BruteforceAssignment.cs

using System;
void Main()
{
	// define the cost matrix
	int[,] costMatrix = {{17, 10, 21},
					{15, 12, 19},
					{13, 10, 17}};
	int numRows = costMatrix.GetLength(0);
	int numCols = costMatrix.GetLength(1);

	// generate all possible assignments
	var assignments = GetPermutations(Enumerable.Range(0, numCols), numRows).ToList();

	// calculate the cost of each assignment
	int minCost = int.MaxValue;
	List<int> minAssignment = new List<int>();
	foreach (var assignment in assignments)
	{
		int cost = 0;
		for (int i = 0; i < numRows; i++)
		{
			cost += costMatrix[i, assignment.ToList()[i]];
		}
		Console.WriteLine($"Assignment: {string.Join(", ", assignment)}, Cost: {cost}");
		if (cost < minCost)
		{
			minCost = cost;
			minAssignment = assignment.ToList();
		}
	}

	Console.WriteLine($"\nMinimum cost: {minCost}, Assignment: {string.Join(", ", minAssignment)}");
}
static IEnumerable<IEnumerable<T>> GetPermutations<T>(IEnumerable<T> list, int length)
{
	if (length == 1)
	{
		return list.Select(x => new[] { x });
	}

	return GetPermutations(list, length - 1)
		.SelectMany(x => list.Where(y => !x.Contains(y)),
			(t1, t2) => t1.Concat(new[] { t2 }));
}

Assignment: 0, 1, 2, Cost: 46
Assignment: 0, 2, 1, Cost: 46
Assignment: 1, 0, 2, Cost: 42
Assignment: 1, 2, 0, Cost: 42
Assignment: 2, 0, 1, Cost: 46
Assignment: 2, 1, 0, Cost: 46

Minimum cost: 42, Assignment: 1, 0, 2

import itertools

# define the cost matrix
cost_matrix = [[17, 10, 21], [15, 12, 19], [13, 10, 17]]

# generate all possible assignments
assignments = list(itertools.permutations(range(len(cost_matrix[0]))))

# calculate the cost of each assignment
min_cost = float('inf')
min_assignment = []
for assignment in assignments:
    cost = sum([cost_matrix[i][assignment[i]] for i in range(len(assignment))])
    print(f"Assignment: {assignment}, Cost: {cost}")
    if cost < min_cost:
        min_cost = cost
        min_assignment = assignment

print(f"\nMinimum cost: {min_cost}, Assignment: {min_assignment}")