GA-ABC hybridization for profit maximization of green 4DTSPs with discrete and continuous variables

A medical or business representative visits and spends some time (termed ‘stay time’) at different cities(nodes) for optimum return. With the rapid development of infrastructures worldwide, several vehicles and routes are available for transport between nodes. Road logistic transportation contributes 10.5% of the total emitted carbon emission (CE). Heuristic method is applied to solve NP-hard problems with continuous/discrete variables. With these facts, here, CE/tour-time constrained profit maximization of green four-dimensional Travelling Salesman Problems (PMG4DTSPs) with both discrete (travel times between nodes) and continuous (stay-times at nodes) variables are formulated. A travelling salesman visits all cities using the appropriate routes and conveyances, spend some time at each node, and some revenues are earned and spent depending upon the stay time at those visiting places. There are toll plazas on the roads, where three types of fixed charges (online, offline, mixed online–offline) are collected as road taxes to maintain the roads. A green constraint is incorporated in the form of limited emission. Several connecting routes between the cities and conveyances at each city are available for travel. Thus, these (PMG4DTSPs) are mixed integer linear programming problems (NP-hard), and to solve them, two hybrid heuristic methods, GA-ABC and GA-PSO, are developed and used. Here, GA and ABC/PSO are used for optimum routing plan and stay time, respectively. The models are illustrated numerically. Multi-path justification, online payment advantage and tour time limitation effect on profit are presented. Some behavioural studies of Green models and managerial decisions are presented.