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.