import numpy as np
from scipy.optimize import minimize
import math
 
def total_travel_time(d):
    """
    Given an array d of 7 horizontal displacements (in leagues) for each segment,
    compute the total travel time.
 
    Each segment has a fixed vertical distance of 10 leagues. The travel distance in
    segment i is L_i = sqrt(d_i^2 + 10^2). The speeds for the 7 segments are:
        v0 = 10, v1 = 9, v2 = 8, v3 = 7, v4 = 6, v5 = 5, v6 = 10 (leagues/day).
 
    The total time is:
        T = sum_{i=0}^{6} L_i / v_i.
    """
    speeds = [10, 9, 8, 7, 6, 5, 10]
    T = 0.0
    for i in range(7):
        L = math.sqrt(d[i]**2 + 10**2)
        T += L / speeds[i]
    return T
 
def constraint_sum(d):
    """Constraint: sum of horizontal displacements must equal 100 leagues."""
    return np.sum(d) - 100
 
# Initial guess: equally distribute 100 leagues among 7 segments.
d0 = np.full(7, 100/7)
 
# Set up the equality constraint and bounds (each d_i must be nonnegative)
cons = {'type': 'eq', 'fun': constraint_sum}
bnds = [(0, None) for _ in range(7)]
 
# Use SLSQP to minimize the total travel time
res = minimize(total_travel_time, d0, method='SLSQP', bounds=bnds, constraints=cons, options={'ftol': 1e-12})
 
if res.success:
    optimal_d = res.x
    T_opt = res.fun
    # Recover the cumulative horizontal x–positions: x0 = 0, then x_i = sum_{j=1}^i d_j, and x7 = 100.
    x_positions = np.concatenate(([0], np.cumsum(optimal_d), [100]))
 
    print("Optimal horizontal displacements d_i (leagues):")
    print(optimal_d)
    print("\nCumulative x–positions:")
    print(x_positions)
    print("\nMinimum travel time = {:.13f} days".format(T_opt))
else:
    print("Optimization did not converge.")

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