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In this article, a new concept of LR-type interval-valued intuitionisticfuzzy numbers (LR-type IVIFN) has been introduced. The theory has also beenenriched by demonstrating diagrammatic representations of LR-type IVIFNs andestablishing arithmetic operations among these fuzzy numbers. The total orderproperties of lexicographic criteria have been used for ranking LR-type IVIFNs.Further, a linear programming problem having both equality as well asinequality type constraints with all the parameters as LR-type IVIFNs andunrestricted decision variables has been formulated. An algorithm to find aunique optimal solution to the problem using the lexicographic ranking methodhas been developed. In the proposed methodology, the given linear programmingproblem is converted to an equivalent mixed 0-1 lexicographic non-linearprogramming problem. Various theorems have been proved to show the equivalenceof the proposed problem and its different constructions. The model formulation,algorithm and discussed results have not only developed a new idea but alsogeneralized various well-known related works existing in the literature. Anumerical problem has also been exemplified to show the steps involved in theapproach. Finally, a practical application in production planning is framed,solved and analyzed to establish the applicability of the study.

Developing solution algorithm for LR-type fully interval-valued intuitionistic fuzzy linear programming problems using lexicographic-ranking method