Linear sum assignment (SciPy) and balancing the costs

Question

I am having difficulty using scipy.optimize.linear_sum_assignment to evenly distribute tasks (costs) to workers, where each worker can be assigned multiple tasks. The cost matrix represents the workload of each task for each worker.

We want to minimize the total costs of all workers, while evenly distributing the costs of each worker.

In this example, we have 3 workers named a, b and c. Each worker can be assigned a total of 4 tasks, so in the cost matrix we have the agents a_1, a_2 and so forth.

linear_sum_assignment does give us the assignment with the total costs minimized. To simply things, our example uses a cost matrix such that any assignment will give us the same total costs.

However, is the costs is not evenly distributed across the 3 workers. In our example, the costs for the 3 workers are 65, 163 and 192 respectively.

Is it possible to minimize the costs as much as possible, while distributing the costs per worker more evenly across the 3 workers?

from scipy.optimize import linear_sum_assignment
import numpy as np

worker_capacities = [
    "a_1", "a_2", "a_3", "a_4",
    "b_1", "b_2", "b_3", "b_4",
    "c_1", "c_2", "c_3", "c_4",
]
n_tasks = len(worker_capacities)
c = np.array([
 [27, 42, 65, 33, 67, 45, 60, 76,  6,  6, 43, 26],
 [27, 42, 65, 33, 67, 45, 60, 76,  6,  6, 43, 26],
 [27, 42, 65, 33, 67, 45, 60, 76,  6,  6, 43, 26],
 [27, 42, 65, 33, 67, 45, 60, 76,  6,  6, 43, 26],
 [27, 42, 65, 33, 67, 45, 60, 76,  6,  6, 43, 26],
 [27, 42, 65, 33, 67, 45, 60, 76,  6,  6, 43, 26],
 [27, 42, 65, 33, 67, 45, 60, 76,  6,  6, 43, 26],
 [27, 42, 65, 33, 67, 45, 60, 76,  6,  6, 43, 26],
 [27, 42, 65, 33, 67, 45, 60, 76,  6,  6, 43, 26],
 [27, 42, 65, 33, 67, 45, 60, 76,  6,  6, 43, 26],
 [27, 42, 65, 33, 67, 45, 60, 76,  6,  6, 43, 26],
])

_, assignments = linear_sum_assignment(c)
print("Costs for worker a:", sum(c[i][j] for i, j in enumerate(assignments[0:4])))
print("Costs for worker b:", sum(c[i+4][j] for i, j in enumerate(assignments[4:8])))
print("Costs for worker c:", sum(c[i+8][j] for i, j in enumerate(assignments[8:12])))

gives the output

Costs for worker a: 65
Costs for worker b: 163
Costs for worker c: 192

Answer 1

The linear_sum_assignment method doesn't support constraints or a custom objective, so I don't think this is possible.

However, you could formulate your problem as a mixed-integer linear programming problem (MILP) and solve it by means of PuLP¹. In order to evenly distribute the total costs per worker, you could minimize the difference between the maximum and the minimum total costs per worker. Here's a possible formulation:

Sets:

- workers = ["a", "b", "c"]
- tasks   = [1, 2, ..., 12]

Variables:

- x[w,t] = 1 iff worker w is assigned to task t, 0 otherwise
- min_val
- max_val

Model:

min  max_val - min_val

s.t. 

# each worker is assigned to exactly n_tasks_per_worker tasks
sum(x[w,t] for t in tasks) == n_tasks_per_worker for all w in workers

# each task can only be assigned once
sum(x[w,t] for w in workers) == 1 for all t in tasks

# evenly distribute the tasks
sum(x[w,t] for t in tasks) <= max_val for all w in workers
sum(x[w,t] for t in tasks) >= min_val for all w in workers

The code is straightforward:

import pulp
import numpy as np

workers = ["a", "b", "c"]
n_workers = len(workers)
n_tasks_per_worker = 4
n_tasks = n_workers * n_tasks_per_worker

c = np.array([[27, 42, 65, 33, 67, 45, 60, 76,  6,  6, 43, 26],
              [27, 42, 65, 33, 67, 45, 60, 76,  6,  6, 43, 26],
              [27, 42, 65, 33, 67, 45, 60, 76,  6,  6, 43, 26]])

# create the model
mdl = pulp.LpProblem("even_assignment")

# decision variables
x = {}
for w in workers:
    for t in range(n_tasks):
        x[w, t] = pulp.LpVariable(f"x[{w}, {t}]", cat="Binary")

max_val = pulp.LpVariable("max_val", cat="Continuous")
min_val = pulp.LpVariable("min_val", cat="Continuous")

# objective: minimize the difference between the maximum and the minimum
#            costs per worker
mdl.setObjective(max_val - min_val)

# constraint: each worker gets assigned exactly n_tasks_per_worker
for w in workers:
    mdl.addConstraint(sum(x[w, task] for task in range(n_tasks)) == n_tasks_per_worker)

# constraint: each task can only be assigned once
for task in range(n_tasks):
    mdl.addConstraint(sum(x[w, task] for w in workers) == 1)

# constraint: evenly distribute the tasks
for i_w, w in enumerate(workers):
    assignment_cost = sum(x[w,task]*c[i_w,task] for task in range(n_tasks))
    mdl.addConstraint(assignment_cost <= max_val)
    mdl.addConstraint(assignment_cost >= min_val)

# solve the problem
mdl.solve()

# Output
for i_w, w in enumerate(workers):
    worker_cost = sum(x[w, t].varValue*c[i_w, t] for t in range(n_tasks))
    print(f"costs for worker {w}: {worker_cost:.2f}")

This gives me

costs for worker a: 165.00
costs for worker b: 167.00
costs for worker c: 164.00

---

[1] To be exact, PuLP isn't a solver, it's just a modelling framework that can pass MILPs to MILP Solvers like CBC, SCIP, HiGHS etc.