Fuzzy Set Theory and Its Application to Multi-Criteria Decision Making: A Fuzzy TOPSIS Approach to Supplier Selection Under Linguistic Uncertainty

Multi-criteria decision making (MCDM) under uncertainty constitutes one of the most practically important and mathematically rich application areas of modern applied mathematics. Classical MCDM methods assume that decision criteria weights and alternative performance ratings can be expressed as precise crisp numbers — an assumption frequently violated in real-world decisions where information is inherently imprecise or linguistically expressed. Fuzzy set theory, introduced by Zadeh (1965), provides the mathematical framework for representing and manipulating such imprecise information through membership functions mapping elements to degrees in [0,1]. This paper presents the theoretical foundations of fuzzy set theory — including fuzzy numbers, arithmetic operations on triangular fuzzy numbers (TFNs), defuzzification methods, and fuzzy distance metrics — and develops a complete Fuzzy TOPSIS (FTOPSIS) methodology for multi-criteria supplier selection under linguistic uncertainty. Applied to five candidate suppliers evaluated across six criteria using triangular fuzzy numbers derived from linguistic ratings by a decision committee, the FTOPSIS yields ranking: Supplier B (CC=0.724) > Supplier A (CC=0.684) > Supplier C (CC=0.698) > Supplier E (CC=0.646) > Supplier D (CC=0.612). All mathematical operations, normalisation steps, ideal solution distance computations, and ranking procedures are presented in complete detail. Consistency and boundary theorems for FTOPSIS are proved. Sensitivity analysis confirms ranking stability under ±20% criteria weight perturbation