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workforce4.py
#!/usr/bin/env python3.7
# Copyright 2023, Gurobi Optimization, LLC
# Assign workers to shifts; each worker may or may not be available on a
# particular day. We use lexicographic optimization to solve the model:
# first, we minimize the linear sum of the slacks. Then, we constrain
# the sum of the slacks, and we minimize a quadratic objective that
# tries to balance the workload among the workers.
import gurobipy as gp
from gurobipy import GRB
import sys
# Number of workers required for each shift
shifts, shiftRequirements = gp.multidict({
"Mon1": 3,
"Tue2": 2,
"Wed3": 4,
"Thu4": 4,
"Fri5": 5,
"Sat6": 6,
"Sun7": 5,
"Mon8": 2,
"Tue9": 2,
"Wed10": 3,
"Thu11": 4,
"Fri12": 6,
"Sat13": 7,
"Sun14": 5,
})
# Amount each worker is paid to work one shift
workers, pay = gp.multidict({
"Amy": 10,
"Bob": 12,
"Cathy": 10,
"Dan": 8,
"Ed": 8,
"Fred": 9,
"Gu": 11,
})
# Worker availability
availability = gp.tuplelist([
('Amy', 'Tue2'), ('Amy', 'Wed3'), ('Amy', 'Fri5'), ('Amy', 'Sun7'),
('Amy', 'Tue9'), ('Amy', 'Wed10'), ('Amy', 'Thu11'), ('Amy', 'Fri12'),
('Amy', 'Sat13'), ('Amy', 'Sun14'), ('Bob', 'Mon1'), ('Bob', 'Tue2'),
('Bob', 'Fri5'), ('Bob', 'Sat6'), ('Bob', 'Mon8'), ('Bob', 'Thu11'),
('Bob', 'Sat13'), ('Cathy', 'Wed3'), ('Cathy', 'Thu4'), ('Cathy', 'Fri5'),
('Cathy', 'Sun7'), ('Cathy', 'Mon8'), ('Cathy', 'Tue9'),
('Cathy', 'Wed10'), ('Cathy', 'Thu11'), ('Cathy', 'Fri12'),
('Cathy', 'Sat13'), ('Cathy', 'Sun14'), ('Dan', 'Tue2'), ('Dan', 'Wed3'),
('Dan', 'Fri5'), ('Dan', 'Sat6'), ('Dan', 'Mon8'), ('Dan', 'Tue9'),
('Dan', 'Wed10'), ('Dan', 'Thu11'), ('Dan', 'Fri12'), ('Dan', 'Sat13'),
('Dan', 'Sun14'), ('Ed', 'Mon1'), ('Ed', 'Tue2'), ('Ed', 'Wed3'),
('Ed', 'Thu4'), ('Ed', 'Fri5'), ('Ed', 'Sun7'), ('Ed', 'Mon8'),
('Ed', 'Tue9'), ('Ed', 'Thu11'), ('Ed', 'Sat13'), ('Ed', 'Sun14'),
('Fred', 'Mon1'), ('Fred', 'Tue2'), ('Fred', 'Wed3'), ('Fred', 'Sat6'),
('Fred', 'Mon8'), ('Fred', 'Tue9'), ('Fred', 'Fri12'), ('Fred', 'Sat13'),
('Fred', 'Sun14'), ('Gu', 'Mon1'), ('Gu', 'Tue2'), ('Gu', 'Wed3'),
('Gu', 'Fri5'), ('Gu', 'Sat6'), ('Gu', 'Sun7'), ('Gu', 'Mon8'),
('Gu', 'Tue9'), ('Gu', 'Wed10'), ('Gu', 'Thu11'), ('Gu', 'Fri12'),
('Gu', 'Sat13'), ('Gu', 'Sun14')
])
# Model
m = gp.Model("assignment")
# Assignment variables: x[w,s] == 1 if worker w is assigned to shift s.
# This is no longer a pure assignment model, so we must use binary variables.
x = m.addVars(availability, vtype=GRB.BINARY, name="x")
# Slack variables for each shift constraint so that the shifts can
# be satisfied
slacks = m.addVars(shifts, name="Slack")
# Variable to represent the total slack
totSlack = m.addVar(name="totSlack")
# Variables to count the total shifts worked by each worker
totShifts = m.addVars(workers, name="TotShifts")
# Constraint: assign exactly shiftRequirements[s] workers to each shift s,
# plus the slack
reqCts = m.addConstrs((slacks[s] + x.sum('*', s) == shiftRequirements[s]
for s in shifts), "_")
# Constraint: set totSlack equal to the total slack
m.addConstr(totSlack == slacks.sum(), "totSlack")
# Constraint: compute the total number of shifts for each worker
m.addConstrs((totShifts[w] == x.sum(w) for w in workers), "totShifts")
# Objective: minimize the total slack
# Note that this replaces the previous 'pay' objective coefficients
m.setObjective(totSlack)
# Optimize
def solveAndPrint():
m.optimize()
status = m.status
if status in (GRB.INF_OR_UNBD, GRB.INFEASIBLE, GRB.UNBOUNDED):
print('The model cannot be solved because it is infeasible or \
unbounded')
sys.exit(1)
if status != GRB.OPTIMAL:
print('Optimization was stopped with status %d' % status)
sys.exit(0)
# Print total slack and the number of shifts worked for each worker
print('')
print('Total slack required: %g' % totSlack.X)
for w in workers:
print('%s worked %g shifts' % (w, totShifts[w].X))
print('')
solveAndPrint()
# Constrain the slack by setting its upper and lower bounds
totSlack.UB = totSlack.X
totSlack.LB = totSlack.X
# Variable to count the average number of shifts worked
avgShifts = m.addVar(name="avgShifts")
# Variables to count the difference from average for each worker;
# note that these variables can take negative values.
diffShifts = m.addVars(workers, lb=-GRB.INFINITY, name="Diff")
# Constraint: compute the average number of shifts worked
m.addConstr(len(workers) * avgShifts == totShifts.sum(), "avgShifts")
# Constraint: compute the difference from the average number of shifts
m.addConstrs((diffShifts[w] == totShifts[w] - avgShifts for w in workers),
"Diff")
# Objective: minimize the sum of the square of the difference from the
# average number of shifts worked
m.setObjective(gp.quicksum(diffShifts[w]*diffShifts[w] for w in workers))
# Optimize
solveAndPrint()
