The fuzzy linear assignment problem (FLAP) is an extension of the classical linear assignment problem (LAP) to situations in which uncertainty in the cost coefficients is represented by fuzzy numbers. FLAP applications range from the assignment of workers to tasks to multiple-criteria decision analysis in fuzzy environments and many other engineering applications. Most FLAP formulations assume that all cost coefficients are fuzzy numbers of the same type (e.g. triangular, trapezoidal). The standard solution approach is the defuzzification of the cost coefficients, thus transforming the FLAP into a crisp LAP that can be solved by classical assignment algorithms such as the Hungarian method. It is known that defuzzification methods suffer from lack of discrimination when comparing fuzzy numbers which may lead to suboptimal assignments. The solution approach proposed in this paper is based on the theory of algebraic assignment problems and total orderings in the set of all fuzzy numbers, and it allows to solve FLAPs with different types of fuzzy numbers. More specifically, the FLAP is transformed into a lexicographic linear assignment problem (LLAP) which is solved in its place. We show, both theoretically and numerically, how this transformation overcomes the limitations present in existing approaches.
Design and fabrication of a freeform mirror generating a uniform illuminance distribution in a rectangular region
