lakemodel_function.py

import math
import numpy as np
from scipy.optimize import brentq


def lake_problem(
        b=0.42,         # Decay parameter for P in lake (0.42 = irreversible)
        q=2.0,          # Recycling exponent
        mean=0.02,      # Mean of natural inflows
        stdev=0.0017,   # Future utility discount rate
        delta=0.98,     # Standard deviation of natural inflows
        alpha=0.4,      # Utility from pollution
        nsamples=100,   # Number of Monte Carlo samples to draw
        steps=100,      # Number of time steps
        l0=0, l1=0, l2=0, l3=0, l4=0, l5=0, l6=0, l7=0, l8=0, l9=0,
        l10=0, l11=0, l12=0, l13=0, l14=0, l15=0, l16=0, l17=0, l18=0, l19=0,
        l20=0, l21=0, l22=0, l23=0, l24=0, l25=0, l26=0, l27=0, l28=0, l29=0,
        l30=0, l31=0, l32=0, l33=0, l34=0, l35=0, l36=0, l37=0, l38=0, l39=0,
        l40=0, l41=0, l42=0, l43=0, l44=0, l45=0, l46=0, l47=0, l48=0, l49=0,
        l50=0, l51=0, l52=0, l53=0, l54=0, l55=0, l56=0, l57=0, l58=0, l59=0,
        l60=0, l61=0, l62=0, l63=0, l64=0, l65=0, l66=0, l67=0, l68=0, l69=0,
        l70=0, l71=0, l72=0, l73=0, l74=0, l75=0, l76=0, l77=0, l78=0, l79=0,
        l80=0, l81=0, l82=0, l83=0, l84=0, l85=0, l86=0, l87=0, l88=0, l89=0,
        l90=0, l91=0, l92=0, l93=0, l94=0, l95=0, l96=0, l97=0, l98=0, l99=0):
    # Create a NumPy array of the decision variables
    decisions = np.array([l0, l1, l2, l3, l4, l5, l6, l7, l8, l9, l10, l11, l12, l13,
                          l14, l15, l16, l17, l18, l19, l20, l21, l22, l23, l24, l25,
                          l26, l27, l28, l29, l30, l31, l32, l33, l34, l35, l36, l37,
                          l38, l39, l40, l41, l42, l43, l44, l45, l46, l47, l48, l49,
                          l50, l51, l52, l53, l54, l55, l56, l57, l58, l59, l60, l61,
                          l62, l63, l64, l65, l66, l67, l68, l69, l70, l71, l72, l73,
                          l74, l75, l76, l77, l78, l79, l80, l81, l82, l83, l84, l85,
                          l86, l87, l88, l89, l90, l91, l92, l93, l94, l95, l96, l97,
                          l98, l99])
    nvars = len(decisions)

    # Calculate the critical pollution level (Pcrit)
    Pcrit = brentq(lambda x: x ** q / (1 + x ** q) - b * x, 0.01, 1.5)

    # Generate natural inflows using lognormal distribution
    natural_inflows = np.random.lognormal(
        mean=math.log(mean ** 2 / math.sqrt(stdev ** 2 + mean ** 2)),
        sigma=math.sqrt(math.log(1.0 + stdev ** 2 / mean ** 2)),
        size=(nsamples, nvars)
    )

    # Initialize the pollution level matrix X
    X = np.zeros((nsamples, nvars))

    # Loop through time to compute the pollution levels
    for t in range(1, nvars):
       X[:, t] = (1 - b) * X[:, t - 1] + (X[:, t - 1] ** q / (1 + X[:, t - 1] ** q)) + decisions[
            t - 1] + natural_inflows[:, t - 1]

    # Calculate the average daily pollution for each time step
    average_daily_P = np.mean(X, axis=0)

    # Calculate the reliability (probability of the pollution level being below Pcrit)
    reliability = np.sum(X < Pcrit) / float(nsamples * nvars)

    # Calculate the maximum pollution level (max_P)
    max_P = np.max(average_daily_P)

    # Calculate the utility by discounting the decisions using the discount factor (delta)
    utility = np.sum(alpha * decisions * np.power(delta, np.arange(nvars)))

    # Calculate the inertia (the fraction of time steps with changes larger than 0.02)
    inertia = np.sum(np.abs(np.diff(decisions)) > 0.02) / float(nvars - 1)

    return max_P, utility, inertia, reliability