We develop and analyze a stochastic model of directed intermittent search for a hidden target on a one-dimensional track. A particle injected at one end of the track randomly switches between a stationary search phase and a mobile, non-search phase that is biased in the anterograde direction. There is a finite possibility that the particle fails to find the target due to an absorbing boundary at the other end of the track or due to competition with other targets. Such a scenario is exemplified by the motor-driven transport of mRNA granules to synaptic targets along a dendrite. We first calculate the hitting probability and conditional mean first passage time (MFPT) for finding a single target. We show that an optimal search strategy does not exist, although for a fixed hitting probability, a unidirectional rather than a partially biased search strategy generates a smaller MFPT. We then extend our analysis to the case of multiple targets, and determine how the hitting probability and MFPT depend on the number of targets.