A model for the study and optimization of two heat reservoirs thermal machines is presented. The mathematical model basically consists of the First and Second Laws of Thermodynamics applied to the cycle and entire system, and the heat transfer equations at the source and sink. The internal and external irreversibilities of the cycle are considered by taking into account the entropy generation terms. Several constraints imposed to the system composed by the engine and the two heat reservoirs (namely, engine efficiency, or power output, or heat flux received by the engine, each of them together with imposed internal entropy generation and total number of heat transfer units of the machine heat exchangers) allow us to find the optimum operational conditions, as well as the limited variation ranges for the system parameters. Emphasis is put on coupling between various possible objective functions, namely thermal cost, useful effect, first law efficiency and whole system dissipation. It is for the first time to our knowledge when it has been proved that if one of the possible objective functions is fixed (as a parameter with imposed value), the optima of the other three always correspond to each other for the corresponding stationary state system, with a given optimum heat conductance allocation (one degree of freedom). Other interesting results are also reported in this paper. Some sensitivity studies were developed, too, with respect to various parameters of the model (engine performance, internal entropy generation, total number of heat transfer units).