© 2019 IOP Publishing Ltd.The half-Heusler TiRhSb and ZrRhSb alloys in the formation of face-centered cubic MgAgAs-type structure, which conforms to the F43m space group with 216 as the space number, have been investigated using Generalized Gradient Approximation (GGA) implemented in Density Functional Theory (DFT). The calculated formation enthalpies and the plotted energy-volume curves of different types of structural phases (a, β, and γ) in these alloys indicate that the γ-phase structure is the best energetically suitable structure. In addition, TiRhSb and ZrRhSb alloys have been found as paramagnetic (PM) with the investigation of antiferromagnetic (AFM), ferromagnetic (FM), and paramagnetic (PM) orders in the most stable γ-phase structure. Therefore, their electronic, mechanical, and dynamical properties have been examined in the γ structural phase and paramagnetic order. These alloys have semiconducting nature due to the observed same band gaps in both the majority and minority spin channels of the calculated spin-polarised electronic band structure. These calculated band gaps are 0.75 eV for γ-TiRhSb and 1.18 eV for γ-ZrRhSb. The predicted elastic constants indicate that the alloys in this study are mechanically stable andshow nearly isotropic behavior in the γ structural phase. Also, the minimum and the diffuson thermal conductivites have been determined for these alloys. Finally, the calculated phonon dispersion spectras for the γ-TiRhSb and γ-ZrRhSb half-Heusler antimonide alloys show the dynamic stability of these systems.