We are observing more often extreme climate incidents because of global warming. There is a dire requirement for governments, enterprises, the overall population, and academics to take facilitated activities so as to handle the difficulties forced by environmental change. The most important strategic issue is to design an effective and environmentally concerned logistics system as transportation is one of the fundamental reasons for carbon emanations. Having this goal and mentioned highly important contributions in the field, our paper introduces an unprecedented integrated mathematical model for a green solid transportation system with dwell time to execute the carbon tax, cap and offset regulation. Due to market fluctuations, the supply and demand parameters are not always of crisp nature. Hence, a two-fold (type-2 intuitionistic) uncertainty is incorporated in this study to provide a realistic transportation system. A new ranking defuzzification technique is presented for conversion into a deterministic form. After that, a fuzzy technique and a non-fuzzy technique are used to get the Pareto-optimal solution of the proposed problem. The performances of our findings are discussed with industrial-based application examples. Moreover, a comparative study is explored among the other relevant existing techniques. Managerial insights, conclusions, and avenues of future scopes are offered at the end of this study.

During past few decades, fuzzy decision is an important attention in the areas of science, engineering, economic system, business, etc. To solve day-to-day problem, researchers use fuzzy data in transportation problem for presenting the uncontrollable factors; and most of multi-objective transportation problems are solved using goal programming. However, when the problem contains interval-valued data, then the obtained solution was provided by goal programming may not satisfy by all decision-makers. In such condition, we consider a fixed-charge solid transportation problem in multi-objective environment where all the data are intuitionistic fuzzy numbers with membership and non-membership function. The intuitionistic fuzzy transportation problem transforms into interval-valued problem using $$(\alpha ,\beta )$$( α , β ) -cut, and thereafter, it reduces into a deterministic problem using accuracy function. Also the optimum value of alternative corresponds to the optimum value of accuracy function. A numerical example is included to illustrate the usefulness of our proposed model. Finally, conclusions and future works with the study are described.