Linear Programming for Optimizing Funding Costs. See https://doi.org/10.5281/zenodo.4607219 for documentation.

optcost.py

from scipy.optimize import linprog
import numpy as np
import pandas as pd


def print_metrics(df):
    print('Total staff costs', df.to_numpy().sum())
    print('Management cost ratio')
    print(df.MgtStaffCosts / df.to_numpy().sum())
    print('Partner cost ratio')
    print(df.sum(1) / df.to_numpy().sum())


def setup_constraint_a(df, r):
    N = len(df)
    A = np.full((N, 2*N), -r)
    np.fill_diagonal(A, 1.0 - r)
    b = np.zeros(N)

    return A, b


def setup_constraint_b(df, st_range):
    N = len(df)
    # Upper bound
    A_ub = np.full((N, 2*N), -st_range[1])
    np.fill_diagonal(A_ub, 1.0 - st_range[1])
    np.fill_diagonal(A_ub[:, N:], 1.0 - st_range[1])
    # Lower bound
    A_lb = np.full((N, 2*N), -st_range[0])
    np.fill_diagonal(A_lb, 1.0 - st_range[0])
    np.fill_diagonal(A_lb[:, N:], 1.0 - st_range[0])
    A_lb *= -1

    b = np.zeros(2*N)
    return np.block([[A_ub], [A_lb]]), b


def main():
    np.set_printoptions(precision=2, floatmode='maxprec', suppress=True)

    df = pd.DataFrame(
        [
            ('A', 107214, 25500),
            ('B', 94674, 17000),
            ('C', 40000, 24000),
        ],
        columns=['Company', 'OtherStaffCosts', 'MgtStaffCosts']
    ).set_index('Company')

    """Maximum management staff cost ratio."""
    R = 0.07
    """Allowed staff costs ratio per partner."""
    ST = (0.1, 0.5)

    N = len(df)
    print(df)
    print_metrics(df)

    # Note, linprog minimizes the objective, so we invert it
    c = -np.ones(2*N)
    # Setup constraints a) and b)
    A0, b0 = setup_constraint_a(df, R)
    A1, b1 = setup_constraint_b(df, ST)

    # Fusion in a large constraint matrix
    A = np.block([[A0], [A1]])
    b = np.concatenate((b0, b1))

    # Constraint c) can be given as bounds.
    bounds = [
        (0, xi) for xi in df.MgtStaffCosts.to_numpy()
    ]
    bounds.extend([
        (0, yi) for yi in df.OtherStaffCosts.to_numpy()
    ])

    res = linprog(c, A_ub=A, b_ub=b, bounds=bounds, method='revised simplex')
    print(pd.Series(np.asarray(res.x[:N])))

    print()
    print('optimized:')
    dfopt = df.copy()
    dfopt.loc[:, 'MgtStaffCosts'] = res.x[:N]
    dfopt.loc[:, 'OtherStaffCosts'] = res.x[N:]
    print(dfopt)
    print_metrics(dfopt)


if __name__ == '__main__':
    main()

Update for a new constraint that popped up during discussion.

def setup_constraint_c(df, r):
    '''Total management costs / sum of all costs <= r
        sum(xi)/(sum(xi) + sum(yi)) <= r    
    '''
    N = len(df)
    A = np.zeros((1, 2 * N))
    A[0, :N] = 1.0 - r
    A[0, N:] = -r
    b = np.zeros(1)

    return A, b