The current article is devoted to mathematically handling the inherent uncertainties of various practical problems by introducing the concept of LR-type interval-valued intuitionistic fuzzy numbers (LR-type IVIFN). The theoretic development of this concept has also been enhanced by providing various diagrammatic representations of LR-type IVIFNs and establishing the arithmetic operations among these fuzzy numbers. The notion of α,β\documentclass[12pt]{minimal}
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\begin{document}$$\alpha ,\beta $$\end{document}-cuts and interval arithmetic have been employed to derive the expressions for the arithmetic operations on LR-type IVIFNs. In addition to this, the total order properties of lexicographic-ranking criteria along with considering seven distinct parameters have been used to define the ordering on LR-type IVIFNs. Further, a linear programming problem (LPP) with both equality and inequality type constraints, all parameters in the form of LR-type IVIFNs, and unrestricted decision variables has been modeled. In order to obtain a unique optimal solution for the proposed LPP, the lexicographic ranking-based solution methodology has been developed in which by introducing some binary variables, the original LPP is converted to an equivalent mixed 0-1 lexicographic non-linear programming problem having seven components in the objective function. Various theorems have also been proved to show the equivalence of the original problem and its different proposed constructions. The model formulation, algorithm, and discussed results have not only developed a new idea but also generalized various well-known related works existing in the literature. Moreover, a numerical illustration has been provided to clarify the step-wise procedure of the proposed solution approach. Additionally, to establish the practical significance of the study, a real-world production planning problem is framed, solved, and analyzed under the LR-type interval-valued intuitionistic fuzzy scenario. In last, a comparative study has also been carried out to prove the relevancy of the developed algorithm.
