To calculate the base-stock to meet a 99% type-1 service level, we need the 0.99 fractile of the demand distribution. The safety stock level is the base-stock level minus the mean demand.
For the lognormal case, the author has fit a lognormal distribution and found that the parameters are $\mu=2.645$ and $\sigma=0.83255$. (Note that for a lognormal distribution, $\mu$ and $\sigma$ are not the mean and SD.)
For a lognormal distribution with those parameters, the mean is 19.9169 and the 99th quantile is 97.6902. The difference between the two is the safety stock (approx. 78).
Here's some Python code to calculate all of this:
from scipy.stats import lognorm
import numpy as np
mu = 2.645
sigma = 0.83255
s = sigma
scale = np.exp(mu)
mean_demand = lognorm(s=s, scale=scale).mean()
base_stock = lognorm(s=s, scale=scale).ppf(0.99)
base_stock - mean_demand
(For more on the parameters of
lognorm, see here.)
I didn't work through the "cauterized normal" case in detail, but roughly speaking the logic should be the same. When he says "cauterized normal" I think he means that we take the probability in the tail below 0 and add a probability mass at 0, with that probability.