# Intuitionistic Fuzzy Model of Traffic Jam Regions and Rush Hours for the Time Dependent Traveling Salesman Problem

## Abstract

The Traveling Salesman Problem (TSP) is one of the most extensively studied NP-hard graph search problems. Many researchers published numerous approaches for quality solutions, applying various techniques in order to find the optimum (least cost) or semi optimum solution. Moreover, there are many different extensions and modifications of the original problem, The Time Dependent Traveling Salesman Problem (TD TSP) is a prime example. TD TSP indeed was one of the most realistic extensions of the original TSP towards assessment of traffic conditions [1]. Where the edges between nodes are assigned different cost (weight), considering whether they are traveled during the rush hour periods or they cross the traffic jam regions. In such conditions edges are assigned higher costs [1]. In this paper we introduce an even more realistic approach, the IFTD TSP (Intuitionistic Fuzzy Time Dependent Traveling Salesman Problem); which is an extension of the classic TD TSP with the additional notion of intuitionistic fuzzy sets. Our core concept is to employ intuitionistic fuzzy sets of the cost between nodes to quantify traffic jam regions, and the rush hour periods. Since the intuitionistic fuzzy sets are generalizations of the original fuzzy sets [2], then our approach is a usefully extended, alternative model of the original abstract problem. By demonstrating the addition of intuitionistic fuzzy elements to quantify the intangible jam factors and rush hours, and creating an inference system that approximates the tour cost in a more realistic way [3]. Since our motivation is to give a useful and practical alternative (extension) of the basic TD TSP problem, the DBMEA (Discrete Bacterial Memetic Evolutionary Algorithm) was used in order to calculate the (quasi-)optimum or semi optimum solution. DBMEA has been proven to be effective and efficient in a wide segment of NP-hard problems, including the original TSP and the TD TSP as well [4]. The results from the runs based on the extensions of the family of benchmarks generated from the original TD TSP benchmark data set showed rather good and credible initial results.

## Keywords

Intuitionistic fuzzy sets Traveling Salesman Problem Time Dependent Traveling Salesman Problem Fuzzy costs Jam region Rush hour period Discrete Bacterial Memetic## Notes

### Acknowledgment

This work was supported by National Research, Development and Innovation Office (NKFIH) K124055. Supported by the ÚNKP-18-3 New National Excellence Program of the Ministry of Human Capacities.

## References

- 1.Tüű-Szabó, B., Földesi, P., Kóczy, L.T.: The discrete bacterial memetic evolutionary algorithm for solving the one-commodity pickup-and-delivery traveling salesman problem (2018)Google Scholar
- 2.Atanassov, R.K.: Intuitionistic fuzzy sets. Fuzzy Sets Syst.
**20**, 87–96 (1986)CrossRefGoogle Scholar - 3.Boran, F., Genç, S., Ku, M., Akay, D.: A multi-criteria intuitionistic fuzzy group decision making for supplier selection with TOPSIS method. Expert Syst. Appl.
**36**, 11363–11368 (2009)CrossRefGoogle Scholar - 4.Földesi, P., Botzheim, J.: Modeling of loss aversion in solving fuzzy road transport travelling salesman problem using eugenic bacterial memetic algorithm. Memet. Comput.
**2**(4), 259–271 (2010)CrossRefGoogle Scholar - 5.Applegate, D.L., Bixby, R.E., Chvátal, V., Cook, W.J.: The Traveling Salesman Problem: A Computational Study, pp. 1–81. Princeton University Press, Princeton (2006)Google Scholar
- 6.Biswas, R.: On fuzzy sets and intuitionistic fuzzy sets. Notes Intuitionistic Fuzzy Sets
**3**, 3–11 (1997)MathSciNetzbMATHGoogle Scholar - 7.Gau, W.L., Buehrer, D.J.: Vague sets. IEEE Trans. Syst. Man Cybern.
**23**(2), 610–614 (1993)CrossRefGoogle Scholar - 8.Szmidt, E., Kacprzyk, J.: Intuitionistic fuzzy sets in group decision making. NIFS
**2**(1), 11–14 (1996)zbMATHGoogle Scholar - 9.Moscato, P.: On evolution, search, optimization, genetic algorithms and martial arts—towards memetic algorithms. Technical report Caltech Concurrent Computation Program, Report. 826, California Institute of Technology, Pasadena, USA (1989)Google Scholar
- 10.Kóczy, L.T., Földesi, P., Tüű-Szabó, B.: An effective discrete bacterial memetic evolutionary algorithm for the traveling salesman problem. Int. J. Intell. Syst.
**32**(8), 862–876 (2017)CrossRefGoogle Scholar - 11.Kóczy, L.T., Földesi, P., Tüű-Szabó, B., Almahasneh, R.: Modeling of fuzzy rule-base algorithm for the time dependent traveling salesman problem. In: Proceedings of the IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), (2019, under review)Google Scholar
- 12.Schneider, J.: The time-dependent traveling salesman problem. PhysicaA
**314**, 151–155 (2002)MathSciNetCrossRefGoogle Scholar - 13.Tüű-Szabó, B., Földesi, P., Kóczy, T.L.: Discrete bacterial memetic evolutionary algorithm for the time dependent traveling salesman problem. In: International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU 2018), Cadíz, Spain, pp. 523–533. Springer, Cham (2018)Google Scholar
- 14.Kóczy, L.T., Földesi, P., Tüű-Szabó, B.: A discrete bacterial memetic evolutionary algorithm for the traveling salesman problem. In: IEEE World Congress on Computational Intelligence (WCCI 2016), Vancouver, Canada, pp. 3261–3267 (2016)Google Scholar