Despite extensive research on stochastic parameters, the relations between such parameters and their impacts on system behaviors have not been fully characterized. These relations are affected by various external conditions and complicated internal processes, making them difficult to define with sufficient accuracy. In this paper, we investigate the relationship between the machine utilization parameter and machine failure-rate parameter. Fuzzy set theory is adopted to represent this uncertain relation. The fuzzy relation is formulated in terms of fuzzy triangular coefficients, and a fuzzy-relation-based Petri-net (FR-PN) is established to model the system behaviors and optimize the system performance. The modeling approach is illustrated through a typical example of a flexible manufacturing cell. A fuzzy chance-constrained programming model is established on the basis of the state equilibrium equations of the FR-PN. Our analysis shows that the proposed FR-PN has a greater generalized modeling ability than a conventional fuzzy-parameter-based Petri-net. For the typical flexible manufacturing cell, the unique optimal machine processing-time parameters obtained for various fuzzy coefficients demonstrate that our results are more realistic than those given by the conventional approach. The machine utilization rate obtained by the proposed FR-PN is superior to that given by the fuzzy-parameter-based Petri-net.