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ScienceDirect Procedia Computer Science (2016)81–90 000–000 Procedia Computer Science 11200 Procedia Computer Science 00(2017) (2016) 000–000

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21th 21th International International Conference Conference on on Knowledge Knowledge Based Based and and Intelligent Intelligent Information Information and and Engineering Engineering Systems, KES2017, 6-8 September 2017, Marseille, France Systems, KES2017, 6-8 September 2017, Marseille, France

Multi Multi Agent Agent model model based based on on Chemical Chemical Reaction Reaction Optimization Optimization with with Greedy Greedy algorithm algorithm for for Flexible Flexible Job Job shop shop Scheduling Scheduling Problem Problem a,d,∗ b,d c,d Bilel Bilel Marzouki Marzoukia,d,∗,, Olfa Olfa Belkahla Belkahla Driss Drissb,d ,, Khaled Khaled Gh´ Ghéedira dirac,d a Ecole Nationale des Sciences de l’Informatique, Universit´ e de Manouba, Tunisia a Ecole Nationale des Sciences de l’Informatique, Universit´ e de Manouba, Tunisia b Ecole Sup´ erieure de Commerce de Tunis, Université de Manouba, Tunisia b Ecole Sup´ erieure de Commerce de Tunis, Université de Manouba, Tunisia c Institut Sup´ erieur de Gestion, Université de Tunis, Tunisia c Institut Sup´ erieur de Gestion, Université de Tunis, Tunisia d COSMOS Laboratory, Universit´ e de Manouba, Tunisia d COSMOS Laboratory, Universit´ e de Manouba, Tunisia

Abstract Abstract Scheduling Scheduling in in production production systems systems consists consists in in assigning assigning operations operations on on aa set set of of available available resources resources in in order order to to achieve achieve defined defined objectives.The Flexible Job shop Scheduling Problem (FJSP) is one of the scheduling problems and also an extension objectives.The Flexible Job shop Scheduling Problem (FJSP) is one of the scheduling problems and also an extension of of classical classical Job Job shop shop Scheduling Scheduling Problem Problem (JSP) (JSP) such such that that each each operation operation can can be be processed processed on on different different machine machine and and its its processing processing time time depends depends on the used machine. The FJSP is classified, as most of scheduling problems, NP-Hard in complexity theory and can be on the used machine. The FJSP is classified, as most of scheduling problems, NP-Hard in complexity theory and can be decomposed decomposed into into two two sub-problems: sub-problems: aa routing routing sub-problem, sub-problem, which which consists consists of of assigning assigning each each operation operation to to aa machine machine out out of of aa set set of of alternative alternative machines, machines, and and aa scheduling scheduling sub-problem, sub-problem, which which consists consists of of sequencing sequencing the the assigned assigned operations operations on on all all selected selected machines machines in in order order to attain a feasible schedule with optimized objectives. In this paper, we propose a decentralized model named Multi Agent to attain a feasible schedule with optimized objectives. In this paper, we propose a decentralized model named Multi Agent model model based based on on Chemical Chemical Reaction Reaction Optimization Optimization with with Greedy Greedy algorithm algorithm (MACROG−FJSP) (MACROG−FJSP) to to solve solve the the FJSP FJSP in in order order to to minimize minimize the the maximum maximum completion completion time time (Makespan). (Makespan). Experiments Experiments are are performed performed on on well well known known benchmark benchmark instances instances proposed proposed in in the the literature literature which which are are Fattahi, Fattahi, Kacem, Kacem, Brandimarte Brandimarte and and Hurink Hurink instances instances to to evaluate evaluate the the performance performance of of our our model. model. c 2016 The Authors. Published by Elsevier B.V. c 2017 2016The TheAuthors. Authors.Published Publishedby byElsevier ElsevierB.V. B.V. © Peer-review under responsibility International. Peer-review under under responsibility responsibility of of KES KES International International. Peer-review system, Optimization, Manufacturing, Scheduling, Flexible Job shop Scheduling Problem, Chemical Reaction Keywords: Keywords: Multi-Agent Multi-Agent system, Optimization, Manufacturing, Scheduling, Flexible Job shop Scheduling Problem, Chemical Reaction Optimization metaheuristic, Greedy algorithm. Optimization metaheuristic, Greedy algorithm.

1. 1. Introduction Introduction Scheduling Scheduling in in production production systems systems consists consists in in assigning assigning operations operations on on aa set set of of available available resources resources in in order order to to achieve achieve defined objectives. The Job shop Scheduling Problem (JSP) is considered as one of the most difficult scheduling defined objectives. The Job shop Scheduling Problem (JSP) is considered as one of the most difficult scheduling probproblems. lems. This This problem problem consists consists on on assigning assigning aa set set of of operations operations on on aa set set of of machines machines such such as as each each operation operation must must be be processed processed on on one one machine. machine. The The Flexible Flexible Job Job shop shop Scheduling Scheduling Problem Problem (FJSP) (FJSP) is is an an extension extension of of the the classical classical Job Job shop shop Scheduling Scheduling Problem Problem (JSP) (JSP) where where each each operation operation can can be be processed processed on on different different machines machines and and its its processing processing time time depends on the used machine, thus FJSP is harder than JSP. FJSP is classified, as most of scheduling depends on the used machine, thus FJSP is harder than JSP. FJSP is classified, as most of scheduling problems, problems, NPNP∗ ∗

Corresponding author. Tel.: +216 21 694 840 Corresponding author. Tel.: +216 21 694 840 E-mail address: [email protected], [email protected], [email protected] E-mail address: [email protected], [email protected], [email protected]

c 2016 The Authors. Published by Elsevier B.V. 1877-0509 c 2016 1877-0509 The Authors. Published by Elsevier B.V. Peer-review under responsibility of KES International. Peer-review©under of Published KES International. 1877-0509 2017responsibility The Authors. by Elsevier B.V. Peer-review under responsibility of KES International 10.1016/j.procs.2017.08.174

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Bilel Marzouki et al. / Procedia Computer Science 112 (2017) 81–90

Hard in complexity theory1 . We have done a study on the existing literature which has studied the FJSP and we cite them in the following. The flexible job shop scheduling problem has been studied for the first time in 2 , by authors they have developed a polynomial algorithm to solve FJSP. The metaheuristics are the most used methods to solve the FJSP. Several researches are made based on tabu search 3 , which resolved the resource allocation problem using the rules of priority. The durations are varied resources functions, the assignment problem is solved then we get the classical job shop scheduling problem which is solved by a tabu search method. The neighborhood function used allows to permute two critical operations. Then a reallocation of these critical operations are performed at predefined time intervals. Tabu search was also used in 4,5,6 to solve the FJSP. A genetic algorithm was used to solve the FJSP in 7,8 . A hybrid Genetic Algorithm (GA) and a Variable Neighborhood Descent (VND) for FJSP was introduced in 9 where the GA used two vectors to represent a solution and the disjunctive graph to calculate it. Then, a VND was applied to improve the GA final individuals. A genetic algorithm approach was presented also in 10 to solve the FJSP and its objective is to minimize the makespan, an assignment algorithm of 7 was used to generate initial population. For the selection phase, individual presenting the highest fitness is selected. For the crossover phase, a two-point crossover is applied on two chromosomes. For the mutation phase, a random operation is selected and then recent machine for selected operation is interchanged with the machine that has shortest processing time among the alternative machine set. The proposed approach is tested on benchmark instances of 11 . The obtained results were compared with the results obtained by other approaches show that this approach surpasses other known algorithms for the same problem. A mathematical model and heuristic approaches (integrated and hierarchical) was proposed in 11 for FJSP to solve real size problems. For the integrated approach, they used an algorithm that uses tabu search called ITS (Integrated approach with Tabu Search heuristic) and another algorithm that uses simulated annealing named ISA (Integrated approach with Simulated Annealing heuristic) for the allocation and sequencing problems consecutively. An approach named AIA was proposed in 12 for the FJSP and based on natural immune system. A novel hybrid harmony search (HHS) algorithm based on the integrated approach was presented in 13 for solving the flexible job shop scheduling problem with the criterion to minimize makespan. A hybrid artificial bee colony algorithm was presented in 14 for solving the FJSP with the criterion to minimize the makespan. In the proposed algorithm, first, several dispatching rules and the harmony search algorithm are used in creating the initial solutions. Thereafter, one of the two search techniques is randomly selected with a probability that is proportional to their fitness values. The selected search technique is applied to the initial solution to explore its neighborhood. If a premature convergence to a local optimum happens, the simulated annealing algorithm will be employed to escape from the local optimum. Otherwise, the filter and fan algorithm are utilized. Finally, the crossover operation is presented to enhance the exploitation capability. Distributed methods are also used; So we find the approach of 15 to solve the FJSP and it is based on a tabu search method. A multi-agent approach has been introduced in 16 based on the combination of genetic algorithm and tabu search. Firstly, a Scheduler Agent applies a Neighborhood-based Genetic Algorithm (NGA) for a global exploration of the search space. Secondly, a Cluster agent set used a local search technique to guide the research in promising regions. In 17 , a multi-agent model based on the Chemical Reaction Optimization metaheuristic (CRO) was proposed for solving the flexible job shop scheduling problem with the criteria to minimize the maximum completion time (makespan). This model presented two classes of agents: Interface Agent in charge of creating the initial solutions and n Scheduler Agents responsible of global optimization process where n is the number of jobs of the problem. Experiments are performed on benchmark instances of 7 and 11 to evaluate the performance of this model. A new metaheuristic hybridization approach that is a hybrid of genetic algorithm and tabu search based on clustered holonic multi-agent model was presented in 18,19 for the flexible job shop scheduling problem with tansportation times and many robots (FJSPT-MR). To measure its performance, numerical tests are made using three well known data sets in the literature of the FJSPT-MR, and where new upper bounds are found showing the effectiveness of the presented approach. In 20 , a chemical reaction optimization metaheuristic hybridized with tabu search in multi-agent system was proposed to solve the FJSP in order to minimize the maximum completion time (makespan). This model presented two classes of agents: Interface Agent in charge of creating the initial solutions and n Scheduler Agents responsible of global optimization process based on hybridization of Chemical Reaction Optimization metaheuristic and Tabu Search. Experiments are performed on benchmark instances of 7,3,4 to evaluate the performance of this model. So, distributed methods used for the combinatorial problems and especially for the flexible job shop scheduling problem show their effectiveness. More, CRO is proposed by 21 to optimize combinatorial problems. Due to its ability to escape from the local optima, it has been


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applied for solving many scheduling problems such as grid scheduling in 22 , Multi-Factory Flow Shop Problem in 23 , multi-objective optimization of Flexible Job shop scheduling with maintenance activity constraints in 24 where three minimization objectives are considered simultaneously: the maximum completion time, the total workload of machines and the workload of the critical machine, etc. Experimental comparisons demonstrated that CRO is one of the most powerful optimization algorithms. Moreover, the greedy algorithm is very used to solve the scheduling problems. So we find the work of 25,26,27 which used the greedy algorithm in their work and we see that this heuristic gives better results knowing that the greedy algorithm is known by its speed in execution and by a good exploitation of search space. That’s why, we propose in this paper a multi-agent model based on Chemical Reaction Optimization metaheuristic combined with Greedy algorithm to solve the FJSP in order to minimize the maximum completion time (makespan). We are interested in this work to solve the problem of scheduling jobs in FJSP in order to minimize the maximum completion time (Makespan) using a multi-agent model based on Chemical Reaction Optimization metaheuristic with Greedy algorithm. The remainder of this paper is organized as follows. In the next section, we present the details of the FJSP, the CRO metaheuristic is defined in section 3, we describe our proposed model namely a Multi-Agent model based on Chemical Reaction Optimization with Greedy algorithm for Flexible Job shop Scheduling Problem (MACROG−FJSP) in section 4, we give some experimental results in section 5 and a conclusion and perspectives are given in section 6. 2. Problem Formulation The FJSP can be decomposed into two sub-problems: a routing sub-problem, which consists of assigning each operation to a machine out of a set of alternative machines, and a scheduling sub-problem, which consists of sequencing the assigned operations on all selected machines in order to attain a feasible schedule with optimized objectives. There are two kinds of FJSP; total FJSP problem (TFJSP) and partial FJSP (PFJSP). For the TFJSP, each job can be operated on every machine from the set m; for the PFJSP, there is a problem constraint for the operating process, namely, one operation of a job cannot be processed by all the machines. The Flexible Job Shop Problem can be formulated as follows 28 : -

A set of n jobs to be performed on m machines Mk , k = 1, 2, ..., m, Each job i consists of a sequence of N J operations Oi j , i = 1, 2, . . . n ; j = 1, 2, ... , N J , Execution of each operation Oi j requires a resource from a set of alternative machines. Each machine can perform only one operation at a time, Assigning an operation Oi j with a machine Mk causes the occupation of this machine throughout the execution time of the operation, denoted by pi jk - Operation preemption is not allowed.

To explain the FJSP, a sample problem of three jobs and three machines is shown in Table 1, where the numbers present the processing times and the tags ”-” mean that the operation cannot be executed on the corresponding machine. 3. Chemical Reaction Optimization metaheuristic Chemical Reaction Optimization is a meta-heuristic developed by 21 for optimization inspired by the nature of chemical reactions. Chemical modification of a molecule is started by a collision. There are two types of collision: uni-molecular and inter-molecular collision. The first type describes the situation when the molecule hits on some external substances; the second type is where the molecule collides with other molecules. The corresponding reaction change is called an elementary reaction; we consider four kinds of elementary reactions: On-wall collision ineffective, Decomposition, Inter-molecular collision ineffective and Synthesis. The two collisions ineffective implement local search (intensification) whereas decomposition and synthesis provide the diversification effect. The mixture between intensification and diversification make an effective search of the global minimum in the solution space. Each molecule has several attributes:


84 Table 1. A simple instance of the FJSP

Job

Operation sequence

Operation O11 O12 O13

M1 5 2 4

M2 6 3

M3 2 -

J1

O11 ,O12 ,O13

J2

O21 ,O22 ,O23

O21 O22 O23

1 2

3 4 3

4 3 5

J3

O31 ,O32 ,O33

O31 O32 O33

3

3 3 2

2 2

- The molecular structure (ω) captures a solution of the problem; it may be a number, a vector or a matrix. - PE: Potential energy is defined as the objective function PE ω = f (ω). - KE: Kinetic energy is the positive number and it quantifies the system tolerance to accept a worse solution than the existing one. - NumHit: is a record of the total number of hits (collisions) a molecule has taken. - MinStruct: the molecular structure with the minimum PE. - Min PE: when a molecule achieved its MinStruct. - MinHit: is the number of hits when a molecule performs MinStruct. There are four basic types of reactions, each of which occurs in each iteration of CRO, they work to manipulate solutions (explore the solution space) and redistribute the energy among the molecules and the buffer. On-wall ineffective collision represents the situation when the molecule collides with a wall of the container. Decomposition refers to the situation in which a molecule hits a wall and breaks into several parts. The idea of decomposition is to allow the system to explore other areas of the solution space after enough local searches by the ineffective collisions. Intermolecular ineffective collision represents the situation when the molecules collide with each other. Synthesis happens when two molecules hit against each other and fuse together. One molecule is produced. This function is the opposite of decomposition. The idea behind Synthesis function is the diversification of solutions. 4. Multi Agent model based on Chemical Reaction Optimization metaheuristic with Greedy algorithm for FJSP We propose a multi-agent model based on Chemical Reaction Optimization and Greedy algorithm to solve the Flexible Job shop Scheduling Problem named MACROG-FJSP ”Multi-Agent model based on Chemical Reaction Optimization metaheuristic with Greedy algorithm for Flexible Job shop Scheduling Problem” to minimize the makespan or the maximum completion time (Cmax) while satisfying the various constraints such as the temporal constraints (precedence constraints) and the resource constraints. The architecture of our model consists of two classes of agents: an Interface Agent who is responsible of creating the initial solutions and n Scheduler Agents where n is the number of jobs in the problem who are responsible of the optimization phase based on CRO and greedy algorithm. Nextly, we describe in detail each class of agents. 4.1. Interface Agent This agent is considered as an interface between the user and the program. It launches the program and creates the n Scheduler Agents where n corresponds to the number of jobs, and then it generates initial solutions according to a predefined number of populations. It supplies the necessary information for every Scheduler Agent to begin the


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global optimization phase; which is based on the Chemical Reaction Optimization metaheuristic (CRO); then send the information for every Scheduler Agent to begin the greedy algorithm, finally it posts the final result. The Interface Agent has two types of knowledge. The static knowledge such as: -

The jobs and the operations to realize so its durations in the various resources. The initial solutions created by the Interface Agent itself. The stopping criterion for the global optimization phase defined as ”nb-iteration”. The necessary variables for the execution of our program. The found solutions after the phase of optimization. The best found solution after the phase of optimization. β the threshold of diversification. α the maximal number of iteration allowed between two successive improvements

And the dynamic knowledge as: -

The current initial solution and its makespan (Cmax) NumHit: the number of iteration for the on wall function MinHit: the value of the iteration which offers us an improved solution KE: the kinetic energy.

4.2. Scheduler Agent This class of agent aims to make the phase of optimization based on the CRO metaheuristic then the execution of greedy algorithm. The static knowledge of the Scheduler Agent are cited as following: - The random solution chosen by the Interface Agent - The job’s operations associated to the agent and its various durations. In the other hand, the dynamic knowledge of the Scheduler Agent are: - The current solution and its makespan. - The inactive time interval on every resource. 4.3. The optimization phase 4.3.1. The global optimization phase The Interface Agent sends one/two initial solutions to Scheduler Agents for making the global optimization phase based on the CRO metaheuristic and greedy algorithm, see algorithm 1. 4.3.2. The Diversification Techniques To guarantee better resolution of flexible job shop problem, we need to explore more search space. That’s why diversification phase should be performed. When a better solution cannot be found after a certain threshold α, the diversification phase is triggered. When the number of iterations is increased without a better solution after a certain threshold, it means that the best found solution was not replaced by one of its neighbors for a certain time, which is a sign that our method is probably trapped in a local optimum. In this situation, the decomposition must be performed with another initial solution and the values of MinHit and NumHit are reset. The other way to set the diversification phase is to execute synthesis function; after each execution of the ”On Wall Function”, the value of KE is reduced by the value of KElossrate, so after some iteration KE becomes lower than β so the Synthesis function to be executed. The Decomposition. The aim of this function is to find two new solutions from the initial solution. This treatment is repeated twice and for each iteration we obtain a new solution, see algorithm 2. Synthesis. The principle of this function is to find a solution from both random solutions, this function is the opposite

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Algorithm 1 Main Algorithm 1.Input:n,m,nbop[],dure[][],α,β,MinHit,NumHit,popsize, KElossrate, initialKE,Molecoll 2.Generate-initial-solution 3.While the stopping criterion not met { 4. if (b>Molecoll) and (NumHit-MinHit> α) then { 5. Trigger Decomposition NumHit =0; MinHit=0; } 6. else if (b>Molecoll) and (NumHit-MinHit ≤ α) then { 7. Trigger On wall ineffective collision 8. NumHit++ ; if (f(s’1) is the best) then MinHit++ } 9. if (b ≤ Molecoll) and (Ke ≤β) then { 10. Trigger Synthesis } 11. else if (b ≤ Molecoll) and (Ke >β) { 12. Trigger inter-molecular ; KE=KE-KElossrate} 13.} End while 14.while the stopping criterion not met { 15. send (Interface Agent,initial-solution,Scheduler Agent) 16. Trigger Greedy algorithm 17.} End while Algorithm 2 Decomposition 1.for ( int i = 1; i end-date (i-1) 14. place(i, r) } 15. Foreach not-affected-operations 16. place (i,r) with satisfying precedence constraint 17. place (i,r)/ i job which corresponds to the current Scheduler Agent 18. } of the decomposition function, see algorithm 3. 4.3.3. The Intensification Techniques The intensification phase is one of the most important phases to better exploit the search space by executing the ”On Wall Function” and ”Inter Ineffective Collision”. On-wall Ineffective Collision. The idea of this function is to find one new solution from the initial solution. Every Scheduler Agent receives the random solution from the Interface Agent and tries to move the operations of the job associated to it. The treatment of this function is the same for the decomposition.


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Algorithm Algorithm 3 Synthesis 3 Synthesis 1.If1.If (the (the Scheduler Scheduler Agent Agent belongs belongs to to thethe first first half half of of thethe solution) solution) then then { { 2. 2. aff=same-Affectations aff=same-Affectations (Scheduler (Scheduler Agent,s1) Agent,s1) 3. 3. send send (Scheduler (Scheduler Agent, Agent, aff,aff, following following Scheduler Scheduler Agent) Agent) } } 4.If4.If (the (the Scheduler Scheduler Agent Agent belongs belongs to to thethe second second half half of of thethe solution) solution) then then { { 5. 5. Foreach Foreach i ini in operations operations 6. 6. if (iif = (i 1) = 1) then then 7. 7. attribute1(i,r1,aff) attribute1(i,r1,aff) 8. 8. if (iif (i 1) 1) then then 9. 9. attribute2(i,r2,aff) attribute2(i,r2,aff) 10.10. send send (Scheduler (Scheduler Agent, Agent, aff,aff, following following Scheduler Scheduler Agent) Agent) } } Table Table 2. The 2. The parameters parameters of the of the CRO CRO

Parameter Parameter Molecoll Molecoll b b a a NumHit NumHit MinHit MinHit buffer buffer InitialKE InitialKE KElossrate KElossrate beta beta popsize popsize nb-iteration nb-iteration Value [0..1] 3 3 0 0 1000 1000 1000 Value 0.50.5 [0..1] 0 0 0 0 200200 1212 2525 1000 Inter-molecular Inter-molecular Ineffective Ineffective Collision. Collision.The The idea idea of of thisthis function function is is to to find find twotwo new new solutions solutions from from both both random random solutions. solutions. The The first first new new solution solution is composed is composed of of thethe first first part part of of thethe first first solution solution andand thethe second second part part of of thethe second second solution. solution. However, However, thethe second second new new solution solution is composed is composed of of thethe first first part part of of thethe second second solution solution andand thethe second second part part of of thethe first first solution. solution. The The treatment treatment of of thisthis function function is the is the same same forfor thethe synthesis. synthesis. Next, Next, wewe addadd a new a new phase phase to to better better explore explore thethe search search space space using using greedy greedy algorithm. algorithm. In In thisthis algorithm, algorithm, wewe start start to to build build thethe solution solution from from oneone operation operation to to another. another. After After adding adding thethe operation operation to to a defined a defined position position of of thethe current current solution, solution, wewe check check thethe different different constraints. constraints. If all If all constraints constraints areare satisfied, satisfied, wewe proceed proceed to to thethe next next operation. operation. Else, Else, wewe delete delete thisthis operation operation in in thethe current current position position andand addadd it to it to another another position position thatthat respects respects thethe different different constraints. constraints. This This treatment treatment is repeated is repeated until until allall operations operations areare affected affected andand wewe find find a feasible a feasible solution. solution. SeeSee algorithm algorithm 4. 4. Algorithm 4 Greedy algorithm Algorithm 4 Greedy algorithm Input : solution ; Set = {operation} 1. 1. Input : solution ω ;ωSet = {operation} while ( Not-empty(Set) and Not-solution.complete 2. 2. while ( Not-empty(Set) and Not-solution.complete () )() ) select (x,Set) ∈ Set 3. 3. select (x,Set) x ∈x Set if solution.add-possible (x)=true then solution.add(x) ∈ Set 4. 4. if solution.add-possible (x)=true then solution.add(x) x ∈x Set else Set.del(x) ∈ Set 5. 5. else Set.del(x) x ∈x Set end while. 6. 6. end while. Output : solution 7. 7. Output : solution ω. ω.

Computational Results 5. 5. Computational Results section, performances proposed model evaluated tested. have made series experIn In thisthis section, thethe performances of of ourour proposed model areare evaluated andand tested. WeWe have made series of of exper3,4,7,11 3,4,7,11 iments benchmark instances proposed literature, namely instances . These . These benchmarks benchmarks represent represent iments onon benchmark instances proposed in in thethe literature, namely instances of of problems problems thatthat thethe number number of of jobjob varies varies in in [2..20] [2..20] andand thethe number number of of resources resources in in [2,15]. [2,15]. AllAll tests tests were were conducted conducted onon a a I3-3110M I3-3110M 2*2.4 2*2.4 GHz GHz with with 4 GB. 4 GB. WeWe useuse thethe Java Java object-oriented object-oriented programming programming language language with with thethe Eclipse Eclipse IDE, IDE, andand forfor thethe development development of of ourour model model MACROG−FJSP, MACROG−FJSP, wewe useuse thethe JADE JADE platform. platform. The The parameters parameters of of thethe CRO CRO areare setset in in table table 2. 2. 5.1.5.1. Results Results onon Kacem Kacem Benchmark Benchmark 7 7 WeWe compared compared thethe results results obtained obtained byby ourour model model (MACROG−FJSP) (MACROG−FJSP) onon benchmark benchmark instances instances of of with with thethe ap-ap17 17 called Multi Multi Agent Agent model model based based onon Chemical Chemical Reaction Reaction Optimization Optimization forfor Flexible Flexible JobJob Shop Shop Problem Problem proach proach of of called


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Table 3. Results on instances of 7 in term of makespan

Name Instance 1 Instance 2 Instance 3 Instance 4

N 4 10 8 10

M 5 7 8 10

UB 11 15 14 7

MACRO−FJSP 16 25 38 21

MACROG−FJSP 11* 20 14* 19


Name SFJS1 SFJS2 SFJS3 SFJS4 SFJS5 SFJS6 SFJS7 SFJS8 SFJS9 SFJS10 MFJS1 MFJS2 MFJS3 MFJS4 MFJS5 MFJS6 MFJS7 MFJS8 MFJS9 MFJS10

UB 66 107 221 355 119 320 397 253 210 516 470 484 564 684 696 786 1433 1914 2908 4960

AIA 66 107 221 355 119 320 397 253 210 516 468 448 468 554 527 635 879 884 1088 1267

ISA 66 107 221 355 119 320 397 253 215 516 488 478 599 703 674 856 1066 1328 1148 1546

ITS 66 107 221 355 137 320 397 253 215 617 548 457 606 870 729 816 1048 1220 1124 1737

MACRO−FJSP 66* 107* 221* 355* 119* 320* 397* 253* 210* 516* 477 464 578 745 708 836 1465 1934 2965 5223

MACROG−FJSP 66* 107* 221* 355* 119* 320* 397* 253* 210* 516* 470 464 564 745 696 836 1454 1934 2953 5223

(MACRO−FJSP), the asterisk (*) indicates that optimal solution was found, see table 3. The results shows that our model reaches the optimum for 50% of instances. The tests demonstrate that our model MACROG−FJSP and MACRO−FJSP provide the same results in 25% of instances, while our model (MACROG−FJSP) surpasses (MACRO−FJSP) in 75% of instances. 5.2. Results on Fattahi benchmark We also test our model MACROG−FJSP on instances of benchmark of 11 and we compare the results obtained by our model with those obtained by 11 called integrated approach with tabu search heuristic (ITS) and integrated approach with simulated annealing heuristic (ISA) and the approach of 12 based on natural immune system and with model of 17 (MACRO−FJSP). The experiments show that our model reaches the optimum for 14 instances of 20 with a percentage equal to 70% of the instances. The results show that our model MACROG−FJSP and the MACRO−FJSP provide the same results in 75 % of instances. The results show also that our model MACROG−FJSP surpasses ITS approach in 40 % of instances and surpasses ISA approach in 30% of instances and provide the same results as AIA in 50% of instances. The asterisk (*) indicates that optimal solution was found, see table 4. 5.3. Results on Hurink benchmark We also compared the results obtained by our model (MACROG−FJSP) on benchmark instances of 4 with those obtained by 16 called A Holonic Multi agent Model Based on a Combined Genetic Algorithm−Tabu Search for the


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Name la01 la02 la03 la04 la05 la16 la17 la18 la19 la20

UB 609 655 554 568 503 905 707 843 796 864

MATSLO+ 609 655 575 579 503 896 708 845 813 863

GATS+HM 609 655 567 568 503 892 707 843 804 857

MACROG−FJSP 609* 655* 573 579 503* 896 707 867 806 863


Name MK01 MK02 MK03 MK04 MK05 MK06 MK07 MK08 MK09 MK10

UB 42 32 204 81 186 86 157 523 307 197

MATSLO+ 40 32 207 67 188 85 154 523 437 380

GATS+HM 40 27 204 64 173 65 144 523 311 222

MACROG−FJSP 40* 32 204* 64* 179 85 172 552 421 358

Flexible Job Shop Scheduling Problem (GATS+HM) and the approach of 15 called Multi Agent model based on the Tabu search metaheuristic and the Local Optimisation approach (MATSLO+), the asterisk (*) indicates that optimal solution was found, see table 5. The experiments show that our model reaches the optimum for 60% of instances. The tests demonstrate that our model MACROG−FJSP and GATS+HM provide the same results in 40% of instances, while the GATS+HM approach surpasses our model in 60% of instances. The results show also that our model MACROG−FJSP and MATSLO+ provide the same results in 60% of instances, while our model (MACROG−FJSP) is better in term of makespan than MATSLO+ in 30% of instances but MATSLO+ surpasses our model in 10 % of instances. 5.4. Results on Brandimarte benchmark We also compared the results obtained by our model (MACROG−FJSP) on benchmark instances of 3 with those obtained by 16 called GATS+HM and the approach of 15 called MATSLO+, the asterisk (*) indicates that optimal solution was found, see table 6. The tests demonstrate that our model reaches the optimum for 60% of instances. The results show that our model MACROG−FJSP and GATS+HM provide the same results in 30% of instances, while the GATS+HM approach surpasses our model also in 70% of instances. We see also that our model MACROG−FJSP and MATSLO+ provide the same results in 30% of instances, while our model (MACROG−FJSP) is better in term of makespan than MATSLO+ in 50% of instances while the MATSLO+ model surpasses our model in 20% of instances.

6. Conclusion and perspectives In this paper, we study the flexible job shop problem and we propose, to solve this problem, a multi-agent model named MACROG−FJSP ”Multi-Agent model based on Chemical Reaction Optimization metaheuristic with Greedy

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algorithm for Flexible Job shop Scheduling Problem” which consists of a society agent and based on the metaheuristic CRO and Greedy algorithm for the optimization phase. Our model is composed of an Interface Agent and Scheduler Agents, where the Interface Agent is considered as an interface between the user and the program and the Scheduler Agents aim to make the phase of optimization by using the CRO metaheuristic with Greedy algorithm. The results obtained in our work now encourage us to continue the study of certain research lines such as the multi-objective FJSP, we can consider other benchmark instances proposed in the literature and adapt other methods to generate initial solutions such as Variable Neighborhood Search (VNS), Particle Swarm Optimization (PSO), etc. References 1. Garey, E.L., Johnson, D.S., Sethi, R.. The Complexity of Flow-shop and Job-shop scheduling. Mathematics of Operations Research 1976; 1:117-129. 2. Bruker, P., Schlie, R.. 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