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In this study, we develop two Ant Colony Optimization (ACO) models as new metaheuristic models for solving the time-constrained Travelling Salesman Problem (TSP). Here, the time-constrained TSP means a TSP in which several cities have constraints that the agents have to visit within prescribed time limits. In our ACO models, only agents that achieved tour under certain conditions defined in respective ACO models are allowed to modulate pheromone deposition. The agents in one model are allowed to deposit pheromone only if they achieve a tour satisfying strictly the above purpose. The agents in the other model is allowed to deposit pheromone not only if they achieve a tour satisfying strictly the above purpose, but also if they achieve a tour satisfying the above purpose in some degree. We compare performance of two developed ACO models by focusing on pheromone deposition. We confirm that the later model performs well to some TSP benchmark datasets from TSPLIB in comparison to the former and the traditional AS (Ant System) models. Furthermore, the agent exhibits critical properties; i.e., the system exhibits complex behaviors. These results suggest that the agents perform adaptive travels by coordinating some complex pheromone depositions.

M. Dorigo proposed Ant Colony Optimization (ACO) as a metaheuristic for solving combination optimization problems [

ACO models have been applied to the Travelling Salesman Problem (TSP) which demands the shortest tour under the condition that travelling agents are allowed to transit each city only once and return to the start city. Then, ACO might be a powerful solving tool for TSP and some dynamic manufacturing problems in the real world [

The time-dependent/constrained TSP is widely studied as an important problem because, in natural conditions, the cost between any two cities can be varied based on the time evolutions [

Previous ACO models reveal that both exploiting and exploring the solution space can be an effective searching manner on the time-dependent/constrained TSP [

To this end, we propose ACO models for the time-constrained TSP in which individual agents judge whether or not they deposit pheromones after each tour. In our time-constrained TSP, several cities have to be visited within individual prescribed time spans; i.e., each agent must find an optimal tour under the constraints of visiting certain cities within respective specified times. This situation means a delivery problem with specified delivery-time constraints.

The ACO models imitate positive feedback of real ants and eventually lead all the ants to a single path. However, further positive feedback might be needed for above time-constrained TSP. Real ants deposit pheromone more often when they encounter profitable resources [

In the first ACO model, agents are allowed to update pheromone if and only if they achieve a tour in which the agents visited all time-constrained cities within a specified period time and that tour was better than any tour each agent found until then. Although the system based on this rule for pheromone update attracts rapidly the agents to one solution, diversity of solutions in the system will be lost because this rule for pheromone update obstructs that the agents deviate from one solution. Real ants allow multiple food locations to be exploited simultaneously when they encounter the ambiguous situation, by upregulating pheromone deposition [

With reference to this feature, we construct the second ACO model in which agents deposit pheromones positively when they finish a tour by visiting not all cities with time constraints but some cities within a specified period of time.

The rule for the deposition of pheromone in the first model means the strict learning procedure corresponding to the requirement of the strict satisfaction of constraints in the mathematical programing. On contrast, the rule for the deposition of pheromone in the second model corresponds to the tolerant learning procedure in soft computing. We found the “upregulated pheromone” in the second ACO model could serve as a key in order to find better solutions.

Ant system (AS) is basic ACO model. Here, we demonstrate concepts of AS in this section.

Firstly, the city that the agent is assigned means the start city and goal city in circuit tour of this agent. The agent

After all agents finish one tour, pheromone amounts

We apply ACO models to the time-constrained TSP. There is the classical time-dependent TSP as a version of TSP in which the transition cost between one city and another city depends on the period of the day [

In the beginning of each trial, the start city is randomly chosen. All agents are arranged on the same start city. Thereafter, several cities are randomly chosen as time-constrained cities. Agents must visit those cities in limited time duration. More specifically, chosen time-constrained city

When the agent

Here, we describe the details of our models. The first one is named as the strict ACO model for time-constrained TSP. In the strict ACO model, agents are allowed to add pheromone on their paths only if they visit all time-constrained cities within limited time duration and update their own best-so-far solution. The second one is named as the tolerant ACO model for time-constrained TSP. In the tolerant ACO model, agents are tolerated to add pheromone positively on their paths if they visit several cities out of all time-constrained cities within limited time duration and update their own best-so-far solution.

We explain submodels for tour iteration of each ACO models. Please note that we adopt synchronous updates in respective ACO models.

The agent

If the agent

If the agent

In case of the strict ACO model,

if

then the agent

Remark that

On the other hand in case of the tolerant ACO model,

if

then the agent

Remark that

We solved the time-constrained TSP using two symmetric TSP dataset (

Parameters for the calculation.

( | | | | ||
---|---|---|---|---|---|

Strict ACO model | (1, 5) | 0.9 | | 4 | - |

Tolerant ACO model | (1, 5) | 0.9 | | 4 | 2 or 3 |

Ant System | (1, 5) | 0.9 | | 4 | - |

Table

The number of trials of the shortest tour of each model. Here, we focus on the tours in which all time-constrained cities are visited within limited time duration. One hundred trials are conducted.

Strict ACO model | Tolerant ACO model | Ant System | |
---|---|---|---|

| 32 | 49 | 11 |

| 32 | 46 | 5 |

| 37 | 27 | 11 |

Figure

Tour period between any two consecutive pheromone depositions for agents.

Finally, we would like to comment on parameter effects by conducting additional analysis using

The number of trials of the shortest tour of each model by replacing a certain equation.

Strict ACO model | Tolerant ACO model | Ant System | |
---|---|---|---|

Equation | 45 | 42 | 0 |

| 24 | 26 | 10 |

In this paper, we developed the ACO models to deal with the time-constrained TSP. We proposed two different models. The one was the strict ACO model in which the agents deposited pheromone if and only if agents found a tour that all the time-constrained cities were visited within limited time duration and that tour was better than any tours individuals achieved until then. The other one was the tolerant ACO model in which the agents sometimes deposited pheromone positively even if they did not achieve above tours strictly. We found that the latter model output better solutions compared with the former model and the AS model.

It is known that the ACO models fall into a local solution [

The tolerant ACO model is not inferior to any other models when some parameters are replaced. However, we might be able to improve our proposed model when considering parameter effects. Proposing ACO models in which agents modify their own parameters adaptively would enable the system to perform flexibly in various conditions. That will become an issue in the future.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

This work was supported by the Okawa Foundation for Information and Telecommunications under the research grant #17-07. We would like to appreciate the grant for our research.