to solve a work scheduling problem to assign
workers to shifts. Here are notes about doing that. This is a common use case,
but isn't explicitly covered in the
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)
variable, and the shift schedule and the
workers' preferences are encoded in the
dict. The problem is defined by
a
dict of dicts, each a boolean variable indicating whether a particular
worker is scheduled for a particular shift. We define a set of constraints to
these worker allocations to restrict ourselves to valid solutions. And among
these valid solutions, we try to find the one that maximizes some benefit
function, defined here as:
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.
The exact trade-off between the different preferences is set in the
dict. With the above numbers, it's equally good for
somebody to have a NEUTRAL shift and a day off as it is for them to have
DISFAVORED shifts. If we really want to encourage the program to work people as
much as possible (days off discouraged), we'd want to raise the DISFAVORED
threshold.
I run this program and I get:
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:
. And then using them to add extra benefit terms, with
values nonlinearly related to
. 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
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:
variable (plotted on the x axis of this
plot) would need to be defined by a set of linear AND constraints to
the true (red) values of this variable from the false (black) values.
As can be seen in this plot, this isn't possible. So this representation does
not work.
How do we fix it? We can use inequality variables instead. I define a different
set of variables
.
The equality variable from before can be expressed as a difference of these
inequality variables:
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:
In this scenario, David would get a boost of 0.4 from giving up his 5th shift,
while Judy would lose a boost of 0.2 from getting her 3rd, for a net gain of 0.2
benefit points. The exact numbers will need to be adjusted on a case by case
basis, but this works.
The full program, with this and other extra features is available
Update 20250309 13:20-03:00 - How to help
A friend of mine living in the USA sent me this link to help the flood victims: 
The "real" thing
Let's get to the heavy, heartbreaker part. I did go to downtown Bah a Blanca, but during night, so let me share some links, most of them in Spanish, but images are images:
Almost exactly four years after I started with this project, yesterday I
presented my PhD defense.
From April 27th to 30th, 2024,
This was the fifth time that a MiniDebConf (as an exclusive in-person event
about Debian) took place in Brazil. Previous editions were in Curitiba
(
2024 edition numbers
During the four days of the event, several activities took place for all
levels of users and collaborators of the Debian project. The official schedule
was composed of:
The final numbers for MiniDebConf Belo Horizonte 2024 show that we had a
record number of participants.
Activities
The MiniDebConf schedule was intense and diverse. On the 27th, 29th and 30th
(Saturday, Monday and Tuesday) we had talks, discussions, workshops and many
practical activities.
On the 28th (Sunday), the Day Trip took place, a day dedicated to sightseeing
around the city. In the morning we left the hotel and went, on a chartered bus,
to the
After a 2-hour tour of the Market, we got back on the bus and hit the road for
lunch at a typical Minas Gerais food restaurant.
With everyone well fed, we returned to Belo Horizonte to visit the city's
main tourist attraction: Lagoa da Pampulha and Capela S o Francisco de Assis,
better known as
We went back to the hotel and the day ended in the hacker space that we set up
in the events room for people to chat, packaging, and eat pizzas.
Crowdfunding
For the third time we ran a crowdfunding campaign and it was incredible how
people contributed! The initial goal was to raise the amount equivalent to a
gold tier of R$ 3,000.00. When we reached this goal, we defined a new one,
equivalent to one gold tier + one silver tier (R$ 5,000.00). And again we
achieved this goal. So we proposed as a final goal the value of a gold + silver
+ bronze tiers, which would be equivalent to R$ 6,000.00. The result was that
we raised R$7,239.65 (~ USD 1,400) with the help of more than 100 people!
Thank you very much to the people who contributed any amount. As a thank you, we 
Food, accommodation and/or travel grants for participants
Each edition of MiniDebConf brought some innovation, or some different benefit
for the attendees. In this year's edition in Belo Horizonte, as with DebConfs, we 
Photos and videos
You can watch recordings of the talks at the links below:
Sponsors
Gold:
This year I was at 
Welcome to post 46 in the 
In today s digital landscape, social media is more than just a communication tool it is the primary medium for global discourse. Heads of state, corporate leaders and cultural influencers now broadcast their statements directly to the world, shaping public opinion in real time. However, the dominance of a few centralized platforms X/Twitter, Facebook and YouTube raises critical concerns about control, censorship and the monopolization of information. Those who control these networks effectively wield significant power over public discourse.
In response, a new wave of distributed social media platforms has emerged, each built on different decentralized protocols designed to provide greater autonomy, censorship resistance and user control. While 
Mastodon was created in 2016 by 
Interestingly, 
Some of you may remember that I recently felt a bit underwhelmed
by the last 
Same as the last goarounds, except the premable here is a 
Dear Debian community,
this is bits from DPL for February.
Ftpmaster team is seeking for new team members
In December, Scott Kitterman announced his retirement from the project.
I personally regret this, as I vividly remember his invaluable support
during the Debian Med sprint at the start of the COVID-19 pandemic. He
even took time off to ensure new packages cleared the queue in under 24
hours. I want to take this opportunity to personally thank Scott for his
contributions during that sprint and for all his work in Debian.
With one fewer FTP assistant, I am concerned about the increased
workload on the remaining team. I encourage anyone in the Debian
community who is interested to consider reaching out to the FTP masters
about joining their team.
If you're wondering about the role of the 
[I d like to stop writing posts like this. I ve been trying to work out what to say now for nearly 2 months (writing the mail to 
Most of my Debian contributions this month were
Another short status update of what happened on my side last
month. One larger blocks are the 
I wanted to follow new content posted to