With the advances in the discovery of molecular targets, there is increasing interest in evaluating targeted therapies for disease subtypes characterized by certain biomarkers. Patients with a certain biomarker could potentially benefit from different experimental drugs, and, therefore, evaluating the relative efficacy of these drugs is an important objective. We consider the design of an early phase platform trial where multiple therapies are evaluated in patients with different biomarkers, with the objective of identifing the best drug at an efficacious and safe dose for a given disease subtype. We use the continual reassessment method to estimate the maximum tolerated dose of a drug and adopt hierarchical Bayesian modelling to estimate the efficacy of a drug administered at multiple doses. Using the continual reassessment method and hierarchical Bayesian modelling as the basis of inference, we propose various algorithms that prescribe the drug–dose for the patients using adaptive randomization. We demonstrate that adaptive randomization puts more patients at the right drug and dose on average than does balanced randomization, with slightly larger variability in distribution, and has no effect on the accuracy of drug–dose selection. Moreover the simulations show advantages of hierarchical Bayesian modelling over the beta–binomial model in scenarios with relatively flat or partially flat dose–efficacy relationships.