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We study how to schedule data sources in a wireless time-sensitive information system with multiple heterogeneous and unreliable channels to minimize the total expected Age-of-Information (AoI). Although one could formulate this problem as a discrete-time Markov Decision Process (MDP), such an approach suffers from the curse of dimensionality and lack of insights. For single-channel systems, prior studies have developed lower-complexity solutions based on the Whittle index. However, Whittle index has not been studied for systems with multiple heterogeneous channels, mainly because indexability is not well defined when there are multiple dual cost values, one for each channel. To overcome this difficulty, we introduce new notions of partial indexability and partial index, which are defined with respect to one channel's cost, given all other channels' costs. We then combine the ideas of partial indices and max-weight matching to develop a Sum Weighted Index Matching (SWIM) policy, which iteratively updates the dual costs and partial indices. The proposed policy is shown to be asymptotically optimal in minimizing the total expected AoI, under a technical condition on a global attractor property. Extensive performance simulations demonstrate that the proposed policy offers significant gains over conventional approaches by achieving a near-optimal AoI. Further, the notion of partial index is of independent interest and could be useful for other problems with multiple heterogeneous resources.