Bilevel multi-objective transportation problem with spherical fuzzy numbers

The COVID-19 pandemic exposed weaknesses in health systems, underscoring the need for robust healthcare that balances prevention and treatment. Vaccine manufacturers have been working tirelessly to produce sufficient vaccines to prevent another wave. Essential raw materials for vaccine production must be procured in a timely and secure manner from multiple sources and supplied to manufacturers to meet market demand. However, parameters like transportation costs, supply, demand and other associated costs remain uncertain and are prone to frequent changes due to unavoidable factors. Conflicting objectives of the decision makers (DMs) at the upper-level and lower-level along with indeterminacy in the transportation parameters are modeled into a novel bilevel multi-objective transportation problem (BLMOTP) with spherical fuzzy numbers (SFNs). These numbers are represented by three independent degrees of membership, neutrality and non-membership on a spherical surface making them an effective tool for decision making in the presence of indeterminacy. The utility of the model is exhibited through a numerical example based on the COVID-19 vaccine market. SFNs are first converted into crisp numbers using score functions. The crisp BLMOTP is then solved using the proposed spherical fuzzy programming approach (SFPA) and intuitionistic fuzzy programming approach (IFPA) available in the literature. A comparative analysis of the obtained solutions shows that the proposed approach is better than IFPA. Sensitivity analysis on relaxation parameters provides managerial insight. The model aims to help the decision makers in efficient cost planning, safety management and smooth administration of the vaccines.