A fundamental question in response-adaptive randomization is: What allocation proportion should we target to achieve required power while resulting in fewer treatment failures? For comparing two treatments, such optimal allocations are well studied in the literature. However, few authors address the question for multiple treatments and the generalization of optimal allocations is necessary in practice. We are interested in finding the optimal allocation proportion, which achieves a required power of a multivariate test of homogeneity in binary response experiments while minimizing expected treatment failures at the same time. We propose such an optimal allocation for three treatments by giving an analytical solution for the optimization problem. Numerical studies show that a response-adaptive randomization procedure that targets proposed optimal allocation is superior to complete randomization. We also discuss some future research topics and additional issues on optimal adaptive designs.