In the literature, there are several models for detecting communities in social networks. In Zardi and Romdhane (2013), we presented a robust method, called maximum equilibrium purity (MEP), in which we defined a new function that qualifies a network partition into communities, and we presented an algorithm that optimises this function. We proved that, unlike modularity-based models, MEP does not suffer from the 'resolution limit' problem. However, MEP operates only on unweighted networks; i.e., networks where all connections are considered equally. Hence, strengths of social ties between network nodes are ignored. Unfortunately, this assumption may not hold in several real-world networks where tie strengths play a major role. In this paper, we present the maximum weighted equilibrium purity algorithm (MWEP), the extension of MEP to weighted networks. Like the original model, the extended model is proved to circumvent the 'resolution limit' problem encountered in community detection. In addition, we have applied our model to real-world and synthetic social networks and experimental results are more than encouraging.