A Quantum-inspired Ant Colony Optimization for solving a sustainable four-dimensional traveling salesman problem under type-2 fuzzy variable

In this paper, a Quantum-inspired Ant Colony Optimization (Qi-ACO) is proposed to solve a sustainable four-dimensional traveling salesman problem (4DTSP). In 4DTSP, various paths with a different number of conveyances are available to travel between any two cities. In this model, we have considered a sustainable 4DTSP in terms of emission as a constraint. Since travel costs and emissions are uncertain/imprecise in nature, so here we consider type-2 variables. Sustainable development in the traveling salesman problem (TSP) sector can be divided into two major sections: economy and environmental. Sustainable TSP development requires balancing to achieve the maximum benefits for these two sectors. For increasing development in sustainable transportation, we need to use some strategies for increasing sustainability. These strategies include improving route and vehicle selection, routing plan, vehicle speed, etc. The novelties of the proposed Qi-ACO algorithm are (i) Qubit generated based on the amount of emission of the vehicle as well as travel cost between two cities, (ii) pheromone initialized and updated depends on the qubit, (iii) quantum-inspired technique makes fast computation. The proposed sustainable 4DTSP is illustrated with some numerical data. The defuzzification of type-2 fuzzy variable based on the Critical value (CV) method is used in this model. The supremacy of the proposed method is established through some statistical tests. The proposed algorithm and its modified form can be easily adapted in ship routing, supply chain problems, and other fields.