Complexity

Volume 2017 (2017), Article ID 9562125, 6 pages

https://doi.org/10.1155/2017/9562125

## Can the Agent with Limited Information Solve Travelling Salesman Problem?

Graduate School of Natural Science and Technology, Okayama University, Okayama, Japan

Correspondence should be addressed to Tomoko Sakiyama

Received 4 January 2017; Revised 14 March 2017; Accepted 29 March 2017; Published 11 April 2017

Academic Editor: Sergio Gómez

Copyright © 2017 Tomoko Sakiyama and Ikuo Arizono. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Here, we develop new heuristic algorithm for solving TSP (Travelling Salesman Problem). In our proposed algorithm, the agent cannot estimate tour lengths but detect only a few neighbor sites. Under the circumstances, the agent occasionally ignores the NN method (choosing the nearest site from current site) and chooses the other site far from current site. It is dependent on relative distances between the nearest site and the other site. Our algorithm performs well in symmetric TSP and asymmetric TSP (time-dependent TSP) conditions compared with the NN algorithm using some TSP benchmark datasets from the TSPLIB. Here, symmetric TSP means common TSP, where costs between sites are symmetric and time-homogeneous. On the other hand, asymmetric TSP means TSP where costs between sites are time-inhomogeneous. Furthermore, the agent exhibits critical properties in some benchmark data. These results suggest that the agent performs adaptive travel using limited information. Our results might be applicable to nonclairvoyant optimization problems.

#### 1. Introduction

Travelling Salesman Problem (TSP) is well-known as one of the combinational optimization problems of finding the best solution out of a finite set of possible solutions. To solve complicate problems, for example, TSP, there is heuristics as optimization algorithms inspired by real systems of animate beings [1]. It appears that, in such optimization algorithms, the advanced intelligence, such as sophisticated cognitive ability and predictive ability, is demanded [1–3]. However, real organisms such as insects and cells appear to solve complicate optimization problems by using only simple rules [4, 5]. This fact implies that real organisms do not need high ability of spatial cognition for solving complicate optimization problems. Therefore, we can see that agents in heuristic inspired by real systems of animate beings do not need high ability of spatial cognition.

The nearest neighbor (NN) method is the famous method for solving TSP by limited cognitive abilities based on only local information. Agents obeying the NN method select always the nearest site from current site. In this sense, the NN method can be regarded as the algorithm using only neighbor sites. The NN method has been conveniently used for a long period, since it can derive reasonable solutions by only local information. Actually, it appears that animate beings use the NN method to solve TSP such as a foraging activity through repeatedly visiting a series of locations [6].

However, the NN method cannot necessarily output suitable solutions in TSP [7]. To obtain a suitable solution in TSP, agents might need to occasionally ignore the NN method. Actually, animate beings appear to flexibly change movement strategies based on local environmental situations [8–10]. For example, bees appear to use a NN strategy at small spatial scale when neighbor resources are close to each other, but to flexibly use more efficient optimization strategies at larger spatial scales when neighbor resources are far from each other [10].

It is known that individuals in living systems coordinate their behaviors using local information and behave adaptively in response to various environments [11]. In addition to that, they appear to adopt negative information and make a profit for a long-term span. This mechanism is also applied to the cognitive model of human [12]. It seems that each individual generates an action that is required for not a short-term span but a long-term span [13].

With reference to such an action of animate beings, we developed a heuristic algorithm for solving TSP, in which individuals occasionally tuned their rules based on local environments while they basically obeyed the NN method. In our heuristic algorithm, it is assumed that agents can estimate topological distances to a few neighbor sites, where topological distances are regarded as neighbor distances [14]. Agents in our heuristic algorithm change site-selection rules to estimate not local efficiencies but global efficiencies for shortening total travel length. If two sites belong to neighbor sites and relative distances between these sites and current site are not much different, agents have a tendency to select farther site than closer site. Selecting farther site means that agents adopt a negative strategy in the short run.

Because we assumed that agents in heuristics could not recognize global tour lengths, agents had to make decisions only using local information. Here, we developed two different experimental conditions. One was simple symmetric TSP. The other one was asymmetric TSP. Here, symmetric TSP means common TSP, where costs between sites are symmetric and time-homogeneous. On the other hand, asymmetric TSP means TSP where costs between sites are time-inhomogeneous. Particularly, we suppose that the energy spent in the moving from the current site to the next site is increased with a load, that is, the number of sites which the agent had already visited. Such a situation has been investigated in the single machine scheduling problems with sequence dependent setup times by Bigras et al. [15]. In this situation, the agent in heuristic needs to select farther sites in the beginning of its tours due to the energy spent increased with the load. Then, unlike symmetric TSP which has the time-homogeneous travel cost, the increment travel cost has been considered in asymmetric TSP. Simulating increment travel cost condition analogizes with the situation of animate beings that agents collect cargos at each visit to a location [16]. We confirmed that agents showed effective tours in both conditions of symmetric TSP and asymmetric TSP.

#### 2. Materials and Methods

##### 2.1. Model Description: Rule Change Algorithm

In this algorithm (Rule Change (RC) algorithm), the agent firstly follows the NN algorithm on each time step. In a word, the agent tends to select the nearest neighbor site. However, the agent selects the other site if following situation is satisfied: If , then the agent chooses the farthest site among* n*-neighbor sites, where is random number satisfying , is the distance between current site and the nearest site for the agent, is the distance between current site and the farthest site among* n*-neighbor sites for the agent.Here, the aim of this research is to investigate that agents enable achieving optimal searching as a result by using negative cues. To this end, we allowed agents to choose either the nearest or the farthest of the neighbor sites.

##### 2.2. Simulation Experiment

We solved symmetric TSP and asymmetric TSP using several dataset from the TSPLIB [17]. In this paper we only used benchmark data containing less than one hundred cities and described distances as Eucrid-2D (EDGE_WEIGHT_TYPE: EUC_2D).

We adopted two experimental conditions. One was simple symmetric TSP. The total cost obtained after one tour was dependent on each route length the agent followed. The other one was asymmetric TSP by considering the agent’s crop capacity. Here, “agent’s crop capability” in this paper meant the load increasing due to time duration from a current site to the next site. The energy spent for time duration from a current site to the next site was increased with the load, that is, the number of sites which the agent had already visited. In this paper, we defined cost between current site and next site for the agent as (time steps) × (distances between these two sites, i.e., route lengths). Therefore, even though the agent followed same routes, total cost obtained after one tour would be different depending on whether the agent rotated clockwise or anticlockwise.

We developed three control algorithms for comparison; the NN algorithm, the FN algorithm, and the FN_{0.5} algorithm. In the FN (farthest neighbor) algorithm, the agent always selected the farthest site among* n*-neighbor sites. In the FN_{0.5} algorithm, the agent selected the nearest/farthest site among* n*-neighbor sites with probability 0.5/0.5. All algorithms were used for both experimental conditions (symmetric/asymmetric TSP).

We set agents for -site benchmark problems in all algorithms. Each agent was, respectively, assigned to every site as starting site. We conducted one hundred trials using each algorithm.

##### 2.3. Simulation Procedure: The RC Algorithm versus the NN Algorithm

We supposed local neighbors for the agent based on topological distances. The parameter and were initially set to 0.90, 2 respectively (, ). By setting as initial value, we set the agent’s detection ability as low-capacity detection. Also, it might be natural for the agent to consider that distances between the nearest site and farthest site among* n*-neighbors are not much different when ratio was close to 1.00. We therefore set ratio = 0.90 as initial value. When the RC algorithm output worse solutions than the NN algorithm, we changed parameters as the following manners until we could obtain better solutions using the RC algorithm than using the NN algorithm: (ratio = 0.90, ) → (ratio = 0.80, ) → (ratio = 0.90, ) → (ratio = 0.80, ) → ⋯ → (ratio = 0.80, ).When we could not get better solutions by using the RC algorithm even after (ratio = 0.80, ), we stopped calculations and concluded that the RC algorithm output worse solutions than the NN algorithm.

In respect to the RC algorithm versus the FN/FN0.5 algorithms, we used the same set of parameters (ratio,* n*) as those of the RC algorithm versus the NN algorithm.

On each trial, summation of total cost from each agent was calculated as where means total cost of each agent . Then, we obtained average summation by conducting 100 trials as follows: where indicates summation of total cost in trial .

#### 3. Results

Tables 1 and 2 illustrate the results. As seen from those tables, the RC algorithm performed better solutions compared with the NN, FN, and FN_{0.5} algorithms for each experimental condition (symmetric/asymmetric). All data were compared using Mann–Whitney* U* test. Then, the value is defined as the probability of obtaining a result equal to or different from what was actually observed, when the null hypothesis that the average of summation of total cost based on the RC algorithm the average of summation of total cost based on other algorithms is true. Note that the RC algorithm could not exhibit dominantly better solutions than the NN algorithm for symmetric experimental condition when rat99 was used as benchmark data. However, based on the relation of 146101 < 146122 and value = 0.35 in average summation of total cost from each agent in the case of rat 99, it seems that the RC algorithm could exhibit relatively better solutions than the NN algorithm. In consequence, it could be concluded that the RC algorithm is superior to the NN, FN, and FN_{0.5} algorithms from the standpoint of summation of total cost.