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Patients with chronic diseases, such as cancer or epilepsy, are often followed through multiple stages of clinical interventions. Dynamic treatment regimes (DTRs) are sequences of decision rules that assign treatments at each stage based on measured covariates for each patient. A DTR is said to be optimal if the expectation of the desirable clinical benefit reaches a maximum when applied to a population. When there are three or more options for treatments at each decision point and the clinical outcome of interest is a time-to-event variable, estimating an optimal DTR can be complicated. We propose a doubly robust method to estimate optimal DTRs with multicategory treatments and survival outcomes. A novel blip function is defined to measure the difference in expected outcomes among treatments, and a doubly robust weighted least squares algorithm is designed for parameter estimation. Simulations using various weight functions and scenarios support the advantages of the proposed method in estimating optimal DTRs over existing approaches. We further illustrate the practical value of our method by applying it to data from the Standard and New Antiepileptic Drugs study. In this analysis, the proposed method supports the use of the new drug lamotrigine over the standard option carbamazepine. When the actual treatments match the estimated optimal treatments, survival outcomes tend to be better. The newly developed method provides a practical approach for clinicians that is not limited to cases of binary treatment options. [ABSTRACT FROM AUTHOR]