To understand the function of genetic regulatory networks in the development of cellular systems, we must not only realise the individual network entities, but also the manner by which they interact. Multi-valued networks are a promising qualitative approach for modelling such genetic regulatory networks, however, at present they have limited formal analysis techniques and tools. We present a flexible formal framework for modelling and analysing multi-valued genetic regulatory networks using high-level Petri nets and logic minimization techniques. We demonstrate our approach with a detailed case study in which part of the genetic regulatory network responsible for the carbon starvation stress response in Escherichia coli is modelled and analysed. We then compare and contrast this multivalued model to a corresponding Boolean model and consider their formal relationship.