import numpy as np

t_star = 8 + 50/60
tas = np.linspace(7.0, 10.0, 10)
tts = np.array([15.0, 17.0, 17.0, 21.0, 23.0, 29.0, 27.0, 23.0, 19.0, 15.0])
if t_star not in tas:
    t_star_id = np.argmax(tas > t_star)
    tas = np.insert(tas, t_star_id, t_star)
    t_star_tt = (tts[t_star_id-1] + tts[t_star_id]) / 2
    tts = np.insert(tts, t_star_id, t_star_tt)

alpha = 15
beta = 5
gamma = 20
schedule_costs = (
    beta * (t_star - tas) * (t_star > tas)
    + gamma * (tas - t_star) * (tas > t_star)
)
costs = alpha * tts / 60 + schedule_costs

mu = 1
exp_costs = np.exp(-costs / mu)

probs = list()
for i in range(len(tas)-1):
    m = min(exp_costs[i], exp_costs[i+1])
    M = max(exp_costs[i], exp_costs[i+1])
    prob = (tas[i+1] - tas[i]) * (m + (M-m) / 2)
    probs.append(prob)
probs = np.array(probs) / np.sum(probs)
probs = np.insert(probs, 0, 0)
cum_probs = np.cumsum(probs)