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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

20

20

20

20

40

40

40

10

10

10

10

10

10

10

10

40

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


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

)

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


How is flexibility implementedat the level of the individual?

The Division of Labor andthe Flexibility of Social Roles


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


σσσσi1σσσσi1σσσσi1



3

1

2

Fixed threshold model


The Division of Labor and the Flexibility of Social Roles

An Example of Control of theDivision of Labor Implying theExistence of Fixed Response

Thresholds

0

1

2

3

4

5

6

7

8

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

0

1

2

3

4

5

6

7

8

9

10

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


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


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



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



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)



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


Num

ber o

f act

s p

er m

ajor

durin

g th

e s

imul

atio

n


0

2

4

6

8

10

0

25

50

75

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



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




+-+-+-

+-+-+-

+-+-+-


The Division of Labor and theFlexibility of Social Roles

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

:


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)


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


Dynamics of response thresholds

0

250

500

750

1000

0 1000 2000 3000

time

ind. 5

ind. 4

ind. 3

ind. 2

ind. 1

0

250

500

750

1000

0 1000 2000 3000

time

ind. 5

ind. 4

ind. 3

ind. 2

ind. 1

Task 2

Res

pons

e th

resh

old

Res

pons

e th

resh

old

Task 1


0,00

0,25

0,50

0,75

1,00

0 1000 2000 3000

time

Prop

ortio

n of

tim

e

Prop

ortio

n of

tim

e

Task 2Task 1

0,00

0,25

0,50

0,75

1,00

0 1000 2000 3000

time

Dynamics of the proportion of time spent executing the 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


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


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


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!

division of labour and task allocationpeople.scs.carleton.ca/~arpwhite/courses/5002/notes/si... ·...

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