Division of Labour and Task Allocation

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1 Division of Labour and Task Allocation

2 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

3 Response Function T θ () s = s n s n + θ n Hmm, looks familiar or T () θ θ Remember the Army Ant tanh? s = 1 e s /

4 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- e Nln(1-p) Same as, s = N, θ = -1/ln(1-p) This is common: Again shows that random behavior can generate emergent organization

5 Automata Theory Learning response threshold model is similar to Automata Theory DL rp Thresholds move rather than probabilities

6 Swarm Intelligence: Bonabeau et al Increasing θ

7 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)

8 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

9 Division of Labor and Task Allocation AM, EE141, Swarm Intelligence, W3-2

10 The Division of Labor and its Control The control of task allocation The most obvious sign of the division of labor is the existence of castes. We distinguish between three kinds of castes: physical, behavioral, and temporal (temporal polyethism). The individuals belonging to different castes are usually specialized for the performance of a series of precise tasks.

12 Minor Major Physical Castes Se lf-grooming Minor worker De alat e quee n Male Behavioral repertoires of majors and minors Carry or roll egg Carry or roll larva Fe e d larva solids Carry or roll pupa Assist eclosion of adult Minor worker De alat e quee n Male Forage In Pheidole guilelmimuelleri the minors show ten times as many different basic behaviors as the majors Lay odor t rail Fe e d inside ne st Agression (drag or attack) Carry de ad larva or pupa Feed on larva or pupa Lick wall of ne st Antennal tipping Guard nest entrance 0 0,2 0,4 0,6 0 0,2 0,4 0,6 Wilson, E.O. (1976)

13 Behavioral Castes Allocation of the daily activities in a colony of desert harverster ants (Portal, AZ) From D. Gordon, Ants at Work, 1999

14 Temporal Polyethism Cleaning cells Tending brood Behavioral changes in worker bees as a function of age 10 Tending Queen Percentage of time spent in each activity Eating pollen Feeding & grooming nestmates Ventilating nest Shaping comb Storing nectar Packing pollen Foraging Patrolling Young individuals work on internal tasks (brood care and nest maintenance). Older individuals forage for food and defend the nest Resting Age of bee (days)

15 The Division of Labor and its Control Flexibility of social roles The division of labor is flexible. The number of individuals belonging to different castes and the nature of the tasks to be done are subject to constant change in the course of the life of a colony. The proportions of workers performing the different tasks varies in response to internal or environmental perturbations.

16 The Division of Labor and its Control The changes in the population of a bee colony in the course of a season AGE (DAYS) 40 May 27- Jun 7 Jun 20- Jul 1 Jul Aug Sep Oct I NCREASI NG PERI OD MAJOR PERI OD CONTRACTI NG PERI OD Egg Larvae Pupae Nurse bee House bee Field bee

17 How is flexibility implemented at the level of the individual?

18 The control of task allocation Task 1 Task 2 Task 3 The Division of Labor and the Flexibility of Social Roles? How is dynamic task allocation achieved?

19 The Division of Labor and its Control Flexibility of social roles How does the colony control the proportions of individuals assigned to each task, given that no individual possesses any global representation of the needs of the colony? The flexibility of the division of labor depends on the behavioral flexibility of the workers. A mechanism arising from the concept of a response threshold allows the production of this flexibility.

20 The Division of Labor and the Flexibility of Social Roles The control of task allocation σ i1 σ i1 σ i1 σ i1 σ i1 σ i1 σ i1 σ i1 σ i1 Fixed threshold model

21 An Example of Control of the Division of Labor Implying the Existence of Fixed Response Thresholds

22 Guy Theraulaz

23 The idea of a response threshold The Division of Labor and its Control 9 8 Pheidole pubiventris Social behavior Self-cleaning 10 9 P. guilelmimuelleri Social behavior Self-cleaning 7 8 Behavioral Acts/Major/Hour Behavioral Acts/Major/Hour ,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 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 Fraction of majors

24 An example of a response threshold The Division of Labor and its Control 1 θ i,1 Task 11 Task 11 θ j,1 T(s) Response Probability 0,75 0,5 0,25 Minors Majors T (s) = θ i s 2 s 2 + θ i 2 0 0, Stimulus s : intensity of the stimulus associated with the task θ i : response threshold θ minors θ majors

25 Properties of Mechanisms Arising from Fixed Response Thresholds

26 Two castes of workers (physical, behavioral, or age-based castes) Fixed Threshold Model with 1 Task and 2 Distinct Castes Equiprobable exposure of individuals to the stimuli associated with the task n 1 = number of workers of type 1 n 2 = number of workers of type 2 n 1 + n 2 = N f = n 1 /N : proportion of type 1 workers in the colony N 1 = number of workers of type 1 engaged in carrying out the task N 2 = number of workers of type 2 engaged in carrying out the task x 1 = N 1 / n 1 : proportion of workers of type 1 engaged in carrying out the task x 2 = N 2 / n 2 : proportion of workers of type 2 engaged in carrying out the task

27 Dynamics of the proportion of active workers in each caste Fixed Threshold Model with 1 Task and 2 Distinct Castes Equiprobable exposure of individuals to the stimuli associated with the task t x 1 t x 2 s 2 = (1-x 1 ) - p x 1 s 2 + θ 2 1 s 2 = (1-x 2 ) - p x 2 s 2 + θ 2 2 x 1 = N 1 / n 1 : proportion of type 1 workers engaged in carrying out the task x 2 = N 2 / n 2 : proportion of type 2 workers engaged in carrying out the task s : intensity of the stimuli associated with the task θ i : response threshold of workers of type i p : probability per unit time that an active individual abandons the task on which he is engaged 1/p : average time spent by an individual in working on a task before abandoning it

28 Dynamics of demand associated with the task Fixed Threshold Model with 1 Task and 2 Distinct Castes Equiprobable exposure of individuals to the stimuli associated with the task α t s = δ (N 1 + N 2 ) N δ : fixed increase in stimulus intensity per unit time α : scale factor measuring the effectiveness of performance on the same task of type 1 and type 2 individuals

29 Fixed Threshold Model with 1 Task and 2 Distinct Castes Equiprobable exposure of individuals to the stimuli associated with the task Change with time in the number of majors carrying out a task when the task demand is doubled at t = 500 (proportion of majors = 0.2) time Increasing of the demand

30 Number of acts per major as a function of the proportion of majors in a colony for different sizes of colony Parameters of the simulation 80 Fixed Threshold Model with 1 Task and 2 Distinct Castes Equiprobable exposure of individuals to the stimuli associated with the task N=10 N=100 N = 10 et 100 θ 1 = 8, θ 2 = 1 α = 3 δ = 1 p = 0.2 Number of acts per major during the simulation N= ,25 0,5 0,75 1 Fraction of majors

31 Comparison between simulations and experimental results Fixed Threshold Model with 1 Task and 2 Distinct Castes Equiprobable exposure of individuals to the stimuli associated with the task Parameters of the simulation N = 10 et 100 θ 1 = 8, θ 2 = 1 α = 3 δ = 1 p = 0.2 Number of acts per major during the simulation Pheidole guilelmimuelleri Pheidole pubiventris simulation N=10 simulation N= Number of acts per major during the real experiment ,25 0,5 0, Fraction of majors

32 Fixed Threshold Robotic Experiment 1 task (foraging), 1 type of robots (homogeneous group), two behavioral castes (active and inactive) Demand: maintaining the nest energy (software) above a given level Video tape!

33 Theoretical contribution - 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 Fixed Threshold Robotic Experiment + 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 measurement of 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. Social insect 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

34 What s next? Fixed Threshold Robotic Experiment W. Agassounon, A. Kenjale 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

35 Variable Threshold Model for Controlling the Division of Labor

36 The Division of Labor and the Flexibility of Social Roles The control of task allocation σ i1 σ i1 σ i σ i1 σ i1 σ i σ i1 σ i1 σ i Variable threshold model

37 Guy Theraulaz

38 Origins of the Division of Labor 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 the adult stages, nor any control or physiological determination of the role an individual will play in the colony as an adult. Individual behavior is very flexible; all individuals are able to perform the whole range of tasks which determine the survival of the colony. The integration and coordination of individual activities is achieved through the interactions which occur between the members of the colony, and between the members of the colony and the local environment.

39 Origins of the Division of Labor in the Polist Wasps 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 a consequence of the individuals activities. Within an individual, the act of setting out on a foraging task has the effect of lowering the response threshold for larval stimulation. There is therefore a self-exciting process bringing about the specialisation of individuals which have carried out foraging tasks; this process can be described by means of a variable threshold model.

40 Properties of Mechanisms Arising from Adaptive Response Thresholds

41 Individual behavioral algorithm Variable Threshold Model with 1 Task θ i θ i - ξ when i performs the task θ i θ i + ϕ when i does not perform the task T (s) = θ i s 2 s 2 + θ i 2 Parameters of individuals s : intensity of stimuli associated with the task θ i : response threshold of individual i to the task: θ i [θ min,θ max ] 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)

42 Dynamics of the demand associated with a task Variable Threshold Model with 1 Task N Σ i = 1 α t s = δ ( X i ) N Parameters of the demand associated with a task δ : fixed increment per unit time in the demand associated with a task α : effect of an individual act (the size of the relevant task is reduced by α per act) N : total number of individuals in the colony X i : proportion of time : during which the individual i carries out the task

43 Description of the algorithm Variable Threshold Model with 1 Task θ i,1 Task θ i,1 YES Execute task ξ action duration = 1/p Change over time in the proportion of time during which an individual i carries out the task NO θ i,1 +ϕ t x i = T θi (s)(1-x i ) - p x i p: probability of abandoning the task in hand (mean task duration = 1/p)

47 Light gray specialist removed

48 The Control of the Division of Labor in a variable Threshold Model with 2 Tasks

49 Individual behavioral algorithm Variable Threshold Model with 2 Tasks θ ij θ ij - ξ when i performs task j θ ij θ ij + ϕ when i does not perform task j T (s) = θ ij s j 2 s j 2 + θ ij 2 Parameters of individuals s j : intensity of the stimuli associated with the task θ ij : response threshold of the individual to task j, θ ij [θ min,θ max ] 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)

50 Variable Threshold Model with 2 Tasks Dynamics of the demand associated with a task N α t s j = δ ( X ij ) N Σ i = 1 Parameters of the demand associated with a task δ : fixed increment per unit time in the demand associated with a task α : effect of an individual act (the size of the relevant task is reduced by α per act) N : total number of individuals in the colony X ij : proportion of time during which the individual i carries out the task j

51 Dynamics of response thresholds Variable Threshold Model with 2 Tasks Task 1 Task Response threshold ind. 1 ind. 2 ind. 3 ind. 4 Response threshold ind. 1 ind. 2 ind. 3 ind. 4 ind. 5 ind time time

52 Variable Threshold Model with 2 Tasks Dynamics of the proportion of time spent executing the tasks Task 1 Task 2 1,00 1,00 Proportion of time 0,75 0,50 Proportion of time 0,75 0,50 0,25 0,25 0, , time time

53 An Example of the Application of a Variable Threshold Model to a Problem of Adaptive Task Allocation

54 Application to a Problem of Adaptive Task Allocation The case of an express mail company Simulations are carried out on a grid with 5X5 zones 4 neighboring zones are taken into account in the calculations; the boundary conditions are periodic 5 agents Different task = different zone -> demand specific to each of the zones!

55 Application to a Problem of Adaptive Task Allocation Simulation details At each iteration, the demand is increased by 50 units in each of 5 randomly selected zones. The agents are consulted in random order, and each computes its probability P i,j of 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 time proportional to the distance between its current position and the zone to which it is moving. When an agent moves to a zone, the demand associated with that zone remains at 0 while it is there.

56 Updating the agents response thresholds Application to a Problem of Adaptive Task Allocation Each time an agent decides to search for a letter in a zone j : θ i,j θ i,j - ξ 0 θ i,n(j) θ i,n(j) - ξ 1 θ i,k θ i,k + ϕ, pour k j, k {n(j)} θ ij θ i,n(j) : response threshold of agent i to a demand coming from zone j : response threshold of agent i to a demand coming from zones adjacent to zone j {n(j)} : the set of zones adjacent to j ξ 0 ξ 1 ϕ : learning coefficient associated with zone j : learning coefficient associated with zones adjacent to j : forgetting coefficient associated with other zones

57 Application to a Problem of Adaptive Task Allocation The threshold function P ij : probability that an individual i, located in zone z(i) will respond to a demand s j in zone j P ij = s j s j + αθ ij + βd z(i),j θ i [θ min,θ max ] : response threshold of agent i to a demand coming from zone j d z(i),j : distance between z(i) and j α > 0, β > 0 : modulation parameters

58 specialist removal threshold demand

59 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

60 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

61 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.

62 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!

SC741 W12: Division of Labor Part I: Fixed- and Variable- Threshold Algorithms

SC741 W12: Division of Labor Part I: Fixed- and Variable- Threshold Algorithms Outline Division of labor in natural systems Ants Bees, wasps Models and mechanisms Fixed-threshold mechanisms Variable-threshold