A MULTI-AGENT MODEL BASED ON RESIDUAL RESOURCES

Ga¨el Hette, Sylvia Estivie, Emmanuel Adam and Ren´e Mandiau

Univ Lille Nord de France, F-59000 Lille, France

UVHC, LAMIH, F-59313 Valenciennes, France

CNRS, FRE 3304, F-59313 Valenciennes, France

Keywords: Multi-agent system, Task allocation, Residual resources, Resources pooling.

Abstract:

We propose a model to solve the disturbances which could occur during the execution of agents task schedules

in a dynamic multi-agent system. Our model is based on pooling residual resources to reallocate tasks in case

of incoming tasks or disturbances during the execution of the system. In our context, initial task schedules of

agents are deﬁned such that, during some given time windows, agents can perform additional tasks with their

residual resources. In this paper, we propose a model for this problem. In order to validate this model, we also

propose a ﬁrst resolution method attempt and an experimental study of this framework.

1 INTRODUCTION

In this paper we consider the case of task realloca-

tion over a multi-agent system, where each agent has

non shareable resources to achieve tasks. These re-

sources are still available during the execution of such

system instances, nevertheless these resources are not

used continuously by the agents. As a consequence,

we focus on the context of a resource pooling system,

where agents can achieve tasks for others by using

their own allocated resources for time periods during

which these resources are not needed by them. We

denote these kind of resources as residual resources.

We propose to use multi-agent task allocation tech-

niques (Chevaleyre et al., 2006) to reallocate tasks

during the execution of a deﬁned schedule. We fo-

cus on a special case of task reallocation problem for

which global schedule modiﬁcation is not allowed.

We aim to proposea framework to deal with incoming

tasks or task failures by pooling residual resources.

Considering an agent facing a loss of its allocated re-

source, our approach is based on a model and a set

of methods to ﬁnd corresponding available resources

(according to other agents schedule) and insert addi-

tional tasks into another agent’s schedule, in order to

solve this problem. We assume that resource allo-

cation and task schedule are such that agents could

perform additional tasks with their residual resources

during given time windows. So, the main idea of

this work is to consider that agents pool their resid-

ual resources in order to enable task reallocation with

a view to execute tasks for other agents. We focus on

the case of task reallocation during the execution of a

deﬁned schedule. The main constraints of our prob-

lem are that any modiﬁcation of the initial resource

allocation is forbidden and no major modiﬁcation of

agents schedule is allowed. As a matter of fact, we

propose in this paper a framework for this problem.

In order to validate this model, we also propose a ﬁrst

resolution method attempt and an experimental study

of this framework. Section 2 details our problem of

adaptive task scheduling within the context of a re-

source pooling system and gives a brief state of the

art. In Section 4 we propose a computational model

and procedures to deal with the problem of adaptive

task scheduling. In order to validate our model, Sec-

tion 5 concludes with an experimental study.

2 OUR PROBLEM

We consider a multi-agent system with an initial re-

source allocation amongst agents computed in order

to accomplish an associated task schedule. Each task

of this schedule has to be performed by a speciﬁc

agent within a given time window, and implies the

use of some of its allocated resources. This problem

includes situated agents moving in an environment,

agents and their allocated resources share the same

locations. Thus when agents move from a location to

another, they are accompanied by their allocated re-

sources. Furthermore, the agent schedules deﬁne a

set of tasks to execute under location constraints. In

addition, we assume that the initial resource alloca-

tion and task schedule are such that some allocated

Hette G., Estivie S., Adam E. and Mandiau R..

A MULTI-AGENT MODEL BASED ON RESIDUAL RESOURCES.

DOI: 10.5220/0003698600850090

In Proceedings of the 4th International Conference on Agents and Artiﬁcial Intelligence (ICAART-2012), pages 85-90

ISBN: 978-989-8425-96-6

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

resources are not fully used by agents and agents can

perform additional tasks during given ﬁnite time win-

dows. Moreover, the problem occurs in a dynamic

environment where tasks appear during the execution.

These dynamic tasks are new task orders or initially

scheduled tasks that agents fail to perform due to the

system dynamics. So, the main idea of this work is

to consider that agents pool their residual resources

in order to execute additional tasks for others and to

be able to delegate the achievement of tasks to other

agents. As we focus on the case of task reallocation

during the execution of a deﬁned schedule, the main

constraint of our problem is that we cannot modify

the resource allocation. We must rather use this al-

location to assign the execution of additional tasks to

agents based upon their resources and task schedule.

Background. We present the two most similar prob-

lems to our, as they include situated agents and their

solutions imply schedule such that some allocated re-

sources are unused by agents during given time pe-

riods. The Vehicle Routing Problem with Time Win-

dows (VRPTW) (Laporte and Osman, 1995) has been

widely studied in both operationalresearch (Solomon,

1987; Cordeau et al., 2001) and multi-agent sys-

tem (Fischer et al., 1999; Barbucha and Jedrzejow-

icz, 2009). The VRPTW concerns the design of

least cost routes over a vehicles ﬂeet, in order to

deliver goods from a central depot to a set of geo-

graphically scattered points during given time win-

dows. The Dynamic Pickup and Delivery Problem

(DPDP) (Berbeglia et al., 2010) is another vehicle

routing problem of interest. This problem concerns

the transport of people or goods from an origin to a

destination. Now we give an overview of the differ-

ent approaches for these problems. These approaches

have led us in our study even if they differ in terms

of constraints on the modiﬁcation of the resource al-

location and task schedule of agents. As a matter

of fact, they were not reusable for our work as they

imply major modiﬁcation that are not allowed in our

case. In the multi-agent literature, the use of mar-

ket based allocation procedures is the main approach

adopted to solve VRPTW. These procedures are of-

ten based on Contract Net Protocol (Davis and Smith,

1983; Fischer et al., 1999) or on Vickrey auction

model (Vickrey, 1961; Gorodetski et al., 2003), but

only few propose an approach for dynamic task real-

location. Fischer et al. (Fischer et al., 1999) presented

a multi-agent simulation framework where agents in-

teractions are driven by negotiation protocols. Here,

the problem is similar to our in the sense of resource

pooling and as it proposes a procedure between com-

panies to buy and sell free loading capacities to over-

come disturbances during the execution. Outcomes of

this procedure are interesting because they can lead to

local modiﬁcations of the schedule. But these out-

comes differs from ours, since they lead to global

modiﬁcation of agents schedule in the worst cases and

thus give no guarantee of minimum changes. Studies

concerning the DPDP are based on the same two steps

approach. In the ﬁrst step, the problem is resolved

in its static version according to requests known be-

fore execution. The second step occurs during execu-

tion, when a new request is known. During this step,

heuristic methods such as insertion methods, inter-

change moves and deletion heuristics are performed.

As a result, the previous solution of the schedule is

updated with minor or global schedule modiﬁcation.

All these approaches concern problems for which a

global modiﬁcation of the schedule is allowed. But

we consider a problem where such global modiﬁca-

tion is forbidden. So, in the next section we propose a

model to enable dynamic task reallocation under tem-

poral and location constraints without global modiﬁ-

cation of the schedule.

3 COMPUTATIONAL MODEL

We consider a multi-agent system populated by n sit-

uated agents a

, ..., a

and m resources. The agents

and their allocated resources share the same location.

Thus when agents move from a location to another,

they are always accompanied by their allocated re-

sources. Each task must be achieve at a given loca-

tion of the environment. The knowledges of an agent

are represented by a tuple < E

, S

, R

, O

, D

, P

with :

• E : environment representation,

• S

: the task schedule of agent a

• R

: ﬁnite set of a

allocated resources,

• O

: ﬁnite set of a

task offers,

• D

: ﬁnite set of a

task demands,

• P

: a

preferences to select the best task offer

among others.

Environment and Communication. The environ-

ment of the agents is deﬁned by a spatial grid di-

vided in small regions. This spatial grid is modeled

by an undirected graph E = (V, A), where each vertex

∈ V represents a small region of the grid. The edge

set is deﬁned as A = {(i, j)|i, j ∈ V, i 6= j}. Each pair

of adjacent regions are represented by a valued edge

(i, j) ∈ A. The weight of each edge (i, j) has a non-

negative value t

i, j

and denotes the time travel of an

agent from region i to j. In the context of our study,

we assume that there is no communication constraint

ICAART 2012 - International Conference on Agents and Artificial Intelligence

between agents. In addition, we consider that agent

are able to perform point to point and broadcast com-

munication protocols.

Task Schedule. Let S

= {t

, ...,t

} denote

the task schedule of an agent a

. Each primi-

tive task t

∈ S

is deﬁned by a tuple t

, SDate

, EDate

, Loc

>, where :

• R

: type of resource needed to achieve t

• V

: amount of resource needed to achieve t

• SDate

: start date of t

execution,

• EDate

: end date of t

execution,

• Loc

: location constraint on t

performance.

The constraint Loc

, deﬁnes the location in the envi-

ronment where the task t

must be performed by an

agent a

. Thus, a

must perform the task t

in regard

with the location constraint speciﬁed by Loc

. In ad-

dition, S

= {t

, ...,t

} is deﬁned as a strictly ordered

set, such that:

SDate

< EDate

< SDate

< ... < EDate

Agents schedule are deﬁned such that, during some

time periods, all allocated resources are not needed

by them (i.e. residual resources). We assume that

during such periods, agents can perform additional

tasks by using these residual resources. Thus, each

agent a

has a task offers set O

, as a way to propose to

other agents to execute tasks for them with its residual

resources. According to the dynamics, disturbances

may occur during the execution of schedules. We de-

ﬁne a disturbance as a loss of one or more resources

initially allocated to an agent. So, when a disturbance

occurs, the agent is unable to perform the tasks requir-

ing these resources. To overcome such situations each

agent a

has a task demands set D

, deﬁning the tasks

it cannot perform and needs to delegate to others. At

the beginning, D

the task demands set of an agent a

is empty, whereas its set of task offers O

contains ele-

ments according to its residual resources. These con-

tents are computed along the dynamics of the system.

We deﬁne each element of O

and D

as ﬁnite sets of

data describing the usage constraints of the resources

associated to a task offer and to a task demand.

Task Offers Representation. Here we give a

detailed description of the ﬁnite data set used by

agents to describe their task offers and correspond-

ing usage constraints. Given an agent a

, each

task offer o

∈ O

, is deﬁned by the tuple o

, T

, R

, Sdate

, Edate

, Loc

>, where:

• a

: the agent proposing the task offer,

• T

: identiﬁer of the task offer,

• R

: offered resource type,

• V

: offered resource volume or amount,

• Sdate

: start date of resource availability,

• Edate

: end date of resource availability,

• Loc

: constraint(s) on a

location,

• C

: acceptance criteria to contract a task execu-

tion for another agent with this offer.

Task Offers Acceptance Criteria. The decision

criteria C

relative to a task offer o

∈ O

of an agent

are used by a

during the task reallocation proce-

dure. Given the task demand d

∈ D

of an agent

, these criteria are used by a

to decide to execute a

task for a

in regard with the constraints of o

and d

Considering o

and d

, Pmax

denotes the maximum

postpone total delay allowed over a

schedule and in-

duced by the execution of a task for another agent,

Dmax

denotes the maximum distance induced by the

location constraints of o

and d

. Let Dist

the dis-

tance induced by o

and d

location constraints and

the time delay occurring in a

schedule if it exe-

cutes the task demand d

. Thus the acceptance crite-

ria used by a

to execute the task associated to d

for

are deﬁned by the following condition :

6 Pmax

) ∧(Dist

6 Dmax

)

Task Demands Representation. We deﬁne a task

demand as the ﬁnite data set used by agents to de-

scribe a task they can no longer perform and to de-

ﬁne the constraints and preferences regarding its ex-

ecution by another agent. Given an agent a

, each

task demand d

∈ D

is deﬁned by the tuple d

, T

, R

, V

, Sdate

, Edate

, Loc

, C

> where:

• a

: the agent willing to delegate a task execution,

• T

: identiﬁer of the task demand,

• R

: needed resource type,

• V

: needed resource volume or amount,

• Sdate

: start date of task execution,

• Edate

: end date of task execution,

• Loc

: location constraint(s) on T

performance,

• C

: acceptance criteria to determine correspond-

ing feasible task offers,

Task Demands Acceptance Criteria. Let o

∈ O

the task offer of an agent a

and d

∈ D

the task de-

mand of an agent a

. The acceptance criteria C

rela-

tive to a task demand d

are used by a

during the task

reallocation procedure to determine if o

satisﬁes the

constraints of d

(i.e. if o

could be a feasible task of-

fer in regard with the execution constraints speciﬁed

by d

). Considering a

and d

, Pmax

denotes the

maximum postponement allowed for the execution of

A MULTI-AGENT MODEL BASED ON RESIDUAL RESOURCES

, Dmax

denotes the maximum distance induced

by the location constraints of d

and o

and allowed

to contract the task offer o

of agent an a

. Let us

consider Dist

the distance induced by d

and o

lo-

cation constraints, and P

the postponement induced

if a

executes d

. Thus the decision criteria used by

to determine if o

is a feasible offer are deﬁned by

the following condition :

6 Pmax

) ∧(Dist

6 Dmax

)

Agent Preferences. Given the task demand d

∈ D

of an agent a

, we deﬁne a method to determine the

best offer among a set of feasible offers and with re-

gard to both of their usage constraints. Thus, the pref-

erences Pref

associated to d

deﬁne the selection

method used in the determination of the best offer.

This selection method uses a multi-criteria aggrega-

tion function (Bellosta et al., 2008). These prefer-

ences deﬁne criteria regarding distance travel et deliv-

ery time delay. Thus, an offer inducing a short travel

distance will be preferred to a longer distance offer.

The same selection method is applied for postpone-

ments criteria. The expression of preferences over

alternative solutions is based on a preferences aggre-

gation deﬁned on these criteria. Each alternative so-

lution o

is deﬁned by a pair of criteria (D

, R

denoting the alternative o

which induces a distance

and a delivery postponement R

. The set of

alternatives can be represented by the set of pairs

, R

). Upon this set, we use a lexicographical

order to determine the best solution. Thus, we deﬁne

a total order over the alternatives set such that :

((D

, R

) ≻ (D

, R

))

iff (D

< D

) ∨((D

= D

) ∧(R

< R

))

Task Reallocation Procedure. In the case of dis-

turbances occurring during the system execution, the

main goal of the task reallocation procedure is to ﬁnd

residual resources allowing the execution of a task

under temporal and location constraints. We propose

a simple heuristic method for such problems (Algo-

rithm 1 delegateTask) and used to validate our model.

This heuristic method is based on a greedy algorithm

which aims to reduce the total amount of informa-

tion exchanges and to shorten the time needed to ﬁnd

a solution by reducing the total number of queried

agents. This algorithm determines successive small

sets of agents to query until a feasible solution is

found. Thus, when an agent needs to delegate a task

execution, at ﬁrst it makes a local search by asking

his nearest neighboring agents about their task offers.

If no corresponding task offer is found, then the agent

performs a more global research by selecting and ask-

ing agents in a farther neighborhood. The notion of

Algorithm 1: delegateTask(demand).

1: contract ← false

2: fSet ← null

3: qSet ← getQueryAgentSet(i=0)

4: while (qSet 6= null) ∧ ¬ contract do

5: fSet ← feasibleOfferSet(qSet, demand)

6: contract ← contractOffer(fSet, demand)

7: fSet ← getQueryAgentSet(i+1)

8: end while

neighboring agents deﬁnes how to determine the suc-

cessive sets of agents to query about their task offers

(method getQueryAgentSet). The set of feasible task

offers, corresponding to the demand d

, is selected

upon the procedure feasibleOfferSet and according to

the acceptance criteria C

of d

. This set of feasi-

ble task offers is used as input of the procedure con-

tractOffer in order to propose and conclude a task

delegation with another agent. The procedure con-

tractOffer determines the best offer o

according to

the preferences of d

. Then a task delegation request

is sent to a

the agent proposing o

. In return, a

ac-

cepts or refuses the request if the realization of d

im-

plies a delay in its schedule. If the request is refused,

the procedure is repeated with the next best offer, un-

til the feasible offer set is empty or a task delegation

is concluded.

4 EXPERIMENTAL STUDY

In this Section, we present an experimentalstudy with

an implementation of our framework and two sim-

ple task reallocation methods, in order to validate our

model for dynamic task allocation.

A Transportation Network Application. This

work is a part of an industrial project for an urban

pooling capacity transportation network. Our experi-

mental study is a simulation of the multi-agent system

used to monitor the network execution. The simu-

lation is based on a set of agents representing trans-

portation vehicles. The agents schedule are computed

as the solution instance of a VRPTW. Time and space

are discretized. Time is divided in 10 minutes rounds.

The environment is described by a spatial grid di-

vided into a set of 89 non overlapping areas. During

rounds, agents travel in areas according to their sched-

ule. Agents can handle a limited amount of freight.

During the execution, the amount of freight handled

by agents evolves. Thus, at given time rounds, agent

load capacity are not fully used and freight transporta-

tion capacity is still available. During schedule ex-

ecution, failures are randomly generated in order to

observe the system behavior. These failures imply the

ICAART 2012 - International Conference on Agents and Artificial Intelligence

inability of an agent to execute a freight delivery ini-

tially planned in it schedule. For example, truck driver

may be missing at the beginning of the day, a vehicle

may suffer of breakdown during its travel,... In the

context of our experimental study, we consider two

different failure types. Given t the time round of a

failure detection, the ﬁrst type of failure induces the

inability to execute a task scheduled in round t + n,

whereas the second induces the inability to execute

a task scheduled in round t. Our experimentations

concern the second type of failure, which implies to

quickly ﬁnd a solution.

Experimental Protocol. Our experiments have

been led over two sets of 10 delivery schedules of two

different types. Both of the delivery schedules types

involve a set of 100 agents. The ﬁrst delivery sched-

ule type (denoted type I) includes 800 delivery orders

over a 5 hours period. The second delivery schedule

type (denoted type II) includes 1600 delivery orders

overs a 10 hours period. The delivery schedules of

type II occur in a far more congested road time pe-

riod than the schedules of type I. For a number of fail-

ures ranging from 1 to 90 (one failure maximum per

agent), we conducted 100 random draws for each dif-

ferent range of failures. Two different task realloca-

tion procedures have been applied separately to solve

each of the schedules thus obtained.

Task Reallocation based on Contract Net Protocol.

This reallocation procedure (denoted global research

in the following), starts with the broadcast of a mes-

sage containing the task demand. Then all the agents

reply with a refusal message or a proposal message

containing a non empty list of their available task of-

fers matching the constraints of the task demand. The

preferred feasible offer is selected and contracted ac-

cording to agent preferences and acceptance criteria.

Task Reallocation based on Local Research. In

regard with our applicative context, the location con-

straints of task offers and task demands refer to ar-

eas of the spatial grid modeling the environment. So,

given an agent a

, we deﬁne the nearest neighboring

agents of a

as the set of agents located in areas adja-

cent to a

location area and agents located in the area

of a

. Starting from the nearest neighboring set, the

farther neighboring sets of agents are deﬁned step by

step by selecting agents located in the adjacent areas

of the previous neighboring set.

Experimental Settings. The main settings used in

our experimental study concern the value of agents

acceptance task offers and task demands acceptance

criteria. Here we give the acceptance criteria assign-

ment used in our experiment. We consider in this

case, that given an agent a

the acceptance criteria

of all its task offers and task demands have the same

value. More formally, given an agent a

, ∀o

∈ O

and ∀d

∈ D

the assignment of a

acceptance crite-

ria are deﬁned such that Pmax

= Pmax

= Pmax and

Dmax

= Dmax

= Dmax. Distribution of the assign-

ment values of our experiments are given in table 1.

Table 1: Acceptance criteria assignments distribution.

number of agents Pmax (minutes) Dmax (km)

33 30 5

33 60 10

34 90 ∞

4.1 Experimental Results

Proportion of Resolved Failures. For a number of

failures ranging from 1 to 10 the mean proportions

of resolved failures with the local research are sim-

ilar to those obtained with the global research. The

related ratio of the local research over the global re-

search method have almost equal values for a num-

ber of failures ranging from 1 to 20. According to

ratio values, the local research procedure is less ef-

ﬁcient for a number of failures exceeding 30%. In

regard with the observations made upon the propor-

tion of resolved failures and their corresponding ratio,

we assume that road congestion periods have little in-

ﬂuence on the mean proportion of resolved failures.

Indeed, the entire ratio result values are similar what-

ever the research method and type of schedule stud-

ied.

Amount of Information Exchanges. In regard

with the deﬁnition of the task reallocation procedures,

the amount of information exchanges depends on the

number of messages exchanged. This number of mes-

sages is deﬁned by the number of agents queried

about their task offer and also by the number of agents

requested to accept additional task until a solution is

found. In case of a global research the number of

agents queried about their task offers is always equal

to 99 agents, whereas this number presents a very

high variability level in the case of a local research.

The mean number of queried agents needed to solve

a failure varies between 15 and 18 agents. Consider-

ing a number of failures evolving from 1 to 40%, the

mean number of agents requested to accept an addi-

tional task varies from 1 to 1.5 for both procedures.

Computational Cost. We deﬁne the computational

cost as the mean CPU time

needed to solve a failure.

Local research is 6 times faster than global protocol

in the worst case and 8.5 faster in the best case. The

The computer used to run the experimentations is

equipped with a 2.66GHz Intel Core i7 processor and a 4Go

1067MHz memory

A MULTI-AGENT MODEL BASED ON RESIDUAL RESOURCES

very high CPU time needed by the global research and

compared to the local research, comes with the size of

the task offers set used to ﬁnd a solution.

Solution Quality. The quality of the solutions is

characterized by three different indicators: the total

postponement over agents initial schedule, the delay

of resolved freight deliveries and the additional travel

distance. Here we focus our study and observations

for a failure proportion varying from 1 to 30%. In-

deed, we consider that for any failure proportion su-

perior to 30% the simulation results concerns excep-

tional cases of disturbances in a transportation net-

work. Under this assumption and according to results

obtained, the mean total postponementinduced by ad-

ditional tasks in agents schedule is higher with the

global protocol than with the local protocol. In ad-

dition, the mean value of postponement is higher with

schedule type II than with type I, this observation rely

on the differences of road congestion characterizing

the two types of delivery schedules. Concerning the

mean postponement of resolved freight deliveries, we

observe higher mean postponement result values with

the global research procedure (from 17 to 27 min-

utes) than with the local research procedure(from 13

to 21 minutes). The mean additional travel distance

of agents obtained with the local research is higher

(from 1.5 km to 2,2 km) than with the global research

procedure (from 1,3 km to 1,5 km). In short, and with

regard to the global research, the local research max-

imizes the additional travel distances, but enables to

minimize the postponement over the delegated deliv-

eries and the agents task schedule.

5 CONCLUSIONS

In this paper, after the presentation of a task reallo-

cation problem and a state of the art, we proposed a

computational model that allows agents to delegate

the execution of tasks to others. This model aims to

capture all the the descriptors needed for task delega-

tion. It gives a description of the various constraints

for the performance of tasks under location and tem-

poral constraints by a set of situated agents. In or-

der to validate our model, we introduced a method

to enable the exploitation of residual resources in

a multi-agent system where agents have non share-

able resources to execute a deﬁned task schedule. To

overcome disturbances during the execution of such

schedule, we propose to reallocate tasks to agents

over time periods where they can perform additional

tasks without modifying their initial schedule. Our

ﬁrst experimental results show that the use of lo-

cal research is more efﬁcient than global search and

gives solutions of equivalent quality, in term of addi-

tional distance and postpone delay. The main lines of

our future research will focus on optimizing the re-

search method by introducing a negotiation protocol.

We also aim to propose a task reallocation procedure

which enable the use of multiple task offers to solve

one failure. Another part of our work will concern

the inﬂuence of acceptance criteria on the method ef-

ﬁciency and the solution quality.

ACKNOWLEDGEMENTS

The present research work has been supported by the

Ile-de-France Region, the ”Minist`ere de l’

Economie,

de l’Industrie et de l’Emploi” and ADVANCITY.

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