Green 4D transportation problems with breakable incompatible items under type-2 fuzzy-random environment

Abstract

As the carbon emission (CE) from a vehicle depends on the factors such as the surface condition of the traveled route, conveyance type, load, speed, curb weight of the vehicles, etc., nowadays, consideration of the appropriate route, conveyance and corresponding CE are given maximum importance in eco-friendly transportation systems. The present study investigates a four-dimensional transportation problem (4DTP) for breakable/damageable items, some of which are pairwise incompatible, with carbon tax policy under crisp and type-2 fuzzy/fuzzy random environments. Here, unit transportation costs are taken as type-2 fuzzy and the products’ availabilities, demands, conveyance capacities are type-2 fuzzy-random. Type-2 fuzzy and fuzzy-random data are respectively transformed to the corresponding crisp and random data using the centroid-based type-2 fuzzy-probabilistic programming with Enhanced Karnik-Mendel (EKM) algorithm. Then a modified interactive algorithm is proposed to convert the above transportation problem into two crisp sub-constrained optimization problems which are solved by Generalized Reduced Gradient method using Lingo 11.0. The models are illustrated with some numerical data. As particular cases, several models are deduced. A real life example is illustrated. Effects of consideration of different paths, conveyances with different capacities, breakability, carbon emission costs, and incompatibilty of items are demonstrated. It is shown that the routes of shortest distance and minimum unit transportation cost are not the optimum paths for the model’s minimum cost. Here, road’s specific constants also play an important role. It is observed that for minimum total cost, there is a trade off between the total carbon emission and total transportation cost. Some managerial decisions are also derived.