The tool is supposed to allocate workers to the shifts to try to cover all the
shifts, give everybody work, and try to match their preferences. I implemented
the tool:
#!/usr/bin/python3
import sys
import os
import re
def report_solution_to_console(vars):
for w in days_of_week:
annotation = ''
if human_annotate is not None:
for s in shifts.keys():
m = re.match(rf'{w} - ', s)
if not m: continue
if vars[human_annotate][s].value():
annotation = f" ({human_annotate} SCHEDULED)"
break
if not len(annotation):
annotation = f" ({human_annotate} OFF)"
print(f"{w}{annotation}")
for s in shifts.keys():
m = re.match(rf'{w} - ', s)
if not m: continue
annotation = ''
if human_annotate is not None:
annotation = f" ({human_annotate} {shifts[s][human_annotate]})"
print(f" ---- {s[m.end():]}{annotation}")
for h in humans:
if vars[h][s].value():
print(f" {h} ({shifts[s][h]})")
def report_solution_summary_to_console(vars):
print("\nSUMMARY")
for h in humans:
print(f"-- {h}")
print(f" benefit: {benefits[h].value():.3f}")
counts = dict()
for a in availabilities:
counts[a] = 0
for s in shifts.keys():
if vars[h][s].value():
counts[shifts[s][h]] += 1
for a in availabilities:
print(f" {counts[a]} {a}")
human_annotate = None
days_of_week = ('SUNDAY',
'MONDAY',
'TUESDAY',
'WEDNESDAY',
'THURSDAY',
'FRIDAY',
'SATURDAY')
humans = ['ALICE', 'BOB',
'CAROL', 'DAVID', 'EVE', 'FRANK', 'GRACE', 'HEIDI', 'IVAN', 'JUDY']
shifts = {'SUNDAY - SANDING 9:00 AM - 4:00 PM':
{'ALICE': 'NEUTRAL',
'BOB': 'NEUTRAL',
'CAROL': 'PREFERRED',
'DAVID': 'PREFERRED',
'EVE': 'PREFERRED',
'FRANK': 'PREFERRED',
'GRACE': 'DISFAVORED',
'HEIDI': 'DISFAVORED',
'IVAN': 'PREFERRED',
'JUDY': 'NEUTRAL'},
'WEDNESDAY - SAWING 7:30 AM - 2:30 PM':
{'ALICE': 'NEUTRAL',
'BOB': 'NEUTRAL',
'CAROL': 'PREFERRED',
'DAVID': 'PREFERRED',
'FRANK': 'PREFERRED',
'GRACE': 'NEUTRAL',
'HEIDI': 'DISFAVORED',
'IVAN': 'PREFERRED',
'EVE': 'REFUSED',
'JUDY': 'REFUSED'},
'THURSDAY - SANDING 9:00 AM - 4:00 PM':
{'ALICE': 'NEUTRAL',
'BOB': 'NEUTRAL',
'CAROL': 'PREFERRED',
'DAVID': 'PREFERRED',
'EVE': 'PREFERRED',
'FRANK': 'PREFERRED',
'GRACE': 'PREFERRED',
'HEIDI': 'DISFAVORED',
'IVAN': 'PREFERRED',
'JUDY': 'PREFERRED'},
'SATURDAY - SAWING 7:30 AM - 2:30 PM':
{'ALICE': 'NEUTRAL',
'BOB': 'NEUTRAL',
'CAROL': 'PREFERRED',
'DAVID': 'PREFERRED',
'FRANK': 'PREFERRED',
'HEIDI': 'DISFAVORED',
'IVAN': 'PREFERRED',
'EVE': 'REFUSED',
'JUDY': 'REFUSED',
'GRACE': 'REFUSED'},
'SUNDAY - SAWING 9:00 AM - 4:00 PM':
{'ALICE': 'NEUTRAL',
'BOB': 'NEUTRAL',
'CAROL': 'PREFERRED',
'DAVID': 'PREFERRED',
'EVE': 'PREFERRED',
'FRANK': 'PREFERRED',
'GRACE': 'DISFAVORED',
'IVAN': 'PREFERRED',
'JUDY': 'PREFERRED',
'HEIDI': 'REFUSED'},
'MONDAY - SAWING 9:00 AM - 4:00 PM':
{'ALICE': 'NEUTRAL',
'BOB': 'NEUTRAL',
'CAROL': 'PREFERRED',
'DAVID': 'PREFERRED',
'EVE': 'PREFERRED',
'FRANK': 'PREFERRED',
'GRACE': 'PREFERRED',
'IVAN': 'PREFERRED',
'JUDY': 'PREFERRED',
'HEIDI': 'REFUSED'},
'TUESDAY - SAWING 9:00 AM - 4:00 PM':
{'ALICE': 'NEUTRAL',
'BOB': 'NEUTRAL',
'CAROL': 'PREFERRED',
'DAVID': 'PREFERRED',
'EVE': 'PREFERRED',
'FRANK': 'PREFERRED',
'GRACE': 'NEUTRAL',
'IVAN': 'PREFERRED',
'JUDY': 'PREFERRED',
'HEIDI': 'REFUSED'},
'WEDNESDAY - PAINTING 7:30 AM - 2:30 PM':
{'ALICE': 'NEUTRAL',
'BOB': 'NEUTRAL',
'CAROL': 'PREFERRED',
'FRANK': 'PREFERRED',
'GRACE': 'NEUTRAL',
'HEIDI': 'DISFAVORED',
'IVAN': 'PREFERRED',
'EVE': 'REFUSED',
'JUDY': 'REFUSED',
'DAVID': 'REFUSED'},
'THURSDAY - SAWING 9:00 AM - 4:00 PM':
{'ALICE': 'NEUTRAL',
'BOB': 'NEUTRAL',
'CAROL': 'PREFERRED',
'DAVID': 'PREFERRED',
'EVE': 'PREFERRED',
'FRANK': 'PREFERRED',
'GRACE': 'PREFERRED',
'IVAN': 'PREFERRED',
'JUDY': 'PREFERRED',
'HEIDI': 'REFUSED'},
'FRIDAY - SAWING 9:00 AM - 4:00 PM':
{'ALICE': 'NEUTRAL',
'BOB': 'NEUTRAL',
'CAROL': 'PREFERRED',
'DAVID': 'PREFERRED',
'EVE': 'PREFERRED',
'FRANK': 'PREFERRED',
'GRACE': 'PREFERRED',
'IVAN': 'PREFERRED',
'JUDY': 'DISFAVORED',
'HEIDI': 'REFUSED'},
'SATURDAY - PAINTING 7:30 AM - 2:30 PM':
{'ALICE': 'NEUTRAL',
'BOB': 'NEUTRAL',
'CAROL': 'PREFERRED',
'FRANK': 'PREFERRED',
'HEIDI': 'DISFAVORED',
'IVAN': 'PREFERRED',
'EVE': 'REFUSED',
'JUDY': 'REFUSED',
'GRACE': 'REFUSED',
'DAVID': 'REFUSED'},
'SUNDAY - PAINTING 9:45 AM - 4:45 PM':
{'ALICE': 'NEUTRAL',
'BOB': 'NEUTRAL',
'CAROL': 'NEUTRAL',
'EVE': 'PREFERRED',
'FRANK': 'PREFERRED',
'GRACE': 'DISFAVORED',
'IVAN': 'PREFERRED',
'JUDY': 'PREFERRED',
'HEIDI': 'REFUSED',
'DAVID': 'REFUSED'},
'MONDAY - PAINTING 9:45 AM - 4:45 PM':
{'ALICE': 'NEUTRAL',
'BOB': 'NEUTRAL',
'CAROL': 'NEUTRAL',
'EVE': 'PREFERRED',
'FRANK': 'PREFERRED',
'GRACE': 'PREFERRED',
'IVAN': 'PREFERRED',
'JUDY': 'NEUTRAL',
'HEIDI': 'REFUSED',
'DAVID': 'REFUSED'},
'TUESDAY - PAINTING 9:45 AM - 4:45 PM':
{'ALICE': 'NEUTRAL',
'BOB': 'NEUTRAL',
'CAROL': 'NEUTRAL',
'EVE': 'PREFERRED',
'FRANK': 'PREFERRED',
'GRACE': 'NEUTRAL',
'IVAN': 'PREFERRED',
'JUDY': 'PREFERRED',
'HEIDI': 'REFUSED',
'DAVID': 'REFUSED'},
'WEDNESDAY - SANDING 9:45 AM - 4:45 PM':
{'ALICE': 'NEUTRAL',
'BOB': 'NEUTRAL',
'CAROL': 'NEUTRAL',
'DAVID': 'PREFERRED',
'FRANK': 'PREFERRED',
'GRACE': 'NEUTRAL',
'HEIDI': 'DISFAVORED',
'IVAN': 'PREFERRED',
'JUDY': 'NEUTRAL',
'EVE': 'REFUSED'},
'THURSDAY - PAINTING 9:45 AM - 4:45 PM':
{'ALICE': 'NEUTRAL',
'BOB': 'NEUTRAL',
'CAROL': 'NEUTRAL',
'EVE': 'PREFERRED',
'FRANK': 'PREFERRED',
'GRACE': 'NEUTRAL',
'IVAN': 'PREFERRED',
'JUDY': 'PREFERRED',
'HEIDI': 'REFUSED',
'DAVID': 'REFUSED'},
'FRIDAY - PAINTING 9:45 AM - 4:45 PM':
{'ALICE': 'NEUTRAL',
'BOB': 'NEUTRAL',
'CAROL': 'NEUTRAL',
'EVE': 'PREFERRED',
'FRANK': 'PREFERRED',
'GRACE': 'PREFERRED',
'IVAN': 'PREFERRED',
'JUDY': 'DISFAVORED',
'HEIDI': 'REFUSED',
'DAVID': 'REFUSED'},
'SATURDAY - SANDING 9:45 AM - 4:45 PM':
{'ALICE': 'NEUTRAL',
'BOB': 'NEUTRAL',
'CAROL': 'NEUTRAL',
'DAVID': 'PREFERRED',
'FRANK': 'PREFERRED',
'HEIDI': 'DISFAVORED',
'IVAN': 'PREFERRED',
'EVE': 'REFUSED',
'JUDY': 'REFUSED',
'GRACE': 'REFUSED'},
'SUNDAY - PAINTING 11:00 AM - 6:00 PM':
{'ALICE': 'NEUTRAL',
'BOB': 'NEUTRAL',
'CAROL': 'NEUTRAL',
'EVE': 'DISFAVORED',
'FRANK': 'NEUTRAL',
'GRACE': 'NEUTRAL',
'HEIDI': 'PREFERRED',
'IVAN': 'NEUTRAL',
'JUDY': 'NEUTRAL',
'DAVID': 'REFUSED'},
'MONDAY - PAINTING 12:00 PM - 7:00 PM':
{'ALICE': 'NEUTRAL',
'BOB': 'NEUTRAL',
'CAROL': 'NEUTRAL',
'EVE': 'DISFAVORED',
'FRANK': 'NEUTRAL',
'GRACE': 'PREFERRED',
'IVAN': 'NEUTRAL',
'JUDY': 'NEUTRAL',
'HEIDI': 'REFUSED',
'DAVID': 'REFUSED'},
'TUESDAY - PAINTING 12:00 PM - 7:00 PM':
{'ALICE': 'NEUTRAL',
'BOB': 'NEUTRAL',
'CAROL': 'NEUTRAL',
'EVE': 'DISFAVORED',
'FRANK': 'NEUTRAL',
'GRACE': 'NEUTRAL',
'IVAN': 'NEUTRAL',
'HEIDI': 'REFUSED',
'JUDY': 'REFUSED',
'DAVID': 'REFUSED'},
'WEDNESDAY - PAINTING 12:00 PM - 7:00 PM':
{'ALICE': 'NEUTRAL',
'BOB': 'NEUTRAL',
'CAROL': 'NEUTRAL',
'FRANK': 'NEUTRAL',
'GRACE': 'NEUTRAL',
'IVAN': 'NEUTRAL',
'JUDY': 'PREFERRED',
'EVE': 'REFUSED',
'HEIDI': 'REFUSED',
'DAVID': 'REFUSED'},
'THURSDAY - PAINTING 12:00 PM - 7:00 PM':
{'ALICE': 'NEUTRAL',
'BOB': 'NEUTRAL',
'CAROL': 'NEUTRAL',
'EVE': 'DISFAVORED',
'FRANK': 'NEUTRAL',
'GRACE': 'NEUTRAL',
'IVAN': 'NEUTRAL',
'JUDY': 'PREFERRED',
'HEIDI': 'REFUSED',
'DAVID': 'REFUSED'},
'FRIDAY - PAINTING 12:00 PM - 7:00 PM':
{'ALICE': 'NEUTRAL',
'BOB': 'NEUTRAL',
'CAROL': 'NEUTRAL',
'EVE': 'DISFAVORED',
'FRANK': 'NEUTRAL',
'GRACE': 'NEUTRAL',
'IVAN': 'NEUTRAL',
'JUDY': 'DISFAVORED',
'HEIDI': 'REFUSED',
'DAVID': 'REFUSED'},
'SATURDAY - PAINTING 12:00 PM - 7:00 PM':
{'ALICE': 'NEUTRAL',
'BOB': 'NEUTRAL',
'CAROL': 'NEUTRAL',
'FRANK': 'NEUTRAL',
'IVAN': 'NEUTRAL',
'JUDY': 'DISFAVORED',
'EVE': 'REFUSED',
'HEIDI': 'REFUSED',
'GRACE': 'REFUSED',
'DAVID': 'REFUSED'},
'SUNDAY - SAWING 12:00 PM - 7:00 PM':
{'ALICE': 'PREFERRED',
'BOB': 'PREFERRED',
'CAROL': 'NEUTRAL',
'EVE': 'DISFAVORED',
'FRANK': 'NEUTRAL',
'GRACE': 'NEUTRAL',
'IVAN': 'NEUTRAL',
'JUDY': 'PREFERRED',
'HEIDI': 'REFUSED',
'DAVID': 'REFUSED'},
'MONDAY - SAWING 2:00 PM - 9:00 PM':
{'ALICE': 'PREFERRED',
'BOB': 'PREFERRED',
'CAROL': 'DISFAVORED',
'EVE': 'DISFAVORED',
'FRANK': 'NEUTRAL',
'GRACE': 'NEUTRAL',
'IVAN': 'DISFAVORED',
'JUDY': 'DISFAVORED',
'HEIDI': 'REFUSED',
'DAVID': 'REFUSED'},
'TUESDAY - SAWING 2:00 PM - 9:00 PM':
{'ALICE': 'PREFERRED',
'BOB': 'PREFERRED',
'CAROL': 'DISFAVORED',
'EVE': 'DISFAVORED',
'FRANK': 'NEUTRAL',
'GRACE': 'NEUTRAL',
'IVAN': 'DISFAVORED',
'HEIDI': 'REFUSED',
'JUDY': 'REFUSED',
'DAVID': 'REFUSED'},
'WEDNESDAY - SAWING 2:00 PM - 9:00 PM':
{'ALICE': 'PREFERRED',
'BOB': 'PREFERRED',
'CAROL': 'DISFAVORED',
'FRANK': 'NEUTRAL',
'GRACE': 'NEUTRAL',
'IVAN': 'DISFAVORED',
'JUDY': 'DISFAVORED',
'EVE': 'REFUSED',
'HEIDI': 'REFUSED',
'DAVID': 'REFUSED'},
'THURSDAY - SAWING 2:00 PM - 9:00 PM':
{'ALICE': 'PREFERRED',
'BOB': 'PREFERRED',
'CAROL': 'DISFAVORED',
'EVE': 'DISFAVORED',
'FRANK': 'NEUTRAL',
'GRACE': 'NEUTRAL',
'IVAN': 'DISFAVORED',
'JUDY': 'DISFAVORED',
'HEIDI': 'REFUSED',
'DAVID': 'REFUSED'},
'FRIDAY - SAWING 2:00 PM - 9:00 PM':
{'ALICE': 'PREFERRED',
'BOB': 'PREFERRED',
'CAROL': 'DISFAVORED',
'EVE': 'DISFAVORED',
'FRANK': 'NEUTRAL',
'GRACE': 'NEUTRAL',
'IVAN': 'DISFAVORED',
'HEIDI': 'REFUSED',
'JUDY': 'REFUSED',
'DAVID': 'REFUSED'},
'SATURDAY - SAWING 2:00 PM - 9:00 PM':
{'ALICE': 'PREFERRED',
'BOB': 'PREFERRED',
'CAROL': 'DISFAVORED',
'FRANK': 'NEUTRAL',
'IVAN': 'DISFAVORED',
'JUDY': 'DISFAVORED',
'EVE': 'REFUSED',
'HEIDI': 'REFUSED',
'GRACE': 'REFUSED',
'DAVID': 'REFUSED'},
'SUNDAY - PAINTING 12:15 PM - 7:15 PM':
{'ALICE': 'NEUTRAL',
'BOB': 'NEUTRAL',
'CAROL': 'PREFERRED',
'EVE': 'DISFAVORED',
'FRANK': 'NEUTRAL',
'GRACE': 'NEUTRAL',
'HEIDI': 'NEUTRAL',
'IVAN': 'DISFAVORED',
'JUDY': 'NEUTRAL',
'DAVID': 'REFUSED'},
'MONDAY - PAINTING 2:00 PM - 9:00 PM':
{'ALICE': 'NEUTRAL',
'BOB': 'NEUTRAL',
'CAROL': 'DISFAVORED',
'EVE': 'DISFAVORED',
'FRANK': 'NEUTRAL',
'GRACE': 'NEUTRAL',
'HEIDI': 'NEUTRAL',
'IVAN': 'DISFAVORED',
'JUDY': 'DISFAVORED',
'DAVID': 'REFUSED'},
'TUESDAY - PAINTING 2:00 PM - 9:00 PM':
{'ALICE': 'NEUTRAL',
'BOB': 'NEUTRAL',
'CAROL': 'DISFAVORED',
'EVE': 'DISFAVORED',
'FRANK': 'NEUTRAL',
'GRACE': 'NEUTRAL',
'HEIDI': 'NEUTRAL',
'IVAN': 'DISFAVORED',
'JUDY': 'REFUSED',
'DAVID': 'REFUSED'},
'WEDNESDAY - PAINTING 2:00 PM - 9:00 PM':
{'ALICE': 'NEUTRAL',
'BOB': 'NEUTRAL',
'CAROL': 'DISFAVORED',
'FRANK': 'NEUTRAL',
'GRACE': 'NEUTRAL',
'HEIDI': 'NEUTRAL',
'IVAN': 'DISFAVORED',
'JUDY': 'DISFAVORED',
'EVE': 'REFUSED',
'DAVID': 'REFUSED'},
'THURSDAY - PAINTING 2:00 PM - 9:00 PM':
{'ALICE': 'NEUTRAL',
'BOB': 'NEUTRAL',
'CAROL': 'DISFAVORED',
'EVE': 'DISFAVORED',
'FRANK': 'NEUTRAL',
'GRACE': 'NEUTRAL',
'HEIDI': 'NEUTRAL',
'IVAN': 'DISFAVORED',
'JUDY': 'DISFAVORED',
'DAVID': 'REFUSED'},
'FRIDAY - PAINTING 2:00 PM - 9:00 PM':
{'ALICE': 'NEUTRAL',
'BOB': 'NEUTRAL',
'CAROL': 'DISFAVORED',
'EVE': 'DISFAVORED',
'FRANK': 'NEUTRAL',
'GRACE': 'NEUTRAL',
'HEIDI': 'NEUTRAL',
'IVAN': 'DISFAVORED',
'JUDY': 'REFUSED',
'DAVID': 'REFUSED'},
'SATURDAY - PAINTING 2:00 PM - 9:00 PM':
{'ALICE': 'NEUTRAL',
'BOB': 'NEUTRAL',
'CAROL': 'DISFAVORED',
'FRANK': 'NEUTRAL',
'HEIDI': 'NEUTRAL',
'IVAN': 'DISFAVORED',
'JUDY': 'DISFAVORED',
'EVE': 'REFUSED',
'GRACE': 'REFUSED',
'DAVID': 'REFUSED'}}
availabilities = ['PREFERRED', 'NEUTRAL', 'DISFAVORED']
import pulp
prob = pulp.LpProblem("Scheduling", pulp.LpMaximize)
vars = pulp.LpVariable.dicts("Assignments",
(humans, shifts.keys()),
None,None, # bounds; unused, since these are binary variables
pulp.LpBinary)
# Everyone works at least 2 shifts
Nshifts_min = 2
for h in humans:
prob += (
pulp.lpSum([vars[h][s] for s in shifts.keys()]) >= Nshifts_min,
f"{h} works at least {Nshifts_min} shifts",
)
# each shift is ~ 8 hours, so I limit everyone to 40/8 = 5 shifts
Nshifts_max = 5
for h in humans:
prob += (
pulp.lpSum([vars[h][s] for s in shifts.keys()]) <= Nshifts_max,
f"{h} works at most {Nshifts_max} shifts",
)
# all shifts staffed and not double-staffed
for s in shifts.keys():
prob += (
pulp.lpSum([vars[h][s] for h in humans]) == 1,
f"{s} is staffed",
)
# each human can work at most one shift on any given day
for w in days_of_week:
for h in humans:
prob += (
pulp.lpSum([vars[h][s] for s in shifts.keys() if re.match(rf'{w} ',s)]) <= 1,
f"{h} cannot be double-booked on {w}"
)
#### Some explicit constraints; as an example
# DAVID can't work any PAINTING shift and is off on Thu and Sun
h = 'DAVID'
prob += (
pulp.lpSum([vars[h][s] for s in shifts.keys() if re.search(r'- PAINTING',s)]) == 0,
f"{h} can't work any PAINTING shift"
)
prob += (
pulp.lpSum([vars[h][s] for s in shifts.keys() if re.match(r'THURSDAY|SUNDAY',s)]) == 0,
f"{h} is off on Thursday and Sunday"
)
# Do not assign any "REFUSED" shifts
for s in shifts.keys():
for h in humans:
if shifts[s][h] == 'REFUSED':
prob += (
vars[h][s] == 0,
f"{h} is not available for {s}"
)
# Objective. I try to maximize the "happiness". Each human sees each shift as
# one of:
#
# PREFERRED
# NEUTRAL
# DISFAVORED
# REFUSED
#
# I set a hard constraint to handle "REFUSED", and arbitrarily, I set these
# benefit values for the others
benefit_availability = dict()
benefit_availability['PREFERRED'] = 3
benefit_availability['NEUTRAL'] = 2
benefit_availability['DISFAVORED'] = 1
# Not used, since this is a hard constraint. But the code needs this to be a
# part of the benefit. I can ignore these in the code, but let's keep this
# simple
benefit_availability['REFUSED' ] = -1000
benefits = dict()
for h in humans:
benefits[h] = \
pulp.lpSum([vars[h][s] * benefit_availability[shifts[s][h]] \
for s in shifts.keys()])
benefit_total = \
pulp.lpSum([benefits[h] \
for h in humans])
prob += (
benefit_total,
"happiness",
)
prob.solve()
if pulp.LpStatus[prob.status] == "Optimal":
report_solution_to_console(vars)
report_solution_summary_to_console(vars)
So for instance each shift that was scheduled as somebody's PREFERRED shift
gives us 3 benefit points. And if all the shifts ended up being PREFERRED, we'd
have a total benefit value of 3*Nshifts. This is impossible, however, because
that would violate some constraints in the problem.
So we have a solution! We have 108 total benefit points. But it looks a bit
uneven: Judy only works 2 days, while some people work many more: David works 5
for instance. Why is that? I update the program with =human_annotate = 'JUDY'=,
run it again, and it tells me more about Judy's preferences:
This tells us that on Monday Judy does not work, although she marked the SAWING
shift as PREFERRED. Instead David got that shift. What would happen if David
gave that shift to Judy? He would lose 3 points, she would gain 3 points, and
the total would remain exactly the same at 108.
How would we favor a more even distribution? We need some sort of tie-break. I
want to add a nonlinearity to strongly disfavor people getting a low number of
shifts. But PuLP is very explicitly a linear programming solver, and cannot
solve nonlinear problems. Here we can get around this by enumerating each
specific case, and assigning it a nonlinear benefit function. The most obvious
approach is to define another set of boolean variables:
vars_Nshifts[human][N]. And then using them to add extra benefit terms, with
values nonlinearly related to Nshifts. Something like this:
So in the previous example we considered giving David's 5th shift to Judy, for
her 3rd shift. In that scenario, David's extra benefit would change from -0.2 to
-0.3 (a shift of -0.1), while Judy's would change from -0.8 to -0.5 (a shift of
+0.3). So the balancing out the shifts in this way would work: the solver would
favor the solution with the higher benefit function.
Great. In order for this to work, we need the vars_Nshifts[human][N] variables
to function as intended: they need to be binary indicators of whether a specific
person has that many shifts or not. That would need to be implemented with
constraints. Let's plot it like this:
How do we fix it? We can use inequality variables instead. I define a different
set of variables vars_Nshifts_leq[human][N] that are 1 iff Nshifts <= N.
The equality variable from before can be expressed as a difference of these
inequality variables: vars_Nshifts[human][N] =
vars_Nshifts_leq[human][N]-vars_Nshifts_leq[human][N-1]
So we can use two linear constraints to make each of these variables work
properly. To use these in the benefit function we can use the equality
constraint expression from above, or we can use these directly:
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