The current problem of developing new kinds of decision support systems for different categories of management personnel is addressed in this study. A critical feature of such systems is their distributed and decentralized nature, which enables the construction of next-generation information systems in the form of Multi-Agent Systems, Internet of Things, or Fog Computing Architectures. Parallel models of the dynamics of artificial neural networks are produced under such realistic circumstances, demonstrating their potential for addressing a variety of issues. The purpose of this study is to conduct a critical analysis of the problem of integrating Artificial Neural Networks with decision support systems using a corpus of relevant scholarly literature. To tackle this question, the Design Science Research methodology was considered. According to this methodology, a literary search strategy was established, scientific literature was collected and analyzed, and key comparisons between different solutions were emphasized. The study resulted in the presentation of the most important findings, outstanding issues, and potential areas of fundamental and applied solutions. A consistent trend toward the development of decision support systems based on integrated neural-network methods has been observed, which is efficient and cost-effective since it enables the creation of distributed and trainable decision support systems.