Effect of physical and chemical pressure on the superconductivity of cage-type compound Lu5Rh6Sn18

${\mathrm{Lu}}₅{\mathrm{Rh}}₆{\mathrm{Sn}}₁₈$ is one of the cage-type superconductors with superconducting transition temperature ${T}_c=4.12$ K. Here we investigate the effect of pressure on the superconductivity in ${\mathrm{Lu}}₅{\mathrm{Rh}}₆{\mathrm{Sn}}₁₈$ by combining high-pressure electrical transport, synchrotron x-ray diffraction (XRD), and chemical doping. Application of high pressure can enhance both the metallicity and the superconducting transition temperature in ${\mathrm{Lu}}₅{\mathrm{Rh}}₆{\mathrm{Sn}}₁₈$. ${T}_c$ is found to show a continuous increase reaching up to 5.50 K at 11.4 GPa. Our high-pressure synchrotron XRD measurements demonstrate the stability of the pristine crystal structure up to 12.0 GPa. In contrast, ${T}_c$ is suppressed after the substitution of La ions in Lu sites, inducing negative chemical pressure. Our study provides valuable insights into the improvement of superconductivity in caged compounds.