We investigate rate selection methods to maximize a utility function of a multihop relay network adopting a Chase combining-type hybrid automatic repeat request scheme. For supporting delay-limited applications, we consider the following two constraints: 1) The number of (re)transmissions used in the whole hops is limited to $L$, and 2) the information-theoretic outage probability should be less than or equal to $\epsilon$. We formulate an optimization problem to maximize the long-term average transmission rate of an $M$-hop relay network by finding the optimal round transmission rate [b/s/Hz] of each hop while satisfying those two constraints. We also consider two reformulated problems and, as the solutions to those problems, propose suboptimal search algorithms, which significantly reduce the complexity, compared with that of the optimal search algorithm corresponding to the solution of the original optimization problem. While the original optimization problem always requires $M$-dimensional exhaustive search, the first reformulated problem requires $M$-dimensional exhaustive search only in a specific condition, and it obtains a closed-form solution otherwise. The second reformulated problem requires 1-D numerical search only in a specific condition, and it obtains a closed-form solution otherwise. We also show that the relative loss of the solution of the first reformulated problem is upper bounded by $\hbox{2}\epsilon$.