In this paper multigraphs \(G=(V,E)\) without loops are under consideration. They are assigned edge colourings satisfying the following two conditions: (i) Each colour appears at each vertex v no more than f(v) times. (ii) Each colour appears at each set of multiple edges joining vertices v and w no more than g(vw) times. These edge colourings are called fg-colourings. \(q^*_{fg}(G)\) denotes the minimum number of colours needed to fg-colour a multigraph G. Various upper bounds on \(g^*_{fg}(G)\) are given. The main result is \[ q^*_{fg}(G)\leq \max_{vw\in G}\lceil d(v)/f(v)+p(vw)/g(vw)\rceil. \] A polynomial time algorithm to fg-colour a given multigraph using \(\leq 2q^*_{fg}(G)\) colours is presented.