The Fuzzy Robust Graph Coloring Problem

Fuzzy graph model can represent a complex, imprecise and uncertain problem, where classical graph model may fail. In this paper, we propose a fuzzy graph model to represent the examination scheduling problem of a university and introduce a genetic algorithm based method to find the robust solution of the scheduling problem that remains feasible and optimal or close to optimal for all scenarios of the input data. Fuzzy graph coloring method is used to compute the minimum number of days to schedule the examination. But problem arises if after the examination schedule is published, some students choose new courses in such a way that it makes the schedule invalid. We call this problem as fuzzy robust coloring problem (FRCP). We find the expression for robustness and based on its value, robust solution of the examination schedule is obtained. The concept of fuzzy probability of fuzzy event is used in the expression of robustness, which in turn, is used for fitness evaluation in genetic algorithm. Each chromosome in the genetic algorithm, used for FRCP, represents a coloring function. The validity of the coloring function is checked keeping the number of colors fixed. Fuzzy graphs with different number of nodes are used to show the effectiveness of the proposed method.