  In this article, a new concept of LR-type interval-valued intuitionistic
fuzzy numbers (LR-type IVIFN) has been introduced. The theory has also been
enriched by demonstrating diagrammatic representations of LR-type IVIFNs and
establishing arithmetic operations among these fuzzy numbers. The total order
properties of lexicographic criteria have been used for ranking LR-type IVIFNs.
Further, a linear programming problem having both equality as well as
inequality type constraints with all the parameters as LR-type IVIFNs and
unrestricted decision variables has been formulated. An algorithm to find a
unique optimal solution to the problem using the lexicographic ranking method
has been developed. In the proposed methodology, the given linear programming
problem is converted to an equivalent mixed 0-1 lexicographic non-linear
programming problem. Various theorems have been proved to show the equivalence
of the proposed problem and its different 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. A
numerical problem has also been exemplified to show the steps involved in the
approach. Finally, a practical application in production planning is framed,
solved and analyzed to establish the applicability of the study.
