| Abstract: |
Linear programming (LP) has been a cornerstone of operations research and mathematical optimization for decades, allowing us to effectively allocate limited resources while satisfying well-established set of constraints and objectives. But often, real-world decision-making environments are not deterministic, and applicability of classical LP frameworks is limited because of imprecision, vagueness and uncertainty in the data and problem structure. Since its introduction by Zadeh in 1965, fuzzy set theory has provided a formal mathematics to model such vagueness by (i) expressing information with degree of membership via membership functions and (ii) relating imprecise terms using linguistic variables. Fuzzy optimization under Linear Programming (LP) gives rise to a class of models known as Fuzzy Linear Programming (FLP), which not only allows us to solve problems having fuzzy objective functions but also those with fuzzy constraints or fuzzy coefficients. This review paper is characterized by an integrated meta-analysis of seminal and post-mid-1980s contributions to the qualitative fuzzy optimization in LP literature, addressing theoretical foundations and methodological advances as well as several application domains (e.g., supply chain management, transportation, production planning, engineering design, finance). We analyze several model frameworks such as Zimmermann's symmetric fuzzy LP, Bellman-Zadeh decision making models, triangular and trapezoidal fuzzy numbers, possibilistic programming and interval based methods. The survey outlines major research trends from 1970 to 2024, provides a critical analysis of the strengths and limitations of existing methodologies, and identifies open research questions. The results indicate that fuzzy optimization is a relatively mature but dynamically evolving area with great potential for theoretical improvement and computational advance, especially in the contexts of multi-objective, large-scale and data-driven paradigms. |
