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This paper presents a comparative performance analysis of some metaheuristics such as the African Buffalo Optimization algorithm (ABO), Improved Extremal Optimization (IEO), Model-Induced Max-Min Ant Colony Optimization (MIMM-ACO), Max-Min Ant System (MMAS), Cooperative Genetic Ant System (CGAS), and the heuristic, Randomized Insertion Algorithm (RAI) to solve the asymmetric Travelling Salesman Problem (ATSP). Quite unlike the symmetric Travelling Salesman Problem, there is a paucity of research studies on the asymmetric counterpart. This is quite disturbing because most real-life applications are actually asymmetric in nature. These six algorithms were chosen for their performance comparison because they have posted some of the best results in literature and they employ different search schemes in attempting solutions to the ATSP. The comparative algorithms in this study employ different techniques in their search for solutions to ATSP: the African Buffalo Optimization employs the modified Karp–Steele mechanism, Model-Induced Max-Min Ant Colony Optimization (MIMM-ACO) employs the path construction with patching technique, Cooperative Genetic Ant System uses natural selection and ordering; Randomized Insertion Algorithm uses the random insertion approach, and the Improved Extremal Optimization uses the grid search strategy. After a number of experiments on the popular but difficult 15 out of the 19 ATSP instances in TSPLIB, the results show that the African Buffalo Optimization algorithm slightly outperformed the other algorithms in obtaining the optimal results and at a much faster speed.

There have been disagreements among computer science experts with regards to what constitutes artificial intelligence and computational intelligence [

On the contrary, artificial intelligence refers to that computer software which enables robots or computer systems perform human tasks with exceptional abilities, accuracy, speed, and capacity. AI encompasses the design of algorithmic and nonalgorithmic methods that involve the use of robots, computer vision, graphics and human-computer interactions, language processing, etc. [

In view of the enormous contributions of AI and computational intelligence (CI) to human development over the years as highlighted above, researchers have devoted much resources and time investigating this area with a viewing to unraveling the untapped potentials inherent in AI and CI, respectively [

In recent times, several heuristics cum metaheuristic methods have been designed to solve the problem of optimization in science, engineering, industrial, and technological problems encountered in many practical fields of human endeavour. Some of these optimization techniques are deterministic while others are stochastic. Deterministic techniques/algorithms have in-built mechanisms that guarantee the exact solution to an optimization problem but many times run into serious problems as the search space gets larger [

Stochastic algorithms use search agent or agents in their search and obtain solutions iteratively without a guarantee of optimal results. However, stochastic algorithms are highly efficient in monomodal and multimodal search environments of any size. Presently, much attention is focused on the stochastic algorithms and, most recently, in hybridization of stochastic algorithms since they tend to be more successful in finding optimal or near-optimal solutions to some difficult real-life situations that require optimization for better results [

Hybrid algorithms are simply a combination of two or more algorithms in such a way that the algorithms are made to cooperate and jointly solve a problem. Hybridization of algorithms are done to harness the unique capabilities of the cooperating algorithms to enhance search efficiency and effectiveness in terms of obtaining optimal or near-optimal solutions, escaping stagnation and ensuring faster computational speed, etc. There are several algorithm-hybridization architectures in literature ranging from master-slave, relay to peer-to-peer paradigms, etc. In all, algorithm-hybridization synergizes algorithms in such a way as to complement one another in order to ensure greater efficiency and effectiveness [

On the one hand, metaheuristics refer to a kind of high level, stochastic, problem-independent, and intelligent manipulators of heuristic information to achieve greater efficiency in their search enterprise [

A heuristic, on the other hand, is an approximate, problem-dependent set of instructions, methods, or principles designed to solve a problem at a reasonable computational cost. Generally, heuristics are, relatively, simple mechanisms designed to determine the cheapest/best/most effective solution among a set of solutions. However, due to the prevalent use of greedy search strategy, heuristics have the problem of premature stagnation. Examples of heuristic algorithms are Lagrangian Relaxation [

The primary difference between heuristics and metaheuristics is that, usually, heuristics are problem-dependent, but the metaheuristics are general-purpose algorithms. Again, metaheuristics have inherent memory capabilities that enable them learn while executing, thus enabling them adapt to any problem, unlike pure heuristic algorithms [

To the best of our knowledge, this is the first time the algorithms used in this comparative analysis are being compared together in one study. The analysis in this paper involves hybrid, metaheuristic, and heuristic algorithms. Moreover, this study aims at adding to the body of knowledge in the ATSP literature which, we observed earlier, is not as researched into as its symmetric counterpart though ATSP has more real-life applications than the symmetric TSP. Moreover, it is hoped that it will be a useful tool in the hands of researchers having to carry out studies that involve the ATSP.

The rest of this paper is organized as follows. Section

The travelling Salesman Problem (TSP) is about the most studied problem among combinatorial optimization problems and is fast becoming the most reliable test bed for newly designed optimization methods [

There are two types of TSP: asymmetric and symmetric. Usually, the symmetric TSP is easier to solve since both to and from journeys are the same in cost/length, as such optimization algorithms simply calculate one length of the journey across different nodes [

In ATSP, there exists a set

This way, the ATSP tour of any three cities

Available literature indicates that metaheuristic approaches used in solving the TSP with a little transformation are effective in providing solutions to the ATSP [

A critical review of literature on the Travelling Salesman Problem reveals that there are lots of studies on the symmetric Travelling Salesman Problem over the past several decades. However, it is rather ridiculous that there exists a paucity of literature on the asymmetric TSP [

This study specifically investigates the performance of six optimization algorithms in literature that have exhibited exceptional performances in solving the ATSP. These algorithms are the African Buffalo Optimization (ABO), Model-Induced Max-Min Ant Colony Optimization (MIMM-ACO), Max-Min Ant System (MMAS), Cooperative Genetic Ant System (CGAS), Improved Extremal Optimization (IEO), and Randomized Insertion Algorithm (RAI). The choice of these algorithms for comparison is informed by their special characteristics, while the ABO and the IEO are standalone metaheuristic algorithms, the MMAS, MIMM-ACO, and CGAS are hybrid metaheuristics, and the RAI is a heuristic algorithm.

The ABO which was inspired by the marvelous organizational ability of herds of buffalos, which, sometimes, are upto 1000 individuals in a single herd, using two primary vocalizations: the

ABO algorithm.

The ABO applies the Modified Karp–Steele algorithm in its solution of the Asymmetric Travelling Salesman Problem [

So far, the observed limitations of the ABO lie in the fact that buffalos are parliamentary in decision-making. That is to say that the choice of majority population of the herd determines their next destination. In the standard ABO variant, the modeling process is not explicit leading to the generation of a new population when there is a case of stagnation occasioned by the decision of the majority of the herd. Another area of weakness is that the frequent reinitialization of the entire population has the tendency to limit the directional search capacity of the buffalos when the algorithm is faced with complex engineering challenges. These observed challenges necessitated the development of the Improved African Buffalo Optimization [

The Cooperative Genetic Ant System (CGAS) [

This information exchange between GA and AS in the end of the current iteration enables the algorithm to choose the best solutions for the next iteration. Such cooperation helps the algorithm to arrive at the global optimal solution and ensures adequate exploration of the search space. The Cooperative Genetic Ant System algorithm is presented in Figure

CGAS algorithm.

The Max-Min Ant System (MMAS) developed by Stuzzle and Hoos [

MMAS algorithm.

To solve ATSP, first, the ants are placed on some randomly selected nodes/vertices and they start constructing their tours from an initial node deliberately exploiting the pheromone trails

The Extremal Optimization algorithm [

Improved extremal optimization algorithm.

There has been a major modification of the classical Extremal Optimization algorithm: the Improved Extremal Optimization [

Model-Induced Max-Min Ant Colony Optimization (MIMM-ACO) [

In solving the ATSP, the algorithm, first, analyzes the ATSP problem and then applies the MIMM-ACO searching system. Information obtained from this phase is then used to direct the search further towards more promising areas of the search space. The pseudocode code of MIMM-ACO is presented in Figure

MIMM-ACO pseudocode.

From Figure

The randomized algorithms make random instead of deterministic decisions through the extensive use of random bits as their input in its search process, thus leading to the generation of random variables. Randomized algorithms are usually faster and simpler than deterministic ones. The Randomized Insertion Algorithm (RAI) uses the arbitrary insertion mechanism which is very close to cheapest insertion strategy in its search for solution to the ATSP. The development of the RAI was borne out of a desire to provide a fast and simple solution to the ATSP. This algorithm starts by constructing an initial solution (see steps 1–4 in Figure

RAI algorithms.

To solve the ATSP, the RAI randomly selects any initial node

The experiments were performed using a desktop with the following configuration: Intel Duo Core ™ 2.00 Ghz, 2.00 Ghz, 1 GB RAM on a Window7 on 15 difficult but popular instances out of the 19 Asymetric Travelling Salesman Problem (ATSP) dataset ranging from 17 to 443 cities available in TSPLIB95 [

The details of the experimental parameters are available in Table _{ITER} is the maximum number of iterations. Please recall that

Experimental parameter setting.

ABO | MIMM-ACO | IEO | MMAS | CGAS | |||||
---|---|---|---|---|---|---|---|---|---|

Parameter | Value | Parameter | Value | Parameter | Value | Parameter | Value | Parameter | Value |

Population | 40 | Ants ( | 10 | Population | Population | Generation | 100 | ||

2.0 | 2.0 | _{ITER} | 200000 | 5.0 | 2.0 | ||||

lp1 | 0.6 | 0.1 | 3.0 | 0.99 | 0.1 | ||||

lp2 | 0.5 | 1.0 | Cost | 1.0 | Ro | 0.33 | |||

N/A | 1.0 | Ǫ | 200 | Best | Φ_{ij} | rand (−1, 1) | Crossover rate | 1.0 | |

N/A | N/A | N/A | N/A | N/A | Known cost | N/A | |||

— | N/A | 0.85 | N/A | N/A | 0.9 | 0.9 | |||

N/A | N/A | Φ | 1/ | N/A | N/A | 200 | 0.3 | ||

N/A | N/A | 1.001 | N/A | N/A | N/A | N/A | 0.2 | ||

N/A | N/A | 1.5 | N/A | N/A | N/A | N/A | _{min} | _{max}/20 | |

N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | _{max} | 1 − ( |

Total no of runs | 50 | — | 50 | — | 50 | — | 50 | — | 50 |

The parameters were obtained after careful parameter-tuning. The parameters used in this experiment are found to give the best results. Please note that, to ensure fairness of comparison among different algorithms, it is necessary to run the experiments in the same machine and using the same programming language.

The comparative experiments were of two parts: the first compared the output of the metaheuristic algorithms in solving the ATSP, while the second compared the ABO performance with that of the RAI which is a heuristic algorithm. The results of the experiments involving the metaheuristic algorithms, namely, the Model-Induced Max-Min Ant Colony Optimization (MIMM-ACO), Max-Min Ant System (MMAS), Improved Extremal Optimization (IEO), Cooperative Genetic Ant System (CGAS), and African Buffalo Optimization algorithm (ABO) are presented in Table

Comparative experimental results of metaheuristics on ATSP.

MIMM-ACO | MMAS | IEO | CGAS | ABO | ||||||
---|---|---|---|---|---|---|---|---|---|---|

TSP case | Rel. err % | CPU time (s) | Rel. err % | CPU time (s) | Rel. err % | CPU time (s) | Rel. err % | CPU time (s) | Rel. err % | CPU time (s) |

Br17 | 0 | 0.01 | 0 | 0.0 | 0 | 0.01 | 0 | 0.01 | 0 | 0.028 |

ft53 | 0 | 3.53 | 0.22 | 3 | 0 | 3.85 | 0.35 | 6.78 | 0 | 0.028 |

ftv33 | 0 | 6.12 | 0 | 10 | 0 | 4.78 | 0 | 28.73 | 0.08 | 0.029 |

ftv35 | 0 | 5.35 | 0 | 15 | 0 | 7.35 | 0 | 21.35 | 0.07 | 0.030 |

ftv38 | 0 | 8.64 | 0 | 11 | 0 | 7.83 | 0 | 29.79 | 0 | 0.026 |

ftv44 | 0 | 9.37 | 0 | 12 | 0 | 8.21 | 0 | 37.63 | 0.06 | 0.032 |

ftv47 | 0 | 7.52 | 0 | 10 | 0 | 9.37 | 0 | 29.7 | 0.06 | 0.029 |

ftv55 | 0 | 6.38 | 0 | 19 | 0 | 5.06 | 0 | 18.41 | 0.12 | 0.029 |

ftv64 | 0 | 15.37 | 0 | 28 | 0 | 16.42 | 0 | 29.25 | 0.0 | 0.041 |

p43 | 0 | 8.35 | 0.08 | 9 | 0.13 | 5.47 | 0 | 7.53 | 0.44 | 0.065 |

ry48p | 0 | 7.83 | 0 | 8 | 0 | 5.45 | 0 | 12.35 | 0.12 | 0.037 |

rgb323 | 0 | 0.01 | 1.3 | 97 | 0.06 | 87.12 | 0.13 | 103.28 | 0 | 2.050 |

rgb358 | 0 | 0.01 | 0.75 | 75 | 0 | 69.65 | 0.35 | 96.49 | 0.18 | 3.043 |

rgb403 | 0 | 0.01 | 1.35 | 104.39 | 0 | 85.32 | 0.31 | 147.83 | 0.08 | 4.741 |

rgb443 | 0 | 0.01 | 1.73 | 91 | 0 | 76.14 | 0 | 143.76 | 0.11 | 10.37 |

Mean | 0 | 15.5 | 0.36 | 32.83 | 0.013 | 26.14 | 0.08 | 52.02 | 0.09 | 1.37 |

Total |

Rel. err = relative error; CPU time = total time taken by the algorithm to obtain result; s = seconds.

Please note that the relative error was obtained by

In Table

On the whole, all the algorithms posted over 94.56% accuracy in solving the problems. These are excellent performances, especially when one realises that these are metaheuristic algorithms, that is to say, they are general-purpose algorithms that were not specifically designed for just the ATSP. One way to explain their exceptional performances could be that they are all hybrid algorithms, except, of course, the ABO. Hybrid algorithms post good performances since they exploit the strength of individual algorithms being hybridized. ABO’s good result could be traceable to its use of less complicated calculation of fitness function coupled with the ability of the buffalos to search both globally and locally at the same time.

In terms of the computational cost which is judged by the amount of computational resources utilized in obtaining the solutions to the ATSP instances under investigation, this is where there is such a gulf in the algorithms performances. Here, the exceptional performer is the ABO. It took the ABO just 20.58 seconds to solve all the ATSP instances under investigation. The next best performer is the MIMM-ACO with 78.51 seconds. These are commanding performances, especially when we consider that it took the other algorithms hundreds of seconds to solve the same number of problems. The excellent performance of the ABO could be due to its use of relatively few parameters. Basically, the algorithm uses two major parameters, the “

It could be observed from Table

The previous analysis of the performance of the metaheuristic algorithms shows that the ABO has an edge over the other metaheuristics. This section is concerned with the comparative assessment of the performance of the ABO with the RAI heuristics in solving the ATSP. The RAI heuristics was especially designed to provide solutions to asymmetric TSP instances. The experimental results are presented in Table

Comparative experimental results.

ATSP cases | No of cities | Opt | ABO | RAI | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Best | Avg | Rel. er % | Time | Best | Avg | Rel. er % | Time | |||

Br17 | 17 | 39 | 39 | 39.98 | 0 | 0.028 | 39 | 39 | 0 | 0.027 |

Ry48p | 48 | 14422 | 14440 | 14455 | 0.12 | 0.037 | 14422 | 14543.20 | 0 | 1.598 |

Ftv33 | 34 | 1286 | 1287 | 1288.4 | 0.08 | 0.029 | 1286 | 1288.16 | 0 | 0.393 |

Ftv35 | 36 | 1473 | 1474 | 1475.8 | 0.07 | 0.030 | 1473 | 1484.48 | 0 | 0.508 |

Ftv38 | 39 | 1530 | 1530 | 1536.4 | 0 | 0.026 | 1530 | 1543.12 | 0 | 0.674 |

Ftv44 | 45 | 1613 | 1614 | 1647.25 | 0.06 | 0.032 | 1613 | 1643.6 | 0 | 1.198 |

Ftv47 | 48 | 1776 | 1777 | 1783 | 0.06 | 0.029 | 1776 | 1782 | 0 | 1.536 |

Ft53 | 53 | 6905 | 6905 | 6920.25 | 0 | 0.028 | 6905 | 6951 | 0 | 2.398 |

Ftv55 | 56 | 1608 | 1610 | 1618.2 | 0.12 | 0.029 | 1608 | 1628.74 | 0 | 2.878 |

Ftv64 | 65 | 1839 | 1839 | 1938 | 0 | 0.041 | 1839 | 1861 | 0 | 5.241 |

P43 | 43 | 5620 | 5645 | 5698 | 0.44 | 0.065 | 5620 | 5620.65 | 0 | 0.997 |

Rbg323 | 323 | 1326 | 1326 | 1417.75 | 0 | 2.050 | 1335 | 1348 | 0.68 | 3874 |

Rbg358 | 358 | 1163 | 1187 | 1299.2 | 0.18 | 3.040 | 1166 | 1170.85 | 0.26 | 6825 |

Rbg403 | 403 | 2465 | 2467 | 2475 | 0.08 | 4.741 | 2465 | 2466 | 0 | 11137 |

Rbg443 | 443 | 2720 | 2723 | 2724 | 0.11 | 10.377 | 2720 | 2720 | 0 | 17126 |

Total | — | — | — | — | — | — |

Opt = optimal values as recorded in TSPLIB; Best = best results obtained by a particular algorithm; Avg = average values obtained after 50 runs; Rel. er (%) = relative error percentage; Time = time taken by the CPU to obtain results.

The two algorithms, under investigation in Table

The excellent performance of both algorithms is further highlighted by the calculation of their cumulative relative errors which is a measure of deviation from the optimal solutions. The cumulative relative error is obtained by summing up the values of the relative errors for each ATSP instance. The cumulative relative error of the ABO is 1.32% and that of RAI is 0.94%. This is also a commendable performance by the ABO in view of the fact that the RAI is a pure heuristic designed primarily to solve the ATSP.

In evaluating the cost implications of obtaining results, the uncommon strength of the ABO becomes outstanding in all instances. It was only in Br17 that the RAI executed slightly faster in 0.027 seconds to ABO’s 0.028 seconds. In the remaining 14 instances, the ABO clearly outperformed the RAI. For instance, while it took ABO 0.037 seconds to obtain result in Ry48p, the RAI used 1.598 seconds. This means that the ABO was over 43.18% faster. This trend continues throughout the remaining ATSP instances under investigation. In fact, the ABO gets progressively faster as the number of ATSP cities increases. Take, for instance, the two largest city instances here which are Rbg403 and Rbg443, while ABO used 4.741 and 10.377 seconds, respectively, the RAI used 11137 and 17126 seconds, respectively. This shows the ABO being over 2,349 and 1,650 times faster, respectively.

As was the case in the comparative performance of the metaheuristics, it can be seen that the ABO outperformed the heuristic algorithm, RAI. Someone may have observed that speed is a function of the hardware configuration, the programmers’ expertise, and a few other factors; nevertheless, an algorithm that has such a straightforward calculation of fitness function and uses very few parameters will undoubtedly obtain results faster than most other algorithms. In all, aside from ABO’s capacity to obtain over 98.5% accuracy to RAI’s 99.06%, it took ABO a total of 20.582 seconds to to RAI’s 38979.448 seconds to execute all the 15 instances under investigation.

This study examined the solutions to the asymmetric Travelling Salesman Problems using computational intelligence techniques. The computational intelligence techniques used include African Buffalo Optimization algorithm (ABO), Improved Extremal Optimization (IEO), Model-Induced Max-Min Ant Colony Optimization (MIMM-ACO), Max-Min Ant System (MMAS), and Cooperative Genetic Ant System (CGAS), as well as the heuristic and Randomized Insertion Algorithm. Experimental results obtained from using these algorithms to solve the ATSP reveal that the MIMM-ACO performed excellently obtaining the optimal solutions to all test instances. However, it was discovered that, to obtain such an excellent result, the MIMM-ACO sacrificed speed. It took the MIMM-ACO 78.51 seconds to solve the 15 ATSP instance, while another algorithm, the African Buffalo Optimization (ABO), obtained 98.6% accuracy at 20.582 seconds. The study, therefore, concludes that since efficiency (speed), trustworthiness (accuracy), versatility, and ease of use are hallmarks of a good computational intelligence methods [

The excellent performance of the MIMM-ACO is traceable to two main factors. First, the algorithm’s ability to replace static biased costs/weights in an ATSP with dynamic ones is something other algorithms struggle to do. This ability stems from the algorithm’s use of partial solutions sampling that each ant has constructed in course of the search and then discarding less fruitful results while holding on to the very best. Moreover, the MIMM-ACO’s use of the assignment problem technique in discarding the nonoptimal solutions from the list of available solutions is a major advantage. Second, MIMM-ACO determines the final output based on the most recent state of the pheromone matrix and combines this using the patch algorithm to micromanage the solutions obtained by the assignment problem. Other algorithms find it hard to outperform the MIMM-ACO’s hybridization of the assignment problem with the patch. This basically explains why the MIMM-ACO results in ATSP remain one of the best over the years [

It is recommended that the other algorithms should be fine-tuned to make them faster. Moreover, the authors recommend the comparison of the performance of ABO with other state-of-the-art algorithms in providing solutions to other optimization problems such as knapsack problem, graph coloring, and urban transport challenges in major cities. Finally, in view of the relevance of the ATSP to our every day activities, it is recommended that more research efforts should be directed towards solving ATSP and its practical applications in transportation, logistics, national security architectural challenges, etc.

As much as the algorithms in this comparative study performed excellently well, it must, however, be observed that good results are a function of the programming language used for the study as well as the machine used for the experiments. Moreover, the choice of benchmark test cases could be a threat that the algorithms performed well in these chosen benchmarks may not be a guarantee that they will do well in other benchmarks. Again, the choice of the comparative algorithms could be a threat that these algorithms performed well against one another may not guarantee their exceptional performance when compared with other newer algorithms.

Finally, it is possible that the programming expertise of our programmer as well as the programming language used in implementing this study could have influenced our experimental output.

The data used to support the findings of the study are available within the article.

The authors declare that they have no conflicts of interest regarding the publication of this research article.

This study was supported by Anchor University Lagos and University Malaysia Pahang under RDU1903122. The researchers have fully acknowledged CREIM UniSZA, for the publication support.