Optimizing Location of Brothers in a Circle: Maximizing Distance for Minimal Constraint

Optimizing Location of Brothers in a Circle: Maximizing Distance for Minimal Constraint ######### In this blog post, we delve into an intriguing problem related to finding the optimal positions of brothers within a circle. Our goal is to maximize their minimum distance from each other while ensuring they all reside within a specific circle. Let's get started! #### Why do we need this solution? The objective is to position the brothers in such a way that their minimum distance is maximized, thus preventing unnecessary conflicts or disagreements. This can be particularly useful in scenarios where team collaboration is crucial. #### Steps for Solving the Problem Here are the key steps involved in solving this problem: 1. Calculate Distance: To calculate the distance between pairs of brothers, we use the formula (x_i - x_j)² + (y_i - y_j)² > r², where small r represents the minimum acceptable distance between two brothers and capital R is the radius of the circle in which they reside. 2. Define Constraints: To ensure that all brothers remain within the given circle, we need to define a constraint for every brother (i). 3. Objective Function: Our objective function aims to maximize the minimum distance between pairs of brothers. We employ linear programming with Pyomo as our solver for this problem. #### Python Code Below you'll find the Python code that calculates the optimal locations of the brothers within a circle: ```python import math import ipopt from pyomo.environ import * n = 20 # number of brothers r = 1.0 # radius of the circle x = Var(range(n), within=Reals(0, 2r)) y = Var(range(n), within=Reals(0, 2r)) model = ConcreteModel() model.x = x model.y = y r_min = param.Parameter(initial=0.1) c1 = constraint.ConstraintList(model.indexes(), rule=lambda i, j: (i != j) & (x[j] - x[i])2 + (y[j] - y[i])2 > r2) model.c1.add_bounds([(-rr, None) for _ in range(n(n-1))]) constraint_circle = constraint.ConstraintList() for i in model.x: constraint_circle.add(sum((model.x[j] - x[i])2 + (model.y[j] - y[i])2) <= r2) objective = Objective(expr=minimize(min([min([(x[j] - x[i])2 + (y[j] - y[i])2 for i in model.indexes() if j != i]) for j in model.indexes()))) model.objective = objective def main(): # Define the problem and solve it solver = ipopt.SolverFactory('ipopt') prob = model.to_ipopt_problem(solver=solver, maxls=50) results = prob.solve(tee=True) print(