACO and PSO is studied in this paper to solve TSP. Enormous benchmark problems are used to test the performance of ACO-PSO and performance is compared with the recently proposed variants of hybrid ACO-PSO.

Organization and Notation of Paper
The remaining part of the paper is organized as follows: PSO is elaborated in Section 3. Section 4 presents ACO. Section 2 presents the Max Min ant system. Section 5 explains the proposed modified version of the Best-Worst Ant System. Similarly, Section 6 discusses hybrid algorithm of particle swarm and ant colony. Section 7 defines results and discussion. Section 8 concludes the study.

Max and Min Ant System
The Max and Min Ant Colony System (MMAS), which is planned in the trial examination and use of the insect framework, makes three upgrades to the insect province framework.

1.
To enhance the capacity of searching, highest value is set for every path of initial pheromone.

2.
In order to update pheromone, just ant with closest path is allowed in an iteration, which is measured as follow.
where L best addresses the briefest length in the current emphasis: 3.
To stay away from untimely assembly of the calculation, the pheromone centralization of every way (τ) is restricted to [ τ min , τ max ] and the worth past this reach is persuasively set to τ min or τ max .

Brief Review on PSO
PSO is considered a clever algorithm which is attained from the birds life way, where the traveling particle is dependent upon the neighborhood and swarm's global position. In the process of optimization, the number of iterations are kept fixed to ensure the convergence of the algorithm, after the selection of best number and best global value from the locals. In addition, every element is given a memory space for the best spot at any point found by utilizing the speed and position for the condition of every particle in the ith emphasis, addressed by ϑ p and ζ p , respectively. The values of best position for the ith particle (Pb) and best global position for the whole folk (Gb) are used to optimize the movement of the whole group. The velocity, position and learning factors of the individual particles are updated according to Equations (1)-(3), respectively.
where the inertial weight ω determines the total swarm population, the learning factors d 1 and d 2 determine the self-seeking ability and collective search ability of an individual particle. ξ 1 and ξ 2 can be between 0 and 1 with random probability.

Ant Colony Optimization
The ant colony algorithm is based on observing real abilities of ants to search for the best possible path (shortest) for food. To describe it mathematically, consider a group of x number of ants which are located at random locations across y number of cities, including primarily pheromone ρ mn (0) at the side of every city. The main city is added with taboo table Z k (0) of each ant, whereby, each ant finding the city to proceed the coming step using probability and can be defined as: where P k mn indicates the likelihood of moving from city m to city n of the kth insect and ρ mn (t) is for the benefit of that of the ith iteration. eta mn (t) is the corresponding of the distance between city m and city n. Parameters α and β are the part of pheromones and way length in picking probabilities, separately. Urban communities outside of the taboo table comprise of the set A t . After the taboo table of all the subterranean insect are satisfied, the all out distance every insect goes through will be determined and the most limited way will be recorded. All the while, the pheromone on each edge will be refreshed by the Equation (5) given beneath.
where ρ mn addresses the amount of the amassed pheromones between city m and city n and k τ mnk is for the benefit of the pheromone created by insect k on the edge mn. In the mean time, p is a number somewhere in the range of 0 and 1 showing the degree of pheromone dispersal. By and large, the aggregate sum of pheromone delivered in one cycle, which is addressed as Q, is restricted to speed up the intermingling speed. In the mean time, L k is for the benefit of the all out length of the kth way. For this situation, the recipe to ascertain ρ k mn is expressed in Equation (7) as

The Proposed Best-Worst Ant System
The best worst ant system (BWAS) idea attempts to enhance the performance of ACO models applying evolutionary algorithm ideas. The suggested BWAS uses the ant system (AS) transition rule which can be stated as Equation (9): ρ rs being the pheromone trail of edge (r, s), η rs being the heuristic esteem, J k (r) being the set of nodes that stay to be visited by ants k, and with α and β being real value weigts. In addition, the typical AS evaporation rule is utilized: ρ rs ←− (1 − τ)ρ rs , ∀ r, s, with τ ∈ [0, 1] belongs to the pheromone fall off parameter. Furthermore, the BWAS has the three after daemon actions that are observed in deep in [31] Best-Worst Performance Update Rule This performance update rule is based on the Population Based Incremental Learning (PBIL) [32] probability array update rule. The offline pheromone trail updating is given below: ρ rs ←− ρ rs + ∆ρ rs , where In Equation (10), the f (C(S global−best )) represents the measure of pheromone to be placed by the global best ant, which relies upon the quality of the solution it produced, C(S global−best ). Additionally, the edges present in the worst current ant are punished: ∀(r, s) ∈ S current−worst and (r, s) / ∈ S global−best , ρ rs ←− (1 − τ)ρ rs

Strategy
Swarm intelligence based ACO-PSO are good algorithm for solving the problems optimally. On the other side TSP is also a discrete optimization problem, however, its control parameters are handled by experience technique. The essential thought of ACO-PSO is to join the upsides of the two calculations and use BWAS to advance the boundaries of ACO, which is applied to tackle TSP. With the mix of both BWAS and PSO, ACO-PSO can set boundaries inside a sensible reach, in this manner upgrading the looking through capacity and accelerating the union.
There are two instatement measures in ACO-PSO. One of the interaction is called PSO introduction. In this cycle, N particles with three boundaries α, β and τ of every individual are haphazardly created to frame a 3 × N cluster. Where, α, β ∈ [0, 15] and τ ∈ [0, 1]. In MMAS instatement, which is the other introduction measure, the underlying pheromone of each side ρ mn (0)= c what's more, pheromone variety ρ mn = 0 are set. At that point haphazardly place N subterranean insects and add the underlying chosen city into untouchable table. The system of ACO-PSO is given in Algorithm 1.
It ought to be seen that g and G are the current cycle and the complete emphasis individually. In the interim, n represents the city number of the issue. Furthermore, to check the searching performance of the ACO-PSO, a few standard test sets are utilized to test the effectiveness of the algorithm.

Algorithm 1 Pseudocode: Hybrid ACO-PSO PSO initialization:
For d = 1 to D MMAS introduction: While (not arrive at the most extreme cycles of MMAS) for i = 1 to i For n = 1 to K Determine target city as per condition (1) Udate Taboo Table  end end Calculate the length of every subterranean insect way; Find the ideal arrangement, the most exceedingly terrible arrangement and the worldwide ideal arrangement of this emphasis; Update worldwide pheromone utilizing condition (5); end; Set the most brief way length as the wellness work esteem; Update the speed and position of every particle; end

Pheromone Trail Mutation
The pheromone trails endure mutations to announce diversity in the search, as done in PBIL with the memoristic structure. To do as such, Equation (11) shows how each line of the pheromone matrix is transformed with probability P m as: with a being an arbitrary value in 0, 1, it being the present iteration, τ threshold being the average of the pheromone trail in the edges forming the global best solution and with mut(·) being a capacity making a more grounded mutation as the iteration counter increases.

Restart of the Search Process When It Gets Stuck
The pheromone matrix is restarted by setting all its components to τ o when the number of edges that are distinctive between the present best and the present worst solutions is lesser than an obvious percentage. A simplified structure of a common BWAS algorithm [31] can be used to make the Algorithm 2: Algorithm 2 Pseudocode: Best worst ant system algorithm.
Give an initial pheromone value, ρ 0 to each edge For n = 1 to k do (in equal) Place insect n in an initial hub s and incorporate s in D n While (insect n not in an objective hub) do Select the following hub to visit, r / ∈ D n by the AS change rule. For D n = 1 to m do Run the nearby pursuit enhancement for the arrangement created by subterranean insect n, r n . s global−best ←− worldwide best subterranean insect visit. s global−worst ←− current most noticeably terrible insect visit.
Pheromone dissipation and Best-Worst pheromone refreshing. If (Stop Condition isn't fulfilled) go to step 2.

Result and Discussion
The proposed outcomes are discussed as follows: The inertia weight is kept at 0.8 and d 1 and d 2 are the learning factors which is equal to 2. Two experimental values, the optimal solution and average value of the 20 independent runs are studied in this paper to evaluate the outcomes. The two variants, PSO and ACO, are taken as comparison algorithm in order to check the performance of ACO-PSO. As is exhibited in over, the outcomes acquired by ACO-PSO is obviously superior to those of ACO and PSO among the 16 benchmark issues. To be more instinctive, we draw the way of every calculation as per the ideal arrangements utilizing the issue Pr124.
It can be demonstrated from the Tables 1 and 2 that the performance of ACO-PSO is much superior to those of simple ACO and PSO not only in the aspect of best results, however, also in worst case. Hence, it can be concluded that BWAS algorithm provides more accurate results than those of MMAS. The analysis is performed using 16 set of different functions from the test environments of IEEE Congress on Evolutionary Computation (CEC) [33]. The mean and standard analysis of Tables 1 and 2 are mentioned in  Tables 3 and 4. Figure 1a-c explain the experimental analysis for PSO (BWAS) in terms of best tour path, best cost and averaging node branching, respectively. Figure 1a. includes the outcomes of the global best tour path based on PSO (BWAS) algorithm. It is clarified that the results present satisfactory convergence, which concludes that BWAS algorithm is more feasible as compare to the current work, generating convergence with high speed.  Figure 4a. shows the better convergence based on best tour path, while Figure 4b,c represent global minimum cost and best average node branching for pr-124. Figure 5a. describes the best possible tour path for ACO (MMAS) based algorithm, the global minimum cost in terms of ACO (MMAS) algorithm is depicted in Figure 5b. Moreover, Figure 5c. consists of best average node branching for pr-124. The integrated results of ACO-PSO (MMAS) are shown in Figure 6a-c. These results present more accurate performance than single ACO (MMAS) and PSO (MMAS). Thus, It is found from the results of the BWAS proposed model are several folds efficient that other existing variants. The Table 5 summarized the performance of the proposed approach in comparison with exist literature.

Conclusions
This paper concludes the results of simple ACO and PSO for BWAS and MMAS algorithms. The outcomes based on simple ACO and PSO show that approaches are limited to solve TSP issues. Thus, a hybrid algorithm based on integration of BWAS based ACO and PSO is proposed, to improve the solution for TSP. The optimization of BWAS parameters is performed in concurrence with output from PSO, and results show a reasonable improvement in shortening the optimized solution path. Hence, in order to increase the searching ability and improving the overall results, multiple standard sets are used as a baseline to compare the performance of the proposed algorithm with conventional PSO and ACO methods. The results show the superiority of the proposed technique in solving TSP.