CI-MM-Dombi operator based on interval type-2 spherical fuzzy set and its applications on sustainable supply chain with risk criteria: using CI-TODIM-MARCOS method

Sustainable supplier selection and optimal quantity transportation (S\(^3\)OQT) play an important role in supply chain management. This research represents a new four-stage solution approach for \(\hbox {S}^3\)OQT where the multi-criteria decision making (MCDM) methods are integrated through an optimization model. In first stage, a new uncertainty interval type-2 spherical fuzzy set (IT2SFS) is introduced to help the decision-makers (DMs) for securing and reliable results in hesitancy situations. We develop a new operator on IT2SFS under Dombi t-norm and t-conorm by integrating Muirhead mean (MM) operator based on Choquet integral (CI). The preferences and priorities to the sustainable criteria based on interaction and interrelationship are represented by CI. Thereafter, the weights of the criteria and sub-criteria are determined by CI-indifference threshold-based attribute ratio analysis (ITARA) method by utilizing the proposed operator. In second stage, to evaluate the weights of the suppliers and to rank these, we construct a new MCDM method CI-TODIM (an acronym in Portuguese of interactive multi-criteria decision-making)-measurement alternatives and ranking according to compromise solution (MARCOS) method by utilizing the proposed operator and then finally design a new ranking function. In third stage, a new model on stochastic multi-objective mixed-integer non-linear solid transportation problem (\(\hbox {SM}^2\)NSTP) is established to identify suitable supplier under sustainable risk criteria, and then, optimal quantity of products are transported from each supplier. Thereafter, we propose TOPSIS-neutrosophic-game theoretic approach (TNGTA) to obtain Pareto-optimal solution. We apply \(\varepsilon \)-constraint method to obtain Pareto-optimal solution from \(\hbox {SM}^2\)NSTP model. In the fourth stage, a comparative study is drawn among the obtained Pareto-optimal solutions that are extracted from TNGTA and \(\varepsilon \)-constraint method. Finally, two MCDM models, CRITIC-TOPSIS and CRITIC-MARCOS, are used to help the DMs for selecting the final Pareto-optimal solution. A real-life example is included to show the applicability and effectiveness of the designed hybrid MCDM-\(\hbox {SM}^2\)NSTP model.