Eq. (5.4) — the first approximation from the direct-route policy does not exceed it

Let $N = n + 1 \ge 2$ and $t_{ij} > 0$ for $i \ne j$. Let $f^{(0)}$ be the direct-route policy (5.2), $f_i^{(0)} = t_{iN}$ for $i \ne N$ and $f_N^{(0)} = 0$, and let $f^{(1)}$ be given by (5.3): $f_i^{(1)} = \min_{j \ne i}[t_{ij} + f_j^{(0)}]$ for $i \ne N$, $f_N^{(1)} = 0$. Then

This is the first step of the monotone decrease (5.5).

Formalization Note The paper prints (5.2) for $i = 1, \dots, N$, which gives $f_N^{(0)} = t_{NN}$. Read literally, (5.4) is then false whenever $t_{NN} > 0$. For $N = 2$: $f_1^{(1)} = t_{12} + f_2^{(0)} = t_{12} + t_{22} > t_{12} = f_1^{(0)}$. The paper's own justification ("$f_i^{(1)}$ represents the minimum time for a path with at most one stop") requires $f_N^{(0)} = 0$, which is what is used here. This is equivalent to the printed (5.2) under $t_{NN} = 0$.