To engineer a cheap, portable and low-power optical gas sensor, incandescent sources are more suitable than expensive quantum cascade lasers and low-efficiency light-emitting diodes. Such sources of radiation have already been realized, using standard MEMS technology, consisting in free standing circular micro-hotplates. This paper deals with the design of such membranes in order to maximize their wall-plug efficiency. Specification constraints are taken into account, including available energy per measurement and maximum power delivered by the electrical supply source. The main drawback of these membranes is known to be the power lost through conduction to the substrate, thus not converted in (useful) radiated power. If the membrane temperature is capped by technological requirements, radiative flux can be favored by increasing the membrane radius. However, given a finite amount of energy, the larger the membrane and its heat capacity, the shorter the time it can be turned on. This clearly suggests that an efficiency optimum has to be found. Using simulations based on a spatio-temporal radial profile, we demonstrate how to optimally design such membrane systems, and provide an insight into the thermo-optical mechanisms governing this kind of devices, resulting in a nontrivial design with a substantial benefit over existing systems. To further improve the source, we also consider tailoring the membrane stack spectral emissivity to promote the infrared signal to be sensed as well as to maximize energy efficiency.