We estimate physical parameters for the late-type massive stars observed as part of the VLT-FLAMES Tarantula Survey (VFTS) in the 30 Doradus region of the Large Magellanic Cloud (LMC). The observational sample comprises 20 candidate red supergiants (RSGs) which are the reddest (($B-V$) $>$ 1 mag) and brightest ($V$ $<$ 16 mag) objects in the VFTS. We use optical and near-IR photometry to estimate their temperatures and luminosities, and introduce the luminosity-age diagram to estimate their ages. We derive physical parameters for our targets, including temperatures from a new calibration of $(J-K_{rm s})_{0}$ colour for luminous cool stars in the LMC, luminosities from their $J$-band magnitudes (thence radii), and ages from comparisons with state-of-the-art evolutionary models. We show that interstellar extinction is a significant factor for our targets, highlighting the need to take it into account in analysis of the physical parameters of RSGs. We find that some of the candidate RSGs could be massive AGB stars. The apparent ages of the RSGs in the Hodge 301 and SL 639 clusters show a significant spread (12-24 Myr). We also apply our approach to the RSG population of the relatively nearby NGC 2100 cluster, finding a similarly large spread. We argue that the effects of mass-transfer in binaries may lead to more massive and luminous RSGs (which we call `red stragglers) than expected from single-star evolution, and that the true cluster ages correspond to the upper limit of the estimated RSG ages. In this way, the RSGs can serve as a new and potentially reliable age tracer in young star clusters. The corresponding analysis yields ages of 24$^{+5}_{-3}$ Myr for Hodge 301, 22$^{+6}_{-5}$ Myr for SL 639, and 23$^{+4}_{-2}$ Myr for NGC 2100.
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