© 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.The ferri-ferro-ferrimagnetic quaternary alloy described in the form as ABpCqDr consisting of spins SiA=1, SiB=1/2, SjC=3/2 and SjD=5/2, respectively, is investigated in detail by the mean field theory (MFT) based on the Bogoliubov inequality for the free energy with the coordination number z= 3. The phase diagrams are mapped on the planes of (R1= JAC/ | JAB| , kTc/ | JAB|) , (R2= | JAD| / | JAB| , kTc/ | JAB|) , (p, kTc/ | JAB|) and (r, kTc/ | JAB|) by studying the thermal variations of the order parameters, i.e., the magnetizations. In addition to the critical temperatures, either second- or first-order phase transition temperatures without distinction, shown by the model, it also exhibits a single or a double compensation temperatures for appropriate values of the system parameters. The Monte Carlo (MC) simulations are also used to confirm the existence of the multi-compensation behaviors in the model as found by the MFT approach by examining the thermal variations of the total magnetization of the system. Our obtained results are compared to those reported in the literature, and reliable agreements are observed.