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.