We propose a new integrated phase I/II trial design to identify the most efficacious dose combination that also satisfies certain safety requirements for drug-combination trials. We first take a Bayesian copula-type model for dose finding in phase I. After identifying a set of admissible doses, we immediately move the entire set forward to phase II. We propose a novel adaptive randomization scheme to favor assigning patients to more efficacious dose-combination arms. Our adaptive randomization scheme takes into account both the point estimate and variability of efficacy. By using a moving reference to compare the relative efficacy among treatment arms, our method achieves a high resolution to distinguish different arms. We also consider groupwise adaptive randomization when efficacy is late-onset. We conduct extensive simulation studies to examine the operating characteristics of the proposed design, and illustrate our method using a phase I/II melanoma clinical trial.