The lead time demand for bathing suits is governed by the discrete random variable shown in Table 11. The company sells an average of $10,400$ suits per year. The cost of placing an order for bathing suits is $30$, and the cost of holding one bathing suit in inventory for a year is $3$. The stockout cost is $3$ per bathing suit. Use marginal analysis to determine the reorder point.
My thought: The key point here is the lead time demand $r$ is discrete, so we cannot apply the usual formula $P(X\geq r) = \frac{hq^{*}}{c_BE(D)} = \frac{3*456}{3*10400}$. So, we have to do it by considering the overstock and understock case, even when we don't know the cost of $1$ bathing suit:
If $d\leq q$, then the system is overstock with $c(d,q) = 30 + 3(q-d)$. Thus, $c_0=3$.
If $d\geq q$, then the system is understock with $c(d,q) = 3(d-q) + 30$. Thus, $c_u=3$.
Therefore, the reorder point $r$ must satisfy: $P(X\leq r) = \frac{3}{3+3} = \frac{1}{2}$. By looking at the table above, $r = \fbox{$190$}$ is the best choice to keep the current service level of $50\%$ and cover the lead time demand.
My question: Is my solution above correct? Or it does not make sense at all, because I would need to factor in the cost of a bathing suit, which affects the values of $c_0$ and $c_u$? Please help with some thought.