With the advent of the Internet of Things (IoT), novel critical applications have emerged that leverage the edge/hub/cloud paradigm, which diverges from the conventional edge computing perspective. A growing number of such applications require a streamlined architecture for their effective execution, often comprising a single edge device with sensing capabilities, a single hub device (e.g., a laptop or smartphone) for managing and assisting the edge device, and a more computationally capable cloud server. Typical examples include the utilization of an unmanned aerial vehicle (UAV) for critical infrastructure inspection or a wearable biomedical device (e.g., a smartwatch) for remote patient monitoring. Task allocation in this streamlined architecture is particularly challenging, due to the computational, communication, and energy limitations of the devices at the network edge. Consequently, there is a need for a comprehensive framework that can address the specific task allocation problem optimally and efficiently. To this end, we propose a complete, binary integer linear programming (BILP) based formulation for an application-driven design-time approach, capable of providing an optimal task allocation in the targeted edge/hub/cloud environment. The proposed method minimizes the desired objective, either the overall latency or overall energy consumption, while considering several crucial parameters and constraints often overlooked in related literature. We evaluate our framework using a real-world use-case scenario, as well as appropriate synthetic benchmarks. Our extensive experimentation reveals that the proposed approach yields optimal and scalable results, enabling efficient design space exploration for different applications and computational devices.