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Division of Labour and Task Allocation
Introduction
• Many species of insects have division of labour• Task allocation at colony level linked to elasticity
of individuals• Model is based upon response thresholds
– Likelihood of reacting to task-associated stimulii– Can add learning:
• Performing a task decreases threshold • Not performing a task increases threshold
Response Function
( )
( ) θθ
θθ
/1 s
ss
esTor
sT nn
n
−
+
−=
=Hmm, looks
familiar
Remember the Army
Ant “tanh”?
Why exponential?
• Imagine N encounters with items:– Probability of processing item is p– Probability of not processing any item is: (1-p)N
– Probability of processing one item is:• P(N) = 1 – (1-p)N = 1- eNln(1-p)
– Same as, s = N, θ = -1/ln(1-p)• This is common:
– Again shows that random behavior can generate emergent organization
Automata Theory
• Learning response threshold model is similar to Automata Theory– DLrp
– Thresholds move rather than probabilities
Increasing θ
Swarm Intelligence: Bonabeau et al
Experimental Evidence
• Robinson and Page– Honey bee workers have different response thresholds
• If honey bee forager takes too long to unload nectar to storer bee, she gives up foraging with prob. proportional to search time.– Starts tremble dance to recruit storer bees
• If storage time small, she’ll recuite other foragers– Starts waggle dance (which inhibits tremble dance)
What can it be used for?
• Task allocation in a multi-agent system– Similar to Market-based Control (Wellman)
• With learning can:– Generate differentiation in task performance
with initially identical agents
Lecture notes
Division of Labor and Task Allocation
AM, EE141, Swarm Intelligence, W3-2
• The most obvious sign of the division of labor is the existence of castes.
• We distinguish between three kinds of castes: physical, behavioral, andtemporal (temporal polyethism).
• The individuals belonging to different castes are usually specialized forthe performance of a series of precise tasks.
The Division of Laborand its Control
The control of task allocation
Physical Castes
Wilson, E.O. (1976)
In Pheidole guilelmimuelleriPheidole guilelmimuellerithe minors show ten timesas many different basicbehaviors as the majors
0 0,2 0,4 0,6
Se lf-grooming
Minor worker
De alat e quee n
Male
Carry or roll e gg
Carry or roll larva
Fee d larva solids
Carry or roll pupa
Assist eclosion of adult
Minor worker
De alat e quee n
Male
Forage
Lay odor t rail
Fe ed inside ne st
Agre ssion (drag or at tack)
Carry de ad larva or pupa
Fe ed on larva or pupa
Lick wall of ne st
Ante nnal t ipping
Guard ne st entrance
Minor
0 0,2 0,4 0,6
Major
Behavioral repertoires of majorsand minors
Behavioral Castes
From D. Gordon, “Ants at Work”, 1999
Allocation of the dailyactivities in a colony ofdesert harverster ants(Portal, AZ)
Temporal PolyethismCleaning cells
Tending brood
Tending Queen
Eating pollen
Feeding & grooming nestmates
Ventilating nest
Shaping comb
Storing nectar
Packing pollen
Foraging
Patrolling
Resting
1 4 7 10 13 16 19 22 25
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6080
Perc
enta
ge o
f tim
e sp
ent i
n ea
ch a
ctiv
ity
Age of bee (days)
Behavioral changes in worker beesas a function of age
Young individuals work oninternal tasks (brood careand nest maintenance).Older individuals forage forfood and defend the nest.
• The division of labor is flexible.
• The number of individuals belonging to different castes andthe nature of the tasks to be done are subject to constantchange in the course of the life of a colony.
• The proportions of workers performing the different tasksvaries in response to internal or environmental perturbations.
Flexibility of social roles
The Division of Laborand its Control
Sep 12- 23
Oct 6- 17
Aug 19- 30
Jul 14- 25
Jun 20- Jul 1
May 27- Jun 7
I NCREASI NG PERI OD MAJOR PERI OD CONT RACT I NG PERI OD
22127 41297 48253 51988 49971 34062
0
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Egg Larvae Pupae Nurse bee
House bee
Field bee
AG
E (D
AYS
)
Sep 12- 23
Oct 6- 17
Aug 19- 30
Jul 14- 25
Jun 20- Jul 1
May 27- Jun 7
I NCREASI NG PERI OD MAJOR PERI OD CONT RACT I NG PERI OD
22127 41297 48253 51988 49971 34062
0
20
40
60
Egg Larvae Pupae Nurse bee
House bee
Field bee
AG
E (D
AYS
)
The changes in the population of a bee colony in the course of a season
The Division of Laborand its Control
How is flexibility implementedat the level of the individual?
The Division of Labor andthe Flexibility of Social Roles
The control of task allocation
3
1
2 ?
Task 1 Task 2 Task 3
How is dynamic task allocation achieved?
How does the colony control the proportions of individuals assignedto each task, given that no individual possesses any globalrepresentation of the needs of the colony?
• The flexibility of the division of labor depends on thebehavioral flexibility of the workers.
• A mechanism arising from the concept of a responsethreshold allows the production of this flexibility.
Flexibility of social roles
The Division of Laborand its Control
σσσσi1σσσσi1σσσσi1
σσσσi1σσσσi1σσσσi1
σσσσi1σσσσi1σσσσi1
3
1
2
Fixed threshold model
The control of task allocation
The Division of Labor and the Flexibility of Social Roles
An Example of Control of theDivision of Labor Implying theExistence of Fixed Response
Thresholds
© Guy Theraulaz
0
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9
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
Fraction of majors
Pheidole pubiventris
Self-cleaning
Social behavior
The idea of a response threshold
Beh
avio
ral A
cts/
Maj
or/H
our
Beh
avio
ral A
cts/
Maj
or/H
our
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0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0
Fraction of majors
P. guilelmimuelleri
Self-cleaning
Social behavior
The Division of Laborand its Control
Task 1Task 1
θθθθj,1
Task 1Task 1
θθθθi,1
An example of a response threshold
0
0,25
0,5
0,75
1
0,1 1 10 100Stimulus
Majors
MinorsT(
s) R
espo
nse
Pro
babi
lity
T (s) = s2
s2 + θi2θ i
s : intensity of the stimulus associated with the taskθi : response threshold
θminors θmajors
The Division of Laborand its Control
Properties of MechanismsArising from Fixed
Response Thresholds
Fixed Threshold Model with 1 Task and 2 Distinct Castes
Two castes of workers (physical, behavioral, or age-based castes)
f = n1/N : proportion of type 1 workers in the colony
N1 = number of workers of type 1 engaged in carrying out the taskN2 = number of workers of type 2 engaged in carrying out the task
x1 = N1 / n1 : proportion of workers of type 1 engaged in carrying out the task
x2 = N2 / n2 : proportion of workers of type 2 engaged in carrying out the task
n1 = number of workers of type 1n2 = number of workers of type 2
n1 + n2 = N
Equiprobable exposure of individuals to the stimuli associated with the task
Dynamics of the proportion of active workers in each caste
s : intensity of the stimuli associated with the taskθ i : response threshold of workers of type ip : probability per unit time that an active individual abandons the
task on which he is engaged1/p : average time spent by an individual in working on a task before
abandoning it
= (1-x1) - p x1s2
s2 + θ12
= (1-x2) - p x2
s2
s2 + θ22
∂t x1
∂t x2
x1 = N1 / n1 : proportion of type 1 workers engaged incarrying out the task
x2 = N2 / n2 : proportion of type 2 workers engaged incarrying out the task
Equiprobable exposure of individuals to the stimuli associated with the task
Fixed Threshold Model with 1 Task and 2 Distinct Castes
Dynamics of demand associated with the task
∂t s = δ – (N1 + N2)α
N
δ : fixed increase in stimulus intensity per unit timeα : scale factor measuring the effectiveness of performance on the same task of type 1
and type 2 individuals
Equiprobable exposure of individuals to the stimuli associated with the task
Fixed Threshold Model with 1 Task and 2 Distinct Castes
0
10
20
30
0 250 500 750 1000
time
Increasing of the demand
Change with time in the number of majors carrying out a task when the task demand isdoubled at t = 500 (proportion of majors = 0.2)
Equiprobable exposure of individuals to the stimuli associated with the task
Fixed Threshold Model with 1 Task and 2 Distinct Castes
0
20
40
60
80
0 0,25 0,5 0,75 1
Fraction of majors
N=1000
N=100
N=10
Number of acts per major as a function of the proportion of majors in a colony fordifferent sizes of colony
Parametersof the simulation
N = 10 et 100θ1 = 8, θ2 = 1α = 3δ = 1p = 0.2
Equiprobable exposure of individuals to the stimuli associated with the task
Num
ber o
f act
s p
er m
ajor
durin
g th
e s
imul
atio
n
Fixed Threshold Model with 1 Task and 2 Distinct Castes
0
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10
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0 0,25 0,5 0,75 1
Fraction of majors
simulation N=100
Pheidole pubiventris
Pheidole guilelmimuelleri
simulation N=10
Comparison between simulations and experimental results
Parametersof the simulation
N = 10 et 100θ1 = 8, θ2 = 1α = 3δ = 1p = 0.2
Num
ber o
f act
s pe
r maj
ordu
ring
the
sim
ulat
ion
Num
ber o
f act
s pe
r maj
ordu
ring
the
real
exp
erim
ent
Equiprobable exposure of individuals to the stimuli associated with the task
Fixed Threshold Model with 1 Task and 2 Distinct Castes
Fixed Threshold Robotic Experiment
• 1 task (foraging), 1 type of robots (homogeneous group), two behavioralcastes (active and inactive)
• Demand: maintaining the nest energy (software) above a given level
Video tape!
Theoretical contribution
+ Fixed thershold algorithm verified in a real robot experiment- Environment too customized (floor lines, global reference with a light beacon,
simulated nest energy with experimenter intervention and global measurementof the demand using external supervisor).
- With one task and active/inactive castes, no equilibrated individual workload(individual with the lower threshold suffers of the most wear and tear).
- Limited effort in co-design and minimalist unit design.
- High investement in manpower (2.5 man/year) and hardware.- No systematic simulations, no modeling: isolated experiment.- No systematic study on threshold distribution.
Autonomous robotics contribution
- The experiment does not add any additional information to a non-embodied(e.g. Montecarlo, point) or embodied simulation
- No quantitative link between artificial (robots) and natural (ants) system
Social insect contribution
Fixed ThresholdRobotic Experiment
• Local perception and local evaluation of the demand• Estimation of the global demand via local communication• Four-level experimental study: analytical models, numerical models, embodied
simulations, real robot experiments
What’s next? W. Agassounon, A. Kenjale
Fixed ThresholdRobotic Experiment
Variable Threshold Modelfor Controlling the Division
of Labor
Variable threshold model
3
1
2
σσσσi1σσσσi1σσσσi1
σσσσi1σσσσi1σσσσi1
σσσσi1σσσσi1σσσσi1
+-+-+-
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The control of task allocation
The Division of Labor and theFlexibility of Social Roles
© Guy Theraulaz
Origins of the Division ofLabor in the Polist Wasps
Polists : primitive eusocial species
• Colonies usually contain only a small number of individuals (ca 20).
• These species do not show morphological differences between castes in theadult stages, nor any control or physiological determination of the role anindividual will play in the colony as an adult.
• Individual behavior is very flexible; all individuals are able to perform thewhole range of tasks which determine the survival of the colony.
• The integration and coordination of individual activities is achieved throughthe interactions which occur between the members of the colony, and betweenthe members of the colony and the local environment.
The role of learning in the differentiation of activities
• The state of the brood triggers the activities of foraging and feeding the larvae.
• In adults there are different response thresholds and these thresholds vary as aconsequence of the individuals’ activities.
• Within an individual, the act of setting out on a foraging task has the effect oflowering the response threshold for larval stimulation.
• There is therefore a self-exciting process bringing about the specialisation ofindividuals which have carried out foraging tasks; this process can be described bymeans of a variable threshold model.
Origins of the Division ofLabor in the Polist Wasps
Properties of MechanismsArising from AdaptiveResponse Thresholds
s : intensity of stimuli associated with the taskθi : response threshold of individual i to the task: θi ∈ [θmin,θmax ]ξ : incremental learning quantity (the threshold of an individual carrying out a task
is reduced by ξ)ϕ : incremental forgetting quantity (the threshold of an individual not carrying out a
task increases by ϕ/elapsed time)
Individual behavioral algorithm
Variable Threshold Model with 1 Task
θi θi - ξ when i performs the task→→→→θi θi + ϕ when i does not perform the task→
T (s) = s2
s2 + θi2θi
Parameters of individuals
Parameters of the demand associated with a task
∂t s = δ – ( Xi )α
N
δ : fixed increment per unit time in the demand associated with a taskα : effect of an individual act (the size of the relevant task is reduced by α per act)N : total number of individuals in the colonyXi : proportion of time during which the individual i carries out the task
Dynamics of the demand associated with a task
Σ
i = 1
N
:
Variable Threshold Model with 1 Task
YES
NO
Taskθθθθi,1
act ion
θθθθi,1
−−−−ξξξξ
θθθθi,1
++++ϕϕϕϕ
Description of the algorithm
∂t xi = Tθi (s)(1-xi) - p xi
duration = 1/p
Change over time in theproportion of time during whichan individual i carries out the task
Execute task
p: probability of abandoning the task in hand (mean task duration = 1/p)
Variable Threshold Model with 1 Task
Light gray specialistremoved
The Control of the Division ofLabor in a variable Threshold
Model with 2 Tasks
Variable Threshold Modelwith 2 Tasks
sj : intensity of the stimuli associated with the taskθij : response threshold of the individual to task j, θij ∈ [θmin,θmax ]p : probability of abandoning task j while it is in progress (mean duration of task = 1/p)ξ : incremental learning quantity (the threshold of an individual carrying out a task is
reduced by ξ)ϕ : incremental forgetting quantity (the threshold of an individual not carrying out a task
increases by ϕ/elapsed time)
Individual behavioral algorithm
θij θij - ξ when i performs task j→→→→θij θij + ϕ when i does not perform task j→
T (s) = sj
2
sj2 + θij
2θij
Parameters of individuals
Parameters of the demand associated with a task
∂t sj = δ – ( Xij )α
N
δ : fixed increment per unit time in the demand associated with a taskα : effect of an individual act (the size of the relevant task is reduced by α per act)N : total number of individuals in the colonyXij : proportion of time during which the individual i carries out the task j
Dynamics of the demand associated with a task
Σi = 1
N
Variable Threshold Modelwith 2 Tasks
Dynamics of response thresholds
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ind. 5
ind. 4
ind. 3
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ind. 1
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ind. 5
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Task 2
Res
pons
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Res
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e th
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Task 1
Variable Threshold Modelwith 2 Tasks
0,00
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time
Prop
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tim
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Prop
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n of
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Task 2Task 1
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Dynamics of the proportion of time spent executing the tasks
Variable Threshold Modelwith 2 Tasks
An Example of the Applicationof a Variable Threshold Modelto a Problem of Adaptive Task
Allocation
• Simulations are carried out on agrid with 5X5 zones
• 4 neighboring zones are taken intoaccount in the calculations; theboundary conditions are periodic
• 5 agents
The case of an express mail company
Application to a Problemof Adaptive Task Allocation
Different task = different zone -> demand specific to each of the zones!
• At each iteration, the demand is increased by 50 units in each of 5 randomlyselected zones.
• The agents are consulted in random order, and each computes its probability Pi,jof responding to the demand coming from each zone.
• If no agent has responded within 5 consultations, the next iteration begins.
• When an agent responds to a demand, it will be unavailable for a timeproportional to the distance between its current position and the zone to which itis moving.
• When an agent moves to a zone, the demand associated with that zone remainsat 0 while it is there.
Simulation details
Application to a Problemof Adaptive Task Allocation
Each time an agent decides to search for a letter in a zone j :
θi,j θi,j- ξ0→
θi,n(j) θi,n(j)- ξ1→
θi,k θi,k+ ϕ, pour k≠ j , k ∉ {n(j)}→
θij : response threshold of agent i to a demand coming from zone jθi,n(j) : response threshold of agent i to a demand coming from zones adjacent to zone j{n(j)} : the set of zones adjacent to jξ0 : learning coefficient associated with zone jξ1 : learning coefficient associated with zones adjacent to jϕ : forgetting coefficient associated with other zones
Updating the agents’ response thresholds
Application to a Problemof Adaptive Task Allocation
Pij : probability that an individual i, located in zone z(i) will respond to a demand sj in zone j
θi ∈ [θmin,θmax] : response threshold of agent i to a demand coming from zone jdz(i),j : distance between z(i) and j α > 0, β > 0 : modulation parameters
The threshold function
Pij = sj + αθij
+ βdz(i),j
sj2
2 2 2
Application to a Problemof Adaptive Task Allocation
threshold
demand
specialist removal
Booth Painting
• Assign trucks to paint booths– Cost of changing paint is high
• Morley and Ekberg developed 4 rules:– Try to take truck with same colour as booth– Take important jobs– Take any job to stay busy– Do not take job if booth down or queue is large
• Used market based control
Mapping to response thresholds
• Booth has response threshold w.r.t arriving truck
• Stimulus:– Response threshold low for truck wanting same
colour as booth, is urgent or queue is empty– Response threshold is high if colour is
different, queue is large or booth is down
Summary
• Division of labour is insects is not rigid– Workers switch tasks
• Simple response threshold model connects individual and colony behaviours
• Variable response thresholds generates differential task allocation and specialization starting from a uniform population
• Can be applied to resource and task allocation in multi-agent systems. Similarity to market-based control.
Possible work …
• Study of response threshold mechanism:– Theory– Dynamics (experimentation)– Integration with other techniques– Generalization to local information usage
• Application of mechanism to:– Dynamic frequency assignment in cell networks– Distributed manufacturing– Robotic soccer– … Good thesis work
here!
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