Derive Rules from Optimal Solution

The main task of the action derive ∈ ActEIA is the derivation of rules from the

optimal solution for the set of recurring tasks calculated by the previous action, which can then be sent to the agents for improving their solution, if necessary. Therefore, AgEIA has to perform four essential steps, constituting sub-actions of

derive:

1. Identif y the solution of the emergent system for the set of recurring tasks 2. Assess the quality of the emergent solution

3. Determine differences between the emergent solution and the optimal solution, if the emergent solution is much worse than the optimal solution

4. Create rules for the agents that do not behave optimally 6.2.5.1 Identification of Emergent Solution

The sub-action identif y ∈ ActEIA is used to identify from the global history of the

system GHistA generated by the action transf orm the solution solemerg ∈ SolP

the emergent system has produced for the set of recurring tasks stored in datrec EIA

at last. The action yields unordered assignments for all of the tasks in the set of recurring tasks of the form ((ta1, Ag1, t1), (ta2, Ag2, t2), . . . , (tak, Agk, tk), which can

then be put together to create the emergent solution. Definition 6.26 (Identify)

The identification of the emergent solution solemerg for a set of recurring tasks is a

function defined as

identif y: DatGHistA

EIA × Dat T asks EIA → Dat Solemerg EIA where • DatGHistA

EIA is the set of all possible global histories of a set of agents A stored

in DatEIA

• DatT asks

EIA is the set of all possible sets of recurring tasks stored in DatEIA

• DatSolemerg

EIA is the set of all possible sets of emergent solutions of a set of agents

stored in DatEIA

However, identifying the emergent solution created by A for the recurring tasks is not trivial. Very often, there will be other tasks mixed into fulfilling the recurring tasks, or in the last run instance not all of the recurring tasks might have occurred, as the size of the clusters representing the recurring tasks can be smaller than k. The fact that the agents fulfill other tasks while fulfilling the recurring tasks means that AgEIA cannot determine the quality based on measuring what really happened. For

6.2 Generic Advisor Model 169

example, between fulfilling two tasks of the recurring task set in a transportation domain, a vehicle agent might have to drive to a far off location to fulfill a not recurring task in a particular run instance. Adding the traveled distance of this agent between the two recurring tasks to the travel cost (if this is the quality criterion) would worsen the emergent solution, although in other run instances the recurring tasks are solved well.

But for such an application a lower bound for the costs that would emerge can be provided, if there were no additional tasks, which is the distance the agent has to travel after the first recurring task to start performing the second one. Such lower bounds are possible to be determined for many applications and many quality criteria. If the quality of such a lower bound is far from the optimum, then advice from AgEIA will be useful for many run instances.

Given that the last solemerg and its quality are already an approximation, the

problem of a recurring task not occurring in the last run instance can now be solved, too. AgEIA determines, which agent fulfills the task in the solemerg by going back

one more run instance (or several). This also allows to determine the correct position of the task in the sequence of tasks the agent performs.

6.2.5.2 Assessing the Solution Quality

In order to determine, if an optimization of the emergent solution is necessary and worthwhile, the sub-action assess ∈ ActEIAcompares the qualities of both the emer-

gent solution and the optimal solution. If the emergent solution is of an acceptable quality compared to the optimal one, i. e. qual(solemerg)/qual(solopt) > qualthresh

for a user defined quality threshold 0 < qualthresh ≤ 1, an optimization of the emergent solution is not worthwhile, e. g. if the emergent solution is within 90% of the optimal one, and the work of AgEIA is done until new information arrives.

Otherwise, AgEIA has to optimize the emergent solution. qualthresh depends on

the concrete application domain and should be carefully evaluated. 6.2.5.3 Determination of Differences Between Solutions

In case that an optimization of the emergent solution is necessary, the sub-action determine ∈ ActEIA compares the differences between both the emergent and the

optimal solution. Therefore, the identified assignments in both solutions are sorted by the order in which the tasks are performed in the solution. This sorting yields the solutions sol_opt0 and sol0_emerg. The sorted solutions are then compared sequen- tially to find the first assignment that differs in both solutions. Given that sol0_opt= ((ta1

1, Ag11, t11), . . . , (tap1, Agp1, t1p)) and solemer0 = ((ta21, Ag21, t21), . . . , (ta2p, Ag2p, t2p)) then

AgEIA identifies the assignment j : ta1j 6= ta2j ∨ Agj1 6= Agj2 such that ∀i < j : ta1i =

ta2_i ∧ Ag1

i = Agi2. Since both solutions are sorted according to the tl-values, this

is essentially the first assignment of a task to an agent for which the agents in A deviated from the optimal solution for the recurring tasks.

to several reasons: First, the order of the optimal solution is usually not completely different from the order of the actual solutions. In many cases, the order is really only different because the tasks were served by different agents (a suboptimal behavior) and therefore will be correct as soon as the advice that changes the assignment is adapted by the agents. Second, the order is highly sensitive to noise tasks that are not part of the recurring tasks. The order can therefore change spontaneously without real impact on the recurring tasks. Third, exception rules that for instance let agents ignore tasks are not in all cases fit to establish a certain order. Exception rules that for instance boost tasks let the agents prefer the tasks that should be serviced before the others.

6.2.5.4 Rule Creation

To force the emergent system to behave in a similar way as the optimal solution in future, the sub-action create ∈ ActEIA instantiates based on the determined

difference of the assignments a new exception rule (see Definition 6.6) for agent Ag_j2 to change its behavior when it encounters a situation that was leading to the suboptimal behavior.

Definition 6.27 (Rule creation)

The creation of a rule is a function defined as

create: Assign × Assign → Rule where

• Assign is a set of assignments • Rule is a set of rules

The event and conditions specified by sit0 respectively dat0 of an exception rule r0 are derived from the descriptions of the assigned tasks. For determining sit0 _and

dat0, AgEIA looks up the assignment (sit, dat, actta2

j) ∈ GHistAg2j, which represents

in the history of Ag2

j the point in time when it choses to perform actta2 j. (sit

0_{, dat}0₎

is then an abstraction of sit and dat. This tuple is application dependent and tries to cover not only ta2

j, but the whole cluster of tasks from the action extract of which

ta2_j is a member of. The types of exception rules that are able to change the behavior of an agent depends on the application domain and the basic MAS. However, the action of the exception rule has to change the sequence of actions an agent performs when the predicate for its current situation is true.

The exception rules created for an agent Agi are stored in the data structure

datRulei

EIA . The possible rule set for this agent is denoted as Dat Rulei

EIA . Consequently,

the possible rules sets for all agents in A with m = |A| are denoted as DatRuleA EIA ⊆ Dat Rule1 EIA × Dat Rule2 EIA × . . . × Dat Rulem EIA (6.5)

6.2 Generic Advisor Model 171 Definition 6.28 (Derive)

The derivation of rules and is a function defined as derive: DatGHistA

EIA × Dat T asks EIA × Dat Solopt EIA × Dat Solemerg EIA → Dat RuleA EIA where • DatGHistA

EIA is the set of all possible global histories of a set of agents A stored

in DatEIA

• DatT asks

EIA is the set of all possible sets of recurring tasks stored in DatEIA

• DatSolopt

EIA is the set of all possible optimal solutions for a set of recurring tasks

and a set of agents stored in DatEIA

• DatSolemerg

EIA is the set of all possible emergent solutions for a set of recurring

tasks and a set of agents stored in DatEIA

• DatRuleA

EIA is the set of all possible rule sets for a set of agents A stored in

DatEIA