CENTRALIZED AND DECENTRALIZED OPTIMISATION

TECHNIQUES FOR THE FLEXIBLE JOB SHOP SCHEDULING

PROBLEM

Meriem Ennigrou

and Khaled Ghédira

1 ur. SOIE, Stratégies d'Optimisation des Informations et de la connaissancE

IPEIM, Institut Préparatoire aux Etudes d'Ingénieurs – ElManar, 2092 El Manar 2 BP 244, Tunisie

2 ur. SOIE, Stratégies d'Optimisation des Informations et de la connaissancE

ENSI, Ecole Nationale des Sciences de l'Informatique, 2010 Campus Universitaire la manouba, Tunisie

Keywords: Multi-Agent System, flexible Job Shop, Scheduling, Tabu Search.

Abstract: This paper proposes two Multi-agent approaches based on a tabu search method for solving the flexible Job

Shop scheduling problem. The characteristic of the latter problem is that one or several machines can

process one operation so that its processing time depends on the machine used. Such a generalization of the

classical problem makes it more and more difficult to solve. The objective is to minimize the makespan or

the total duration of the schedule. The proposed models are composed of three classes of agents: Job agents

and Resource agents and an Interface agent. According to the location of the tabu search core, two versions

have been proposed. The first one places the optimisation method only on the Interface agent whereas the

second associates to each Resource agent its own optimisation process.

1 INTRODUCTION

In the later decades, extensive researches on

scheduling have been reported from both theoretical

and practical issues. Scheduling means allocating a

set of jobs to a finite set of resources over time while

satisfying a set of constraints. An important goal in

the scheduling function is to assure that the work is

completed as early as possible.

Among the most difficult scheduling problems,

we find the Job Shop Scheduling Problem (JSSP).

Finding an optimal solution for such problems in a

reasonable time seems to be very hard, in the

majority of cases, because of their high complexity.

In fact, this problem falls into the category of NP-

hard problems for which exact solving methods are

inappropriate since they explode with problem size.

However, approximate methods are more suitable

for such problems. The latter are either based on

local search techniques such as tabu search and

simulated annealing or on evolutive techniques such

as genetic algorithms and ant systems.

In this paper, we are concerned with an extended

class of the JSSP, namely the flexible Job Shop

(FJSP), to which we propose two Multi-Agent

models based on the tabu search optimisation

method. The objective is to minimize the makespan

or the total duration of the schedule.

2 THE FJSP

A JSSP consists in performing a set of n jobs {J

…, J

} on a set of m resources {R

, …, R

}. Each job

, i=1,…,n, is composed of n

operations that must

be performed on the different resources according to

a predefined order, known as the job process

routing. This one characterizes the precedence

constraints existing between the operations of one

job. In addition, each operation has a processing

time known in advance and can be processed by only

one resource.

Furthermore, each job has to be achieved in a

temporal range defined by its release date, before

which the job cannot be started, and its due date,

before which the job must be completed. This

temporal range defines the temporal constraints of

that job. Moreover, a resource can perform only one

operation at a time, which corresponds to the

Ennigrou M. and Ghédira K. (2005).

CENTRALIZED AND DECENTRALIZED OPTIMISATION TECHNIQUES FOR THE FLEXIBLE JOB SHOP SCHEDULING PROBLEM.

In Proceedings of the Second International Conference on Informatics in Control, Automation and Robotics - Robotics and Automation, pages 31-36

DOI: 10.5220/0001176900310036

 SciTePress

disjunctive constraints, and an operation cannot be

interrupted unless it is finished, i.e. no pre-emption

is allowed. A solution for the JSSP consists in fixing

a start time for each operation satisfying the set of

constraints.

FJSP, first introduced by (Nuijten & Aarts,

1996), is a generalisation of the above mentioned

problem, where each operation can be processed by

more than one resource and has consequently a

processing time depending on the resource used. A

solution consists then not only in sequencing the

operations on the resources and fixing them a start

time but also in allocating them to a resource likely

to achieve them. This problem is also NP-hard.

3 TABU SEARCH

The models we propose in this article are based on a

combinatorial optimisation technique, namely the

tabu search (TS) method, proposed by (Glover,

1986), which is a meta-heuristic based on the local

search principle. Beginning from an initial solution,

the local search consists to choose, at each iteration,

the best solution in the current solution

neighbourhood, even if it does not improve the

quality of the solution. A neighbourhood is

composed of all the solutions obtained by a simple

move on the current solution. These solutions are

named, then, neighbours of the current one. TS have

proved its power to handle JSSPs. Several

researches have used TS and good results have been

obtained. Among the approaches proposed for the

JSSP, the ones proposed by (Mastrollili &

Gambardella, 2000), (Brucker & Neyer, 1998) and

(Chambers & Barnes, 1996).

In order to escape local optima in which the

system can be easily trapped, TS uses a temporary

memorisation structure in which it keeps track of the

last visited solutions: the tabu list. In fact, a solution

is forbidden during a number of iterations equal to

the tabu list size. Then, the best solution among the

ones not forbidden is selected for the next iteration.

Although its efficiency in solving many difficult

problems, TS remains yet hardly adaptable to FJSP

because of the great number of parameters to define:

– initial solution,

– neighbourhood function,

– evaluation of the current solution,

– tabu list size, etc.

Later in this paper, we will describe briefly our

adaptation of the different parameters to the FJSP.

The next section presents the two multi-agent

models proposed and subsequently their global

dynamic.

4 MULTI-AGENT MODELS

Two Multi-Agent models have been proposed for

solving FJSP. The first one consists in centralizing

the optimisation process in a unique agent

responsible for finding the optimal solution in

cooperation with the remainder of the agents which

are responsible for generating successive feasible

solutions at each step of the process. Whereas, the

second one distributes the optimisation process

between a collection of agents cooperating together

in order to find the best possible solution.

Since scheduling problems involve two sorts of

constraints: the ones concerning the jobs, namely

precedence and temporal constraints, and those

concerning the resources, namely disjunctive

constraints, both Multi-Agent models proposed in

this paper are then composed of two agent classes:

Job Agents and Resource Agents responsible for the

satisfaction of the two classes of constraints. In

addition, a third agent class, containing a single

agent, the Interface agent, is added to both models.

The degree of importance of the latter agent in the

solving process differs from the first model to the

second. In fact, in the first model, it contains the

core of the TS method. However, in the second

model, its role is limited to the interface between the

agents and the user. The optimisation process, in this

case, is distributed among the Resource agents.

Each agent in these models has its own

acquaintances (i.e. the agents that it knows and with

which it can communicate), a local memory

composed of its static and dynamic knowledge and a

mailbox in which it stores the messages received

from the other agents. In the remaining of this

section, we describe briefly each agent class in both

models.

4.1 Centralized Optimisation Model

4.1.1 Job Agent

The acquaintances of Job agent are composed of

Resource agents that are likely to fulfil its operations

and of the Interface agent. Its static knowledge

includes its release and due dates, its process routing

and the different processing times of its operations

according to the resources. Whereas its dynamic

knowledge comprises the start times of its operations

and the current resources to which they are

allocated.

The Job agent is satisfied when all its operations

have been affected to potential resources and when

all its constraints are not violated and in this case it

does nothing. Otherwise, it is unsatisfied and it tries

ICINCO 2005 - ROBOTICS AND AUTOMATION

to assign an operation to an eligible resource in

cooperation with its acquaintances.

4.1.2 Resource Agent

The acquaintances of Resource agent are composed

of all Job agents whose operations are likely to be

fulfilled by it and of the Interface agent. Its static

knowledge encloses the list of operations that it can

perform along with their processing times. While its

dynamic knowledge is composed of the operations

currently assigned to it and their start times.

The Resource agent is satisfied when no

overlapping conflict exists between two operations

assigned to it and in this case it does nothing. If not,

it is unsatisfied and it tries to solve these conflicts by

sending one of the conflicting operation to its Job

agent in order to replace it elsewhere.

4.1.3 Interface Agent

The Interface agent acquaintances are composed of

all the agents existing in the system. Its static

knowledge contains:

– The maximal number of iterations allowed

– The tabu list size

Its dynamic knowledge is composed of:

– The tabu list

– The current solution and its makespan

– The best solution encountered so far and its

makespan

– The current number of iterations performed.

As mentioned before, the Interface agent, in this

model, contains the core of the optimisation process.

The Interface agent remains unsatisfied until the

current number of iterations exceeds the maximal

number of iterations allowed. Otherwise, it delivers

the best solution to the user.

4.2 Distributed Optimisation Model

4.2.1 Job Agent

The acquaintances of Job agent are composed of all

Resource agents and the Interface agent. Its static

knowledge includes its release and due dates, its

process routing and the different processing times of

its operations according to the resources. Whereas it

has no dynamic knowledge.

The Job agent is satisfied when all its operations

have been assigned, and in this case it does nothing.

Otherwise, it is unsatisfied and it tries to assign an

operation to an eligible resource in cooperation with

its acquaintances.

4.2.2 Resource Agent

A Resource agent can communicate with all the

other agents existing in the system. Its static

knowledge encloses the entirety of the information

concerning the operations; i.e. their processing

times, their potential resources, their predecessors

and successors according to their job process

routings, etc.; along with the maximal number of

iterations allowed, the number of iterations allowed

between two successive improvements and its own

tabu list size. While its dynamic knowledge is

composed of its tabu list, its current solution and its

cost, the best solution that it has encountered so far

and its cost, the number of iterations that it has

performed, the number of iterations since the last

improvement made and a list of the best solutions

reached by the other Resource agents.

The Resource agent is unsatisfied while the

number of iterations that it has performed is less

than or equal to the maximal number of iterations

allowed. Otherwise, it is satisfied and it does not

anything.

4.2.3 Interface Agent

The Interface agent acquaintances are composed of

all the agents existing in the system. This agent has

no static knowledge. However, its dynamic

knowledge is composed of the list of the best

solutions encountered by the Resource agents and

the global best solution and its cost.

It is satisfied when all the other agents are

satisfied and in this case it delivers the best solution

found to the user. Otherwise, it is unsatisfied and it

does nothing.

In the remaining of this paper, we present the

Multi-Agent global dynamic in the two cases of

initial solution and optimal solution determination.

5 MULTI-AGENT GLOBAL

DYNAMIC

In this section, we describe the global dynamic of

both Multi-Agent systems proposed for the FJSP.

Two main phases compose this global dynamic:

initial solution determination phase and optimisation

phase based on TS. The former is similar in both

models, whereas the latter differs from the

centralized version to the decentralized one. The

following section describes the initial solution

generation process used for generating the initial

solution from which the optimisation starts. Next,

the optimisation processes of both models will be

described.

CENTRALIZED AND DECENTRALIZED OPTIMISATION TECHNIQUES FOR THE FLEXIBLE JOB SHOP

SCHEDULING PROBLEM

5.1 Initial solution determination

phase

The initial solution is the result of agent cooperation.

Initially, the Interface agent creates the different Job

and Resource agents and sends the message

“Determine_Initial_Allocation(Jk)” to Job agents in

order to find an initial allocation for all their

operations. The job agent selects, consequently, the

less loaded resource among the potential resources

and a start time d such thatFor the first operation of a

job (according to the process routing) d is equal to

the release date of the job. Otherwise, d is equal to

the finish time of its precedent operation.

Such an initial allocation satisfies all precedence

and temporal constraints. However, it remains to

verify the disjunctive constraints. Each time an

operation is assigned to a resource, its Job agent

informs the concerned Resource agent through the

message “Operation_allocated(R

, O

, d)”. At the

receipt of this message, the Resource agent R

checks

its satisfaction. In the case it is unsatisfied, i.e. there

is an overlapping conflict between this operation and

another operation that has been already affected to it,

it tries to find another satisfying location on it which

start time d

is the closest possible to d. If such a

location exists, then it informs the Job agent through

the message “Operation_modified(J

, O

, d

)”.

Otherwise, it ejects the operation and sends it to its

Job agent in order to search for another location

through the message “Operation_refused(J

, O

)”.

At this moment, the Job agent sends the operation to

another potential resource.

The process above-mentioned will be repeated as

many times as the operation is not yet assigned and

for a predefined number of iterations. Once this

threshold is exceeded, namely the Job agent has not

found any location on a potential resource, it will

request one of the possible resources to create a

location through the message “Create_location(R

)”. Such a location must satisfy all problem

constraints. Similarly, if the Resource agent fails in

creating such a location, it ejects the operation and

sends it to its Job agent to contact another Resource

agent, and so on. This process stops when a second

predefined threshold has been exceeded.

5.2 Optimisation process

5.2.1 Centralized Optimisation Process

As mentioned before, in this first approach the core

of the TS is implanted on the Interface agent. The

latter generates the neighbourhood of the current

solution and then chooses the best non-tabu move

contained in it using its evaluation mechanism. This

best move is then sent to the other agents in order to

generate the feasible solution obtained by applying

this move on the current solution, and so on.

In the following, we describe the parameters of

the TS used in this model.

Neighbourhood function

A tabu search-based approach complexity depends

essentially on (1) the current solution neighbourhood

size and on (2) the evaluation scheme of this

neighbourhood with which the best solution will be

determined. It seems then interesting to reduce the

size of the neighbourhood in order to reduce

problem complexity.

To present the neighbourhood function of the

centralized optimisation process, we need first

define the notion of critical path. A critical path of a

solution is the path which length is equal to the

schedule one and that is composed of operations

related to by either a precedence constraint, or a

disjunctive constraint.

A critical operation is an operation which

belongs to a critical path. The neighbourhood of a

solution is obtained by two types of moves: Switch

of two adjacent critical operations achieved by the

same resource or migration of a critical operation on

another potential resource.

Neighbourhood evaluation

The best non-tabu neighbour belonging to the

current solution neighbourhood will be selected for

the next iteration. Hence, all neighbours must be

evaluated in order to determine the best one.

However, a global evaluation, i.e. computation of all

start times of all operations, of each neighbour will

need a considerable time. For this reason, only a

subset of operations will be taken into account and

to which start times will be redefined. These

operations are effectively concerned by the move

executed.

Optimisation process

At the end of the first phase detailed earlier, the

Interface agent receives the initial solution and

launches then the second phase based on TS. The

following algorithm presents the core of the

optimisation process implanted in the Interface

agent.

1. tabu_list

←

∅

2. nb_iter

←

3. current_sol

←

initial_solution

4. best_sol

←

current_sol

5. While nb_iter <= nb_iter_max do

6. iter_diversif

←

7. While iter_diversif<=iter_max_diversif &

nb_iter <= nb_iter_max do

8. path

←

critical_path (current_sol)

9. neighbH

←

determine_ neighbH(path)

ICINCO 2005 - ROBOTICS AND AUTOMATION

10. best_N

←

determine_best_N(neighbH)

11. tabu_list

←

add_in_tabu_list (best_N)

12. current_sol

←

perform_move

(current_sol,best_N)

13. if cost(current_sol)<cost(best_sol) then

14. best_sol

←

current_sol

15. iter_diversif

←

End if

16. nb_iter

←

nb_iter +1

17. iter_diversif

←

iter_diversif +1

End while

18.

Diversification

End while

Once the best neighbour among the

neighbourhood of the current solution has been

chosen, the Interface agent sends the operation

concerned by the move chosen to its Job agent in

order to inform it about the move to perform. At the

receipt of this operation, the Job agent sends it to the

Resource agent R

in order to find a location starting

at a date d. The date d is equal to the finish time of

the predecessor of O

. The same process involving

the cooperation between the agents, described in the

first phase, will then be repeated in the case that this

assignment leads to a conflict on resource R

otherwise, no changes are made.

When the number of iterations between two best

solutions exceeds a predefined threshold

“iter_max_diversif”, a diversification phase is

performed. The latter consists in varying the search

in order to explore new regions of the search space.

In our approach, such a phase is characterized by

replacing some operations selected randomly. An

operation is replaced on one of its potential

resources selected also randomly.

5.2.2 Decentralized Optimisation Process

In this approach, we have distributed the

optimisation process among the Resource agents.

Each Resource agent will, from now on, have its

own optimisation process and its individual

parameters of its TS. The Resource agents will send

mutually the best solutions encountered in order to

help each other to diversify their search process.

In the following, we describe the parameters of

the TS used in this model.

Neighbourhood function

The neighbourhood of a current solution in a

Resource agent is obtained only by switching two

operations achieved by this resource or by

transferring an operation currently affected on this

resource to another potential resource.

Neighbourhood evaluation

The best non-tabu neighbour belonging to the

current solution neighbourhood will be selected for

the next iteration. The subset of operations that will

be concerned and to which start times will be

recomputed is the same as defined for the first

version.

Optimisation process

Once the initial solution has been determined, the

Interface agent sends it to each Resource agent in

order to start its local optimisation process. The

Resource agent determines then, the neighbourhood

of the current solution. After evaluating the

neighbourhood, the Resource agent chooses the best

non-tabu neighbour (best_N). When no non-tabu

neighbour exists, the Resource agent continues its

process from a solution already sent by another

Resource agent and which has been stored in the list

of best solutions (list_best_solutions). In the case

that the latter is empty, a diversification phase will

take place. When the neighbour is chosen, the move

will be accomplished and the new solution will be

obtained. In the case that the new current solution

improves the best solution (best_sol) encountered so

far, the Resource agent sends a message to the other

Resource agents in order to add this solution to their

lists of best solutions. When the number of iterations

between two best solutions exceeds a predefined

threshold “iter_max_diversif”, a diversification

phase is performed. The following algorithm

presents the core of the optimisation process

implanted in the Resource agent.

1. tabu_list

←

∅

2. nb_iter

←

3. current_sol

←

initial_sol

4. best_sol

←

current_sol

5. While nb_iter <= nb_iter_max do

6. iter_diversif

←

7. While iter_diversif <= iter_max_diversif

& nb_iter <= nb_iter_max do

8. neighbH

←

determine_neighbH(current_sol)

9. best_N

←

determine_best_N(neighbH)

10. While best_N = nil

& list_best_sol is not empty do

11. alea

←

random_selection(list_best_sol)

12. neighbH

←

neighbH

∪

{alea}

13. tabu_list

←

∅

End while

14. If best_N <> nil then

15. tabu_list

←

add_in_list_tabu (best_N)

16. current_sol

←

perform_move

(current_sol, best_N)

17. If cost(current_sol) < cost (best_sol) then

18. best_sol

←

current_sol

19. iter_diversif

←

20. loop on Resource agents R

21. send_best_sol(R

,best_sol)

End loop

End if

Else

CENTRALIZED AND DECENTRALIZED OPTIMISATION TECHNIQUES FOR THE FLEXIBLE JOB SHOP

SCHEDULING PROBLEM

22. current_sol

←

Diversification(current_sol)

23. iter_diversif

←

24. tabu_list

←

∅

End if

25. nb_iter

←

nb_iter+1

26. iter_diversif

←

iter_diversif+1

End while

27. current_sol

←

Diversification(current_sol)

28. iter_diversif

←

29. tabu_list

←

∅

End while

6 EXPERIMENTATION

Some experiments have been made on various

benchmarks defined by (Chambers & Barnes, 1996),

(Dauzerre & Paulli, 1997), etc. These benchmarks

have a number of jobs varying in the set {10, 15,

20}, the number of resources in the range [5, 20], the

number of operations per job in the range [5, 25] and

the number of potential resources per operation in

the range [1,3]. Consequently, the benchmarks

considered have a total number of operations

ranging in [50, 500].

The parameters used in the TS of each Resource

agent are the following:

− Tabu list size varying in {8,10,15,20,30} for the

centralized process and in {8,10} for the

distributed one.

− Total number of iterations nb_iter_max fixed to

1000 in the centralized approach and to 300 in the

distributed one.

− Number of iterations between two diversification

phases varying in {250,300,350} for the

centralized process and in {20,40} for the

distributed one.

Table 1 presents the results obtained by both

approaches for some instances among the

benchmarks mentioned earlier as same as the lower

and the upper bounds (LB and UB) presented in the

literature for the same instances.

Table 1: Results for Lawrence & al. instances

Benchmark LB UB Centralized

process

Distributed

process

la01 609 609 620 662

la02 655 655 666 704

la03 550 554 575 596

la04 568 568 581 675

la05 503 503 503 541

7 CONCLUSION

In this article, we have presented two Multi-Agent

approaches for solving the FJSP. These approaches

are based on the TS. The Multi-Agent systems

proposed are composed of three agent classes: Job

agents, Resource agents and an Interface agent. In

the first approach, the core of the TS is implanted in

the Interface agent and the other agents cooperate in

order to generate a feasible solution from the best

neighbour of the current solution. In the second

approach, each Resource agent has its own TS and

the Resource agents send mutually their best

solutions encountered. Some experiments have been

made on a plenty of benchmarks. The results in both

approaches show that the solution provided is close

to a range defined by the lower and the upper

bounds given in the literature.

REFERENCES

Brucker, P. & Neyer, J. , 1998. Tabu-search for the multi-

mode job-shop problem. OR Spektrum 20, 21-28.

Chambers, J.B. and Barnes, J.W., 1996. Flexible Job Shop

scheduling by tabu search. Graduate program in

Operations Research and Industrial Engineering, The

university of Texas at Austin, Technical Report series,

ORP96-09.

Dauzerre-Peres, S. and Paulli, J., 1997. An integrated

approach for modeling and solving the general multi-

processor job shop scheduling problem using tabu

search. Annals of Operations Research 70: 281-306.

Glover F., 1986. Future paths for Integer Programming

and Links to Artificial Intelligence,.Computers and

Operations Research, 5:533-549.

Mastrolilli M., Gambardella L.M., 2000. Effective

Neighborhood Functions for the Flexible Job Shop

Problem. Journal of Scheduling, Volume 3, Issue 1.

Pages:3-20.

Nuijten W., Aarts E., 1996. A computational study of

Constraint Satisfaction for multiple capacitated Job

Shop scheduling. European Journal of Operations

Research, 90(2): 269-284.

ICINCO 2005 - ROBOTICS AND AUTOMATION