​Optimality conditions of fuzzy fractional optimization and its application in the problems of energy efficiency optimization

Abstract

Fuzzy-valued fractional optimization is an important topic in uncertain mathematical programming. It is related to efficiency-oriented decision-making problems in management, economics, and engineering. For example, in communication systems, energy-efficiency optimization in non-orthogonal multiple access systems naturally involves fractional objectives and uncertain parameters. However, optimality conditions for fuzzy-valued fractional optimization problems under granular differentiability have not been sufficiently studied. Therefore, this paper investigates Karush--Kuhn--Tucker (KKT) optimality conditions for fuzzy-valued fractional optimization problems under granular differentiability. First, horizontal membership functions are used to characterize fuzzy numbers and incorporate fuzzy parameters into the fractional optimization framework, providing a parameterized basis for granular analysis. Second, under granular differentiability and convexity assumptions, the relationship between the fuzzy-valued fractional optimization problem (FVFOP) and its associated fuzzy-valued optimization problem (FVOP) is established through a Dinkelbach-type transformation. Based on this relationship, KKT optimality conditions are derived for the original FVFOP. Finally, numerical examples are provided to illustrate the feasibility of the proposed theoretical results. The framework is further applied to energy-efficiency optimization in non-orthogonal multiple access systems and fuel-efficiency optimization in multi-generator systems, showing its potential relevance to efficiency-oriented decision-making under fuzzy uncertainty.