Solving the single depot open close multiple travelling salesman problem through a multi-chromosome based genetic algorithm

The multiple travelling salesman problem (MTSP) extends the classical travelling salesman problem (TSP) by involving multiple salesman in the solution. MTSP has found widespread applications in various domains, such as transportation, robotics, and networking. Despite extensive research on MTSP and its variants, there has been limited attention given to the open close multiple travelling salesman problem (OCMTSP) and its variants in the literature. To the best of the author's knowledge, only one study has addressed OCMTSP, introducing an exact algorithm designed for optimal solutions. However, the efficiency of this existing algorithm diminishes for larger instances due to computational complexity. Therefore, there is a crucial need for a high-level metaheuristic to provide optimal/best solutions within a reasonable timeframe. Addressing this gap, this study proposes a first meta-heuristic called multi-chromosome-based Genetic Algorithm (GA) for solving OCMTSP. The effectiveness of the developed algorithm is demonstrated through a comparative study on distinct asymmetric benchmark instances sourced from the TSPLIB dataset. Additionally, results from comprehensive experiments conducted on 90 OCMTSP symmetric instances, generated from the renowned TSPLIB benchmark dataset, highlight the efficiency of the proposed GA in addressing the OCMTSP. Notably, the proposed multi-chromosome-based GA stands out as the top-performing approach in terms of overall performance. Further, solutions to symmetric TSPLIB benchmark instances are also reported, which will be used as a basis for future studies.


Introduction
The Multiple Traveling Salesman Problem (MTSP) represents an extension of the classical Traveling Salesman Problem (TSP).In MTSP, the challenge involves coordinating multiple salesman to efficiently cover a specified set of cities, ensuring each city is visited exactly once, and the salesman collectively return to the starting point, aiming to minimize the total traversal cost or distance.The MTSP is closely connected with diverse optimization problems such as vehicle routing problem (VRP) (Braekers et al., 2016) and task assignment problem (TAP) (Öncan, 2007).Indeed, MTSP can be viewed as a relaxed version the VRP, assuming unlimited vehicle capacity and customers with unit demands.While MTSP shares similarities with the TSP and the TAP, it imposes restrictions on multiple visits to the same city and the formation of subtours.Consequently, a solution to MTSP can be effectively applied to tackle the challenges posed by VRP or TAP.
The MTSP has undergone extensive research and has found applications in diverse fields, including transportation, robotics, and networking.In different practical scenarios, the "salesman" in MTSP can represent entities such as trucks, robots, or drones.The cities to be covered by the salesman correspond to customers in transportation and logistics distribution, critical sites in disaster management, targets in strategic war operations, sensor nodes in wireless sensor networks, and victims in emergency missions (Cheikhrouhou & Khoufi, 2021).Due to its broad applicability, MTSP has garnered significant attention from the research community.Researchers have explored various variants of the problem, such as MTSP with multiple depots (Malik et al., 2007), solid MTSP (Changdar et al., 2017), MTSP with precedence constraints (Sarin et al., 2014), MTSP with time windows (Kara & Bektas, 2006), Open close MTSP (Thenepalle & Singamsetty, 2019) and Open MTSP with load balancing (Thenepalle & Singamsetty, 2021) etc. Due to NP-hardness, no known algorithm is presented that works within polynomial time for solving the MTSP and its variants (Garey & Johnson, 1979).MTSP, being a generalized form of TSP, allows solution techniques developed for TSP to be applicable to MTSP and its variations.These solution approaches can broadly be categorized into two types: heuristics or metaheuristics, which provide solutions that are optimal or near-optimal without guaranteeing their quality, and exact algorithms, which assure optimal solutions.As the problem size grows, achieving an optimal solution becomes increasingly challenging and computationally expensive.Consequently, the application of heuristics or metaheuristics becomes essential for solving practical MTSP models within reasonable time constraints.
Concerning the solution techniques, some of the heuristic/metaheuristics techniques devoted for MTSP by implementing biological characteristics such as Evolutionary approach (Bao et al., 2021;Sofge et al., 2002), Particle swarm optimization (PSO) (Yan et al., 2012), Ant colony optimization (ACO) (Changdar, 2023;Wang et al., 2020;Yousefikhoshbakht, 2013), Hybrid algorithm (Jiang et al., 2020), Genetic algorithm (GA) (Gomes et al., 2021;Király & Abonyi, 2011;Lou et al., 2021;Singamsetty & Thenepalle, 2021), Learning-based metaheuristic approach (Belhor et al., 2023), Efficient routing optimization with discrete penguins search algorithm (Mzili & Riffi, 2023) etc. Various heuristic and metaheuristic approaches have been proposed for the Multiple Traveling Salesman Problem (MTSP) and its variations, each carrying its unique strengths and challenges.For instance, Particle Swarm Optimization (PSO) is known to be prone to local optima, while Ant Colony Optimization (ACO) is computationally demanding, characterized by a slower convergence rate.On the other hand, Genetic Algorithms (GA) tend to exhibit early convergence and can be heavily influenced by the initial population.Despite these complications associated with GA, literature shows to its effectiveness, as it is widely employed for successfully addressing the MTSP and its diverse variants (Xu et al., 2018).
Examining the evolution of Genetic Algorithms (GA), Bagley (1967) was a pioneer in introducing the concept, followed by Holland, who conducted a scientific exploration into the mechanism of the survival of the fittest in 1975.Subsequently, the literature has witnessed significant progress in the development of GA, particularly in its application to addressing the MTSP and its variants, which can be seen in following works (Al-Omeer & Ahmed, 2019;Alves & Lopes, 2015;Brown et al., 2007;Carter & Ragsdale, 2006;Harrath et al., 2019;Kaliaperumal et al., 2015;Király & Abonyi, 2011;Shuai et al., 2019;Tang et al., 2000;Thenepalle & Singamsetty, 2021;Xu et al., 2018;and Yuan et al., 2013).The cited studies motivate us to develop an efficient GA for solving the OCMTSP, aiming to find optimal solutions within a short timeframe.Table 1 summarizes the various instances of MTSP and its variants addressed through different solution methodologies.
The MTSP can be categorized into two main types based on the nature of the salesman routes: open MTSP and open-close MTSP.In the open MTSP, all salesman initiate their routes from the depot city and are not obligated to return to the starting city after completing the assigned cities.On the other hand, the open-close MTSP (OCMTSP) involves salesman starting from the depot city, visiting a specified number of cities, with only certain salesman required to return to the depot city.The goal is to minimize the overall distance or cost covered in the routing process.This model finds practical applications in transportation and logistics distribution, particularly in scenarios where vehicle operations are outsourced.
For instance, consider Fig. 1, which illustrates two variants of the classical MTSP: open MTSP and open-close MTSP.In this representation, numbered circles denote cities to be visited by the salesman, with city 0 serving as the depot city where all salesman commence their routes.Fig. 1 depicts a scenario of open MTSP involving 3 salesman and 10 cities, including the depot city.As shown, salesman 1 covers 3 cities, salesman 2 visits 3 cities, and salesman 3 covers 3 cities.Similarly, Fig. 2 illustrates a scenario of OCMTSP involving 3 salesman (1 internal and 2 external) and 10 cities, including the depot city.As shown, salesman 1 covers 3 cities and returned to depot city, salesman 2 just visits 3 cities, and salesman 3 covers 3 cities.The varying routes and the decision of which salesman return to the depot city contribute to minimizing the overall distance or cost incurred in the routing process.This flexibility in route planning is particularly beneficial in logistics and transportation optimization.Upon reviewing the existing literature, it becomes evident that, with the exception of the OCMTSP model, the other models mentioned have received significant attention.To the best of the author's knowledge, the only investigation into OCMTSP, addressed by an exact lexi-search algorithm (Thenepalle & Singamsetty, 2019), stands as the only study on this specific topic in the literature.However, the existing exact algorithm for OCMTSP demonstrates effectiveness primarily for smaller instances but proves computationally expensive when applied to higher dimensions.Recognizing this limitation, there is a pressing need to devise high-level, efficient metaheuristic approaches capable of producing optimal solutions within practical time constraints.To address this gap, the present study aims to introduce a multi-chromosome-based Genetic Algorithm (GA) for solving OCMTSP.The performance of this proposed approach is thoroughly analyzed through computational experiments, aiming to provide insights into its performance.
The rest of the paper is organized as follows: Section 2 describes a detailed description of the problem statement and its formulation.The preliminary concepts of GA are presented in Section 3. Extensive computational results are reported in Section 4. Finally, conclusions are given in section 5.

Problem Statement
The OCMTSP is formally stated as follows: Let x assumes 1, and 0 ij x = , otherwise.The OCMTSP objective is to find a solution involving p closed paths and q open paths, such that each city is to be covered by just one salesman and the total distance covered by msalesman is minimized.Note that, the lower and upper bound on the number of cities visited by any salesman is 1 and

Assumptions
The below assumptions are used to formulate the OCMTSP: a.There are n cities to be covered by m salesman located at the depot city, of which a predefined p salesman are intended for closed paths and q salesman are employed for open paths.b.All the salesman is asked to begin at the depot city and only p salesman are required to come back to the depot city, whereas the other q salesman is not necessary to return.
c.The values , ,& m p q are predefined.
d.The cities assigned dynamically to each salesman in order to minimize the overall traversal distance.e.Each salesman has to cover atleast 1 city and at most 1 n m − + cities.

Mathematical Model
The OCMTSP can be expressed as a zero-one integer linear programming (0-1 ILP) as follows: In this article, a salesman objective (distance) is established that generally assures that the distance found is reduced.The mathematical model formulated by Thenepalle & Singamsetty, (2019) is considered without any modifications in the present study a.This objective is to minimize overall traversal distance The objective is to minimize overall traversal distance.This objective function promises that the distance involved in covering all the cities is minimized.It can be shown as: ,where bethesetof cities covered by the salesman +Sub tour/illegal tour elimination constraints (8) {0,1} , The 2 nd Constraint guarantees that any feasible solution must include 1 m n q + − − edges.The 3 rd and 4 th Constraints ensure that m salesman departs from the depot city ( ) α and exactly p salesman must return to it.The 5 th and 6 th Constraints guarantee that a salesman visits each city exactly once and can depart from each city at most once.The 7 th Constraint ensures that each salesman covers atleast 1 city and at most 1 n m − + number of cities.The 8 th Constraint is intended to eliminate the sub-tours from the solution.Finally, Constraint (9) denotes the binary variable.

Genetic algorithm
This section provides an overview of both the conventional Genetic Algorithm (GA) and the proposed algorithm.The GA stands out as one of the widely adopted metaheuristic algorithms in evolutionary computation for addressing various combinatorial optimization problems, as highlighted by Goldenberg (1989).Originating from the survival of the fittest strategy introduced by Holland in 1975, GA is an adaptive exploration method.Genetic Algorithms are extensively utilized for solving the Multiple Traveling Salesman Problem (MTSP) and its variants, as evidenced by the following studies: (Tjandra et al., 2022 andWang et al., 2021).Typically, a GA commences with an initial set of solutions constituting the initial population, referred to as chromosomes, where all genetic information is encoded.Each element within the chromosome is treated as a gene.The efficiency of a chromosome is evaluated through a fitness value.The GA process involves selecting two parent chromosomes with high fitness values randomly from the population.Following the selection, a crossover operation is performed between the chosen parent chromosomes to generate two new chromosomes.The resulting chromosomes replace the old ones if they exhibit superior fitness values.To maintain diversity within the population, a mutation operation is applied to the newly generated chromosomes.This cycle of selection, crossover, and mutation iterates, producing novel chromosomes until the size of the new population matches that of the old one.The updated population is then utilized to initiate the next iteration.Notably, the probability of selecting superior chromosomes for crossover is higher, and the newly generated chromosomes are more likely to inherit the characteristics of their parent chromosomes.The search process continues for several generations until predefined conditions are met, constituting a complete iteration of the classical GA.

Proposed GA
To achieve an optimal or suboptimal solution for the OCMTSP, various crucial factors come into play.These factors encompass chromosome representation, population initialization, computation of fitness values, selection mechanisms, crossover operations, mutation operators, as well as the fine-tuning of GA parameters.The diversity in GA approaches stems from variations in how encoding, crossover, and mutation operations are employed, leading to divergence in the search process.Consequently, the redesign of these fundamental operations is imperative to ensure the attainment of optimal or suboptimal solutions.The foundational components of the proposed GA for OCMTSP are outlined as follows:

a. Chromosome representation
To efficiently address the OCMTSP, it is crucial to adopt a suitable chromosome representation.Given that the MTSP is a generalized form of the TSP, it is possible to adapt TSP chromosome representations with minor adjustments for OCMTSP.Various techniques have been proposed for representing TSP solutions as chromosomes, including path representation (Larranaga et al., 1999), matrix representation (Khan et al., 2009), double chromosome representation (Riazi, 2019), and multi-chromosome representation (Singh et al., 2018).
In this study, the OCMTSP solution is depicted using a multi-chromosome representation, a technique previously employed by researchers such as Albayrak and Allahverdi (2011) and Király and Abonyi (2015).This approach involves utilizing the same number of chromosomes as there are salesman.The length of each chromosome, which dynamically changes, is determined by the number of cities assigned to each respective salesman.This choice of representation is made to suit the specific characteristics and constraints of the OCMTSP, aligning with the objectives of this research.Assume that the first chromosome is 1 p , the second is 2 p , and so on.As a result, total number of cities in the multi-chromosome representation equals to  (since depot city is not included in any of the chromosome).Cities are randomly chosen to be assigned to salesman.For example, let there will be 17 cities ( 17  The objective function, as expressed in Eq. ( 1), aligns with our fitness function in this study.In the context of the OCMTSP, the fitness value serves as a measure of the total traversal distance incurred by m salesman while visiting a set of n cities, including the home city.It is noteworthy that, in this context, a higher fitness value indicates a more favourable chromosome.Therefore, the better the chromosome in terms of solving the OCMTSP, the greater the corresponding fitness value.The fitness function effectively captures the essence of optimizing the total traversal distance for each salesman, and thus, maximizing the fitness value signifies superior solutions within the genetic algorithm framework.

c. Selection
In our approach, we employ tournament selection, where 8 individuals compete for survival.The chromosome exhibiting the lowest fitness value successes in the tournament, earning the privilege of being chosen for the generation of new individuals.The selected individual seamlessly integrates into the new population without any modifications.This process ensures that more favourable chromosomes are likely to be chosen, while less desirable ones are discarded.The underlying principle of this selection strategy is to transmit high-quality chromosomes to the next generation, enhancing both evaluation efficiency and the convergence towards optimal and sub-optimal solutions.

d. Mutation Operators
Following the selection process, the mutation operation is promptly executed.Its primary goal is to prevent the GA from becoming trapped in local optima and to enhance the genetic diversity within the population.In this study, we incorporate seven mechanisms-namely, Flip, Swap, Slide, Crossover, Flip + Crossover, Swap + Crossover, and Slide + Crossoveradopted from the work of Király & Abonyi (2015).These mechanisms collectively contribute to enhancing the population of candidate solutions across generations, thereby maintaining genetic diversity within the population.Consequently, this approach mitigates the risk of the algorithm being confined to local optima by facilitating exploration across a broader search space.As previously discussed, there exist various genetic operators in the literature.For instance, a multichromosomal mutation can be derived by combining a series of single-chromosomal mutations.While the majority of these operators can be derived from others, introducing a new representation may necessitate the inclusion of additional genetic operators.The operators outlined below can be constructed from existing simple operators.The terms "In-route mutations" and "Cross route mutations" denote two distinct sets of mutation operators.In-route mutation operators, such as flip or gene sequence inversion, modify only two genes within a chromosome and operate within a single chromosome.Larranaga et al., (1999) serve as reliable sources for these "classical" representations and operators.On the other hand, a cross-route mutation operator simultaneously alters several chromosomes.It is important to note that this operator may bear similarity to the standard crossover operator when utilizing conventional notation and treating chromosomes as individual entities.
To begin, the application of an in-route mutation known as the flip operator is initiated, modifying a set of genes within a chromosome.Subsequently, a Swap operator is employed, transposing the gene sequences between two chromosomes, resulting in the creation of a new offset.Following this, a slide operator is applied, moving the last gene from each chromosome to the beginning of another.In addition to in-route mutations, a cross-route mutation employed in this study is the crossover operator, which executes a one-point crossover between two chromosomes.In this process, two random crossover points are selected, and the tails of the two chromosomes are swapped to generate new offspring.The effective utilization of these basic mutation operators is visually depicted in Figs.4-7.
Utilizing the aforementioned simple mutations, more complex mutations, namely Flip + Crossover, Swap + Crossover, and Slide + Crossover, can be derived.Figure 8 depicts the process when two cross-route operators are sequentially applied, resulting in a complex mutation.Initially, a Swap operator is employed, transposing the gene sequences between two chromosomes and moving the last gene from each chromosome to the beginning of another.Subsequently, another Swap operator is applied, generating the new offset.The algorithm's effectiveness is determined by its parameter values, encompassing the population size, mutation probability rate, and termination criterion.A thorough calibration process was undertaken to establish optimal values for the genetic algorithm parameters.The population size is set at 160, and the iteration number is fixed at 3000.The utilization of crossover and mutation operators with diverse probability values is designed to guide the Genetic Algorithm (GA) toward the convergence of efficient solutions.The termination condition for the current GA is contingent upon reaching the maximum number of generations.The flow diagram of the proposed GA is illustrated in Fig. 9.

Computational Results
In this section, we present the experimental results attained by employing the proposed GA.Subsequently, we provide a comprehensive analysis of the algorithm's performance by comparing it to alternative state-of-art algorithms.Our experimentation involved the utilization of diverse benchmark instances sourced from TSPLIB to assess the algorithm's performance.The implementation of the algorithm was carried out in MATLAB 2023a, executed on a PC equipped with MS Windows 2010 and an Intel Core i3-5005U CPU operating at 2 GHz and an 8GB RAM.The algorithm was subjected to 3000 iterations, population size 160, with a termination condition defined as the stabilization of the best solution over ten consecutive iterations.The reported results represent the mean outcomes derived from 50 independent runs of the algorithm.

Comparative results of OCMTSP over asymmetric TSP benchmark instances
This section presents a comparative study of OCMTSP over asymmetric TSP benchmark instances generated from TSPLIB (http://comopt.ifi.uni-heidelberg.de/software/TSPLIB95/).To measure the performance of the proposed GA, it has been tested over 17 asymmetric TSP benchmark instances generated from br17, ftv33, ftv35, and ftv44.The GA results obtained are compared against with the best know results achieved by Lexi-search algorithm (LSA) (Thenepalle & Singamsetty, 2019) and the TSP solver LKH-3 (http://webhotel4.ruc.dk/~keld/research/LKH-3/).Table 1 comprehensively presents the results of our analysis.Notably, in 3 out of the 17 instances, the solutions obtained through the GA align precisely with those derived from Lexi-search algorithm (LSA) and the TSP solver LKH-3.For the remaining cases, the GA solutions demonstrate proximity to the best-known solutions.This observation is reinforced by examining deviation percentage values calculated using equation ( 10).Additionally, it is noteworthy that the proposed GA exhibits a marked advantage in terms of CPU runtime, ranging from 6.12 seconds to 6.92 seconds.

Experimental results of OCMTSP over symmetric TSP benchmark instances
The author's would like to highlight that as far as their knowledge extends, there exists only a single study addressing the asymmetric OCMTSP.Consequently, no evaluations comparing algorithmic performance against symmetric benchmark test cases have been conducted.However, comprehensive experimentation has been undertaken, involving the evaluation of the proposed Genetic Algorithm (GA) on symmetric TSP benchmark instances.The benchmark dataset comprises a total of 90 distinct test cases, categorized into 15 groups.Each group consists of six distinct test cases, each with varying parametric (i.e. , & m p q ) values.The results of these experiments are presented in Table 2.
The results reveal that the CPU runtime required for the proposed GA to determine the best solution varies between 23.05 seconds and 28.70 seconds, as visually depicted in Fig. 10.It's important to emphasize that CPU runtimes exhibit inconsistency, making it challenging to pinpoint which specific parameters or factors might be influencing these variations in performance.These findings demonstrate the strong performance of the proposed algorithm in terms of computational efficiency.Further, mean absolute deviation (MAD) for each test case has been computed.Based on the obtained, it is evident that the MAD implies that the solutions generated by the proposed GA in ten separate runs exhibit a slight tendency to cluster around the average solution.It should also be emphasized that the results presented here can serve as a valuable reference for future research comparisons.Finally, the best route plans for the instance eil51 in each of the six cases are now depicted in Figs.11-16.

Conclusions
In this research paper, we developed a genetic algorithm that depend on a multi-chromosome approach to tackle the open close multiple traveling salesman problem (OCMTSP).As far as the author's knowledge, this is the second study and first meta-heuristic algorithm designed for addressing the OCMTSP.To assess the effectiveness of our GA, incorporating the multi-chromosome method, we conducted a comparative analysis over 17 asymmetric TSP benchmark instances against other established methods such as LSA, and LKH-3.Our computational results conclusively demonstrate that our proposed GA outperforms all other methods in terms of computational runtime and reasonably performs well in terms of solution quality.Additionally, we conducted a thorough validation of GA effectiveness and performance by testing it on a diverse set of instances 90 symmetric instances with different parametric ( ) . ., & i e m p q values derived from the TSPLIB Library.Based on the statistical metrics employed, it is evident that the proposed algorithm excels in addressing OCMTSP.Notably, it's important to highlight that the proposed GA is versatile and effective for solving both symmetric and asymmetric instances of the problem.As it is new attempt, these findings serve as a valuable benchmark for future investigations, offering insights into the problem's complexity and highlighting potential areas for improvement using state-of-the-art algorithms.However, it is essential to acknowledge the limitations of this research.The proposed algorithm, while efficient, may not guarantee the optimal solutions.Despite the limitations, this work serves as a stepping stone for future research, as we look forward to seeing advancements in the context of OCMTSP scenerios.
Future research directions can focus on various aspects.Firstly, exploring more advanced algorithms, such as meta-heuristic techniques, could lead to further improvements in optimizing OCMTSP.Secondly, incorporating practical constraints into the model could render it more realistic and robust, aligning it better with real-world scenarios.

Fig. 2 .
Fig. 2.An arbitrary open close MTSP solution with 10 cities and 3 salesman ) n = including depot city (α = 1), four salesman (m = 4) are positioned at the depot city.Of the four salesman, two salesman (p = 2) are intended for closed paths and other two (q = 2) are devoted for open paths.Then the multi-chromosome representation of an arbitrary OCMTSP solution with seventeen (n = 17) cities and with four salesman ( Fig.3.For the four salesman involved, the chromosomes are partitioned into four distinct parts, each uniquely represented by a different colour, as depicted in Fig.3.The numerical values within each segment denote the cities covered by the respective salesman.To create a closed tour for each salesman, the home city (1, for instance) is inserted at both the beginning and the end of each segment.It's important to note that the insertion of the home city (1) at the end is only applicable for closed tours but not for open tours.The route plan for the four salesman is outlined below: Salesman 1: 1→ 4 → 9 → 15 → 12 →6→ 1; Salesman 2: 1→ 14→ 5 → 7 →3 → 1 and Salesman 3: 1→ 2 → 10 → 17; Salesman 4: 1→ 8 → 16 → 13→11.Here, closed paths are associated with the salesman 1 and 2, whereas, open paths are assigned to the salesman 3 and 4.

Fig. 3 .
Fig. 3. Illustration of a multi-chromosome representation for a 17-city OCMTSP with 4 salesman b.Evaluation of fitness function

Fig. 10 .
Fig. 10.Proposed GA performance in terms of CPU runtime (in sec.)

Table 1
Comparative results of GA against LSA and LKH-3 for OCMTSP over asymmetric TSP benchmark instances

Table 2
GA performance results on symmetric TSP benchmark instances for OCMTSP

Table 2
GA performance results on symmetric TSP benchmark instances for OCMTSP(Continued)