Multi-Objective Green 4-dimensional transportation problems for breakable incompatible items with different fixed charge payment policies

Abstract

Transportation system approximately contributes one-seventh of total world pollution, which demands to be minimized. Against some costs, developing countries have recently introduced FASTag and advanced payment systems at toll plazas of National Highways for vehicles’ non-stoppage continuous run, which minimizes travel time. Usually, transport companies try to operate at minimum time and cost. Transportation cost goes up when some items are incompatible i.e., not to be transported together. The above objectives are conflicting with each other. Nowadays, due to infrastructural developments, there are several connecting roads between different cities (sources/destinations). Considering all these facts, to find a trade-off between the above objectives, multi-objective incompatible and breakable/damageable multi-items 4-dimensional transportation models for minimum cost, time and carbon emission are developed and solved by different methods. Formulated multi-objective models are reduced to single objective ones by three ways- Weighted sum technique, Max-min Zimmermann technique and Neutrosophic programming technique and then Generalized Reduced Gradient method is used for solutions using Lingo 11.0 software. Applying the appropriate method for removing the non-linearity expressions in the constraint part of the models, the converted linear problems are also solved by using an integer linear programming tool, Cplex 12.9.0. The proposed methods are tested against two standard test functions, and Pareto front solutions by weighted sum technique are compared graphically. A real-life example is presented for incompatible items, and the effects of ‘green corridor’ on fixed charge are discussed. Also, the impact of considerations of different paths and incompatibility among the items on transportation cost, time, and carbon emission are demonstrated through numerical experiments. It is shown that the routes of the shortest distance and minimum unit transportation cost are not always the optimum choices for the model’s minimum cost. Here, road’s specific constants also play an essential role. For minimization of the objectives, trade-off between carbon emission, total transportation cost, and time is demonstrated. To identify the best suitable model among the formulated ones, the ‘TOPSIS’ method is used for ranking, and in this connection, some managerial decisions are derived.