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Procedia Computer Science 00 (2018) 000–000 Procedia Computer Science 00 (2018) 000–000 Procedia Computer Science 126 (2018) 1424–1433

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22nd International Conference on Knowledge-Based and 22ndIntelligent International Conference on Knowledge-Based Information & Engineering Systems and Intelligent Information & Engineering Systems

Solving Solving Distributed Distributed and and Flexible Flexible Job Job shop shop Scheduling Scheduling Problem Problem using a Chemical Reaction Optimization metaheuristic using a Chemical Reaction Optimization metaheuristic Bilel Marzoukia,d,∗ , Olfa Belkahla Drissb,d , Khaled Ghédirac,d Bilel aMarzoukia,d,∗, Olfa Belkahla Drissb,d , Khaled Ghédirac,d Ecole Nationale des Sciences de l’Informatique, Université de Manouba, Tunisia

a Ecole Nationale des Sciences de l’Informatique, Universit´ de Manouba, Tunisia b Ecole Sup´ erieure de Commerce de Tunis, Université dee Manouba, Tunisia b Ecole Sup´ erieure de Commerce de Tunis, Universit´ e de Manouba, Tunisia c Institut Supérieur de Gestion, Université de Tunis, Tunisia c Institut Sup´ erieur de Gestion, Universit´ e de Tunis, Tunisia d SOIE-COSMOS Laboratory, Universit´ e de Manouba, Tunisia d SOIE-COSMOS Laboratory, Universit´ e de Manouba, Tunisia

Abstract Abstract The current industrial production environment is characterized by highly competitive markets, where customer requirements and The current industrial production environment characterized bytime highly competitive customer requirements and expectations are becoming stronger in terms ofisquality, cost and to delivery. Themarkets, modern where production systems are generally expectations are becoming stronger in terms of quality, cost and time to delivery. The modern production systems are generally composed of machines that must participate in the manufacture of several types of products simultaneously and efficiently. The composed machines that must participate the manufacture of several types of products simultaneously and efficiently. The Flexible Jobofshop Scheduling Problem (FJSP) in is one of the scheduling problems where each operation can be processed on different Flexible Scheduling Problem (FJSP) is used one ofmachine the scheduling where eachBut operation can beyears, processed different machine Job andshop its processing time depends on the and thisproblems in a single factory. in the recent manyon companies machine its towards processing depends on the used factories machine and thisallows in a single in the recent years, many companies decide toand move thetime decentralization of their which them factory. to gain But advantages towards their customers. So, decide to move towards the decentralization of their factories which allows them to gain advantages towards their customers. So, we are now interested in Distributed and Flexible Job shop Scheduling Problem (DFJSP) where there is a set of geographically we are now interested in Distributed and Flexible Job shop Scheduling Problem (DFJSP) where there is a set of geographically distributed factories in different locations. Each factory contains m machines on which n jobs must be processed. The Distributed distributed different locations. Each factory are contains machines on which mustproblems be processed. Thethey Distributed scheduling factories problemsinand more specifically the DFJSP muchm more complicated thann jobs standard because involve scheduling problems and more specifically the DFJSP are much more complicated than standard problems because they not only the problem of assigning jobs to machines but also the problem of distribution of jobs in different factories. So, the involve DFJSP not only the of assigning jobs to machines also problemproblems, of distribution of jobs in different factories. DFJSP is harder thanproblem the FJSP. The DFJSP is classified, as but most of the scheduling NP-Hard in complexity theory. InSo, thisthe paper, we is harderathan the FJSP. The DFJSP is classified, as most oftoscheduling in complexity In thisProblem paper, we propose Chemical Reaction Optimization metaheuristic solve the problems, DistributedNP-Hard and Flexible Job shoptheory. Scheduling in propose a Chemical Optimization to solve and Flexible shop Scheduling Problem ina order to minimize theReaction maximum completion metaheuristic time (makespan) amongthe allDistributed factories. To evaluate theJob performance of our algorithm, order minimize the timebenchmark (makespan)instances among all To evaluate the performance of our algorithm, a set of to experiments are maximum performedcompletion on well known infactories. the literature. set of experiments are performed on well known benchmark instances in the literature. c 2018 Authors. Published Published by by Elsevier Elsevier Ltd. © 2018 The The Authors. Ltd. c 2018 The Authors. Published by Elsevier Ltd. This This is is an an open open access access article article under under the the CC CC BY-NC-ND BY-NC-ND license license (https://creativecommons.org/licenses/by-nc-nd/4.0/) (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an and openpeer-review access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection Selection peer-review under responsibility of KES International. Selection and peer-review under responsibility of KES International. Keywords: Manufacturing, Scheduling, Distributed and Flexible Job shop Scheduling Problem, Optimization, Chemical Reaction Optimization Keywords: metaheuristic.Manufacturing, Scheduling, Distributed and Flexible Job shop Scheduling Problem, Optimization, Chemical Reaction Optimization metaheuristic.

∗ ∗

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

c 2018 The Authors. Published by Elsevier Ltd. 1877-0509 c 2018 1877-0509 Thearticle Authors. Published by Elsevier Ltd. This is an open access under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) 1877-0509 © 2018 The Authors. Published by Elsevier Ltd. This is an and open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection peer-review under responsibility ofBY-NC-ND KES International. This is an open access article under the CC license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of KES International. Selection and peer-review under responsibility of KES International. 10.1016/j.procs.2018.08.114

Bilel Marzouki et al. / Procedia Computer Science 126 (2018) 1424–1433

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1. Introduction The organization and management of the production contribute in the success of the projects of the world of the company and the research. In this process, the scheduling function aims to allocate tasks to existing resources in limited quantities while satisfying a set of constraints in order to achieve defined objectives. The Job shop Scheduling Problem (JSP) is considered as one of the most difficult scheduling problems. This problem consists in assigning a set of operations on a set of machines such as each operation must be processed on one machine. The Flexible Job shop Scheduling Problem (FJSP) is an extension of the classical Job shop Scheduling Problem (JSP) where each operation can be processed on different machines and its processing time depends on the used machine. This problem is known to be strongly NP-Hard by Garey et al. [1], that is why most methods of resolution of this problem are based on metaheuristics [2,3,4,5,6,7]. The Distributed and Flexible Job shop Scheduling Problem (DFJSP), that we study in this work, is a extension of the FJSP and there is a set of geographically distributed factories in different locations such that each factory contains m machines on which n jobs must be processed. New researches, such as (Lu et al. 2015 [9]; Chang et Liu 2015 [10]) have shown that companies with multiple production centers are more competitive in a globalized economy. With factories located in different geographical areas, companies can benefit from several advantages such as proximity to customers, can respond quickly to market changes, benefit from significant productivity gains from several factors such as the difference in cost of labor or qualification of personnel in certain areas, etc, that’s why such companies with multiple production centers have increased in recent years. The distributed scheduling problems and more specifically the DFJSP are much more complicated than standard problems because they involve not only the problem of assigning jobs to machines but also the problem of distribution of jobs in different factories. So, the DFJSP is harder than the FJSP. The DFJSP is classified, as most of scheduling problems, NP-Hard in complexity theory. In the literature, there are few works that studied the Distributed and Flexible Job shop Scheduling Problem. This new concept is proposed for the first time by Chan et al. 2006 [11], they proposed a solution based on genetic algorithms with dominant genes to solve the DFJSP in order to minimize the Makespan. The idea of dominant genes (DG) is applied and used to identify and save the best genes in the chromosome during evolution. For the crossover function, the authors performed a simple crossover with one-point crossover method and for the mutation, a pair of gene is selected and randomly swapped. A series of tests of this model are performed and comparisons made with other works and the results show the effectiveness of the proposed model. Giovanni and Pezzella in [12] have proposed a model based on genetic algorithms to solve the Distributed and Flexible Job shop Problem in order to minimize the makespan. For the selection phase, a growing sort of makespan is performed on individuals to choose the chromosomes that will be used for the crossover phase which is based on the one-point crossover and two-point crossover method. For the mutation, the authors select a number of gene pairs in the same chromosome and swapping their positions then he change randomly the assignment of jobs to factories. This approach is tested and compared with other existing approaches on different instances of benchmarks in the literature. Liu et al. in [13] have proposed a heuristic based on a constructive procedure to solve the DFJSP in order to minimize maximum completion time. To evaluate this algorithm, a series of experiments is carried out on instances of benchmark of the literature and the results show the efficiency of this algorithm. Chang and Liu in [10] have proposed a hybridization of genetic algorithms to solve the DFJSP in order to minimize the maximum completion time (makespan). For the crossover phase, the authors used three methods, the two-point crossover, precedence-preserving order-based crossover and uniform crossover methods. The authors divided the mutation operation into two parts, exchange two random machines and change the assignment of operations to the machines. Experiments are performed to evaluate the performance of the used approach on benchmark instances in the literature and the results show the effectiveness of this approach. Lu et al. in [9] have proposed an approach based on genetic algorithms to solve the Distributed and Flexible Job shop Scheduling Problem in order to minimize the makespan. To evaluate the performance of this approach, comparisons are made with other existing approaches and the results show the competitiveness of the proposed approach. We find also works in the literature that have studied the distributed Scheduling Problem [14,15,16,17,18,19,30]. Chemical Reaction Optimization (CRO) is proposed by Lam et al. in 2010 [21] to optimize combinatorial problems. Due to its ability to escape from the local optima, it has been applied also for solving many scheduling problems such as grid scheduling [22], Multi-Factory Flow Shop Scheduling Problem in [23,24,25], multi-objective optimization


1426 Table 1. A simple instance of the DFJSP

job Job 1 Job 2 Job 3 Job 4

operation O11 O12 O21 O22 O23 O31 O32 O33 O41 O42 O43

M1(F1) 1 2 1 2 3 4 1 3 2

M2(F1) 2 3 2 1 1 -

M3(F1) 2 2 2 3 2 5 4 4

M1(F2) 1 3 1 2 2 4 2

M2(F2) 2 2 2 1 2 2 1

M3(F2) 1 4 2 1 4 6 3 5 -

Fig. 1. Gantt diagram of a faisable solution

of Flexible Job shop Scheduling with maintenance activity constraints [26], etc. Experimental comparisons demonstrated that the CRO is one of the most powerful optimization algorithms, and since the Distributed and Flexible Job shop Scheduling Problem is known to be strongly NP-Hard, we propose in this paper to use the chemical reaction optimization metaheuristic to solve the DFJSP in order to minimize the maximum completion time (Makespan). The remainder of this paper is organized as follows. In the next section, we present the problem formulation, we describe our proposed algorithm namely a Chemical Reaction Optimization metaheuristic for DFJSP (CRODFJSP) in section 3, we give some experimental results in section 4 and a conclusion and perspectives are given in section 5.

2. Problem Formulation The Distributed and Flexible Job shop Scheduling Problem consists of a set of factories (F=1, 2, . . . , l ) and a set of jobs (N=1, 2, . . . , i ) they are geographically distributed with a travel time (Dli ) knowing that each factory has a set of machines (Hl ). Each job i consists of a sequence of N j operations Oi j , i=1,2,...N; j=1,2,...,N j , and each operation can be processed on different machine with different processing times (Pi jhl ). The objective of the DFJSP is to find firstly how to allocate these jobs to factories, then determine the production scheduling in each factory in order to minimize the maximum completion time (makespan) among all factories. In DFJSP, we consider the following hypotheses and constraints: (1) It is generally assumed that each machine can only handle one operation at each time; (2) Each operation can only be processed after its precedence operation; (3) Each operation has to be carried out on only one machine; (4) Once a job is allocated to a factory, all of its operations will be processed in that factory; (5) The operations have to be proceeded until completion. To explain the DFJSP, a sample problem of 4 jobs, 3 machines and 2 factories is shown in Table 1, where columns 3 to 8 show the processing times of the operations on the different machines and F1 correspond to factory 1 and F2 correspond to factory 2. The symbol ”-” means that the operation cannot be executed on the corresponding machine and we consider the travel times between the factories and the jobs are negligible. Figure 1 shows a Gantt diagram for a scheduling of the proposed example.


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3. CRO for DFJSP In this section, we present our algorithm based on the Chemical Reaction Optimization metaheuristic (CRO) to solve the Distributed and Flexible Job shop Scheduling Problem (FJSP) named CRODFJSP in order to minimize the makespan or the maximum completion time (Cmax). 3.1. Chemical Reaction Optimization metaheuristic Chemical Reaction Optimization is a metaheuristic developed by Lam et al. [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 ineffective collision, Decomposition, Inter-molecular ineffective collision and Synthesis. Each molecule has several attributes such as: (1) ω: The molecular structure captures a solution of the problem, (2) PE: Potential energy is defined as the objective function PE ω = f (ω), (3) KE: Kinetic energy is the positive number and it quantifies the system tolerance to accept a worse solution than the existing one, (4) NumHit: is a record of the total number of hits (collisions) a molecule has taken, (5) MinStruct: the molecular structure with the minimum PE, (6) Min PE: when a molecule achieved its MinStruct, (7) MinHit: is the number of hits when a molecule performs MinStruct. There four basic types of reactions, each of which occurs in each iteration of CRO, 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 the diversification of solutions.. Inter-molecular 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. The idea behind Synthesis function is the diversification of solutions. 3.2. Global optimization process The global optimization phase is based on the CRO metaheuristic. In the case of unimolecular collision (b > Molecoll), the Decomposition function and On-Wall Ineffective Collision function will be executed, else in the case of inter-molecular collision (b < Molecoll), the Synthesis function and the Inter-Molecular Ineffective Collision function will be executed, this treatment is repeated until reach the maximum number of iteration. The two function Decomposition and Synthesis will be executed when the diversification conditions are met, see algorithm 1. Algorithm 1 Main Algorithm 1.Input:n,m,nbop[],dure[][],α,β,MinHit,NumHit,popsize,KElossrate,initialKE,Molecoll 2.generate-initial-solutions 3.While the stopping criterion not met { 4. Generate b ∈ [0, 1] 5. if (b>Molecoll) and (NumHit-MinHit> α) then { 6. Trigger Decomposition NumHit =0; MinHit=0; } 7. else if (b>Molecoll) and (NumHit-MinHit ≤ α) then { 8. Trigger On-Wall Ineffective Collision 9. NumHit++ ; if (f(s’1) is the best) then MinHit++ } 10. if (b ≤ Molecoll) and (Ke ≤β) then { 11. Trigger Synthesis } 12. else if (b ≤ Molecoll) and (Ke >β) { 13. Trigger Inter-Molecular Ineffective Collision ; KE=KE-KElossrate} 14.End while }

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Fig. 2. Decomposition function

3.3. Determination of initial solution The initial solution algorithm consists in affect firstly the different jobs to all factories using a heuristic which consists in scheduling the current job in a each factory, then we choose the best scheduling. See algorithm 2. Algorithm 2 Generate initial solution 1.begin-date (All factories) =0 2.end-date (All factories) = 0 3.Foreach i in jobs do 4. Foreach f in factories do 5. Foreach Oi j in operations do 6. Choose a random machine r Pi jr f 0 7. end-date (f ) = begin-date (f )+Pi jr f 8. begin-date (f ) = end-date (f ) 9. EndFor 10. end-date(f )=end-date(f )+travel-time(i,f ) 11. EndFor 12.EndFor 13.Choose the factory which gives a best scheduling

3.4. Diversification techniques The diversification phase is an important phase to guarantee better resolution of DFJSP through more exploration of our search space. When a better solution cannot be found after a certain threshold ”α” it’s a sign that our method is probably trapped in a local optimum. So, the diversification phase must be launched. In this situation, the ”decomposition” must be performed. The diversification phase can be also launched by executing the synthesis function. So, after some iterations of Inter-Molecular Ineffective Collision, KE becomes lower than ”β” then the Synthesis function will be executed.

3.4.1. Decomposition function The first function to performed the diversification phase is ”Decomposition”. The objective of this function is to find two new solutions from one initial solution. We choose randomly one initial solution and we choose two randomly jobs in different factories and we swap these two jobs and reassign its operations in initial scheduling, then we choose two others randomly jobs and we do the same instructions. The final solutions should satisfy the various constraints of DFJSP, see figure 2.


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Fig. 3. Synthesis function

Fig. 4. On-Wall Ineffective Collision function

Fig. 5. Inter-Molecular Ineffective Collision function

3.4.2. Synthesis function The second function to establish the diversification phase is ”Synthesis”. The principle of this function is to find a solution from both random solutions, this function is the opposite of the decomposition function. The new found solution is composed of the first part of the first solution and the second part of the second solution, see figure 3. 3.5. Intensification techniques The intensification phase is one of the most important phases to better exploit the search space by executing the ”On-Wall Ineffective Collision” and ”Inter-Molecular Ineffective Collision”. 3.5.1. On-Wall Ineffective Collision The aim of this function is to find one new solution from one initial solution, so we choose a randomly job and we are trying to move its operations in other positions (in inactivity intervals). We start with browsing the operations of job chosen one per one and we choose a new position for each operation. The assignment of operation should satisfy the various constraints such as the temporal constraints (precedence constraints) and the resource constraints, see figure 4. 3.5.2. Inter-Molecular Ineffective Collision The idea of this function is to find two new solutions from both random solutions. The first new solution is composed of the first part of the first solution and the second part of the second solution. However, the second new solution is composed of the first part of the second solution and the second part of the first solution, see figure 5.


1430 Table 2. The parameters of the CRODFJSP

Parameter nbiteration Value 1000

Molecoll

b

a

NumHit MinHit

buffer InitialKE KElossrate

β

0.5

[0..1]

3

0

0

25

N 10 10 10 10 10 15 15 15 15 15 20 20 20 20 20 10 10 10 10 10

M 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 10 10 10 10 10

Table 3. Hurink instances with 2 factories

Instance La01 La02 La03 La04 La05 La06 La07 La08 La09 La10 La11 La12 La13 La14 La15 La16 La17 La18 La19 La20

LB 413 394 349 369 380 413 376 369 382 443 413 408 382 443 378 717 646 663 617 756

0

IGA 413 394 349 369 380 445 412 420 469 445 570 504 542 570 584 717 646 663 617 756

GA-JS 413 394 349 369 380 424 398 406 447 443 548 480 533 542 562 717 646 663 617 756

200

12

CRODFJSP 413* 394* 349* 369* 380* 424 398 420 473 445 600 504 537 576 584 717* 646* 663* 617* 756*

4. Experiments In this section, the performance of our proposed algorithm is evaluated and tested. We have made series of experiments on two groups of instances, the first group consists of instances of classical FJSP from the literature adapted for the Distributed and Flexible Job shop Scheduling Problem namely instances of Hurink et al. [27] and Brandimarte [28], and the second group consists of instances of DFJSP namely instances of Chan et al. [11]. The asterisk (*) indicates that optimal solution was found and LB indicates the lower bound. All tests were conducted on a I3-3110M 2*2.4 GHz with 4 GB. We use the Java object oriented programming language with the Eclipse IDE. The parameters of our algorithm are set in table 2. 4.1. Tests on FJSP instances adapted for DFJSP 4.1.1. Tests on Hurink instances We compare the results obtained by our algorithm CRODFJSP on benchmark instances of Hurink et al. [27] with 2 factories with N is the number of job and M is the number of machines with the approach of Giovanni and Pezzella [12] called an Improved Genetic Algorithm for the Distributed and Flexible Job shop Scheduling Problem (IGA) and the approach of Lu et al. [9] called a genetic algorithm embedded with a concise chromosome representation for Distributed and Flexible Job shop Scheduling Problems (GA-JS). The results show that our algorithm CRODFJSP reaches the optimum for 50% of instances, see table 3. The tests demonstrate that our algorithm CRODFJSP and IGA provide the same results in 70% of instances, while our algorithm CRODFJSP surpasses IGA in 15% of instances. The results show also that our algorithm CRODFJSP and GA-JS provide the same results in 60% of instances but GA-JS surpasses our algorithm in 40 % of instances.


Table 4. Hurink instances with 3 factories


N 10 10 10 10 10 15 15 15 15 15 20 20 20 20 20 10 10 10 10 10

M 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 10 10 10 10 10

LB 413 394 349 369 380 413 376 369 382 443 413 408 382 443 378 717 646 663 617 756

IGA 413 394 349 369 380 413 376 369 382 443 425 408 419 443 451 717 646 663 617 756

GA-JS 413 394 349 369 380 413 376 369 382 443 413 408 398 443 420 717 646 663 617 756

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CRODFJSP 413* 394* 349* 369* 380* 413* 381 403 437 443* 590 483 419 470 451 717* 646* 663* 617* 756*

We also compare the results obtained by our algorithm CRODFJSP on benchmark instances of Hurink with 3 factories such as N correspond to the number of job and M is the number of machines with IGA and GA-JS approaches. The results show that our algorithm reaches the optimum for 60% of instances, see table 4. The tests demonstrate that our algorithm CRODFJSP and IGA provide the same results in 60% of instances. The results show also that our algorithm CRODFJSP and GA-JS provide also the same results in 60% of instances. We also compare the results obtained by our algorithm CRODFJSP on benchmark instances of Hurink with 4 factories such as N is the number of job and M correspond to the number of machines with IGA and GA-JS approaches. The tests demonstrate that our algorithm reaches the optimum for 70% of instances, see table 5. The results show that our algorithm CRODFJSP, IGA and GA-JS provide the same results also in 70% of instances.

4.1.2. Tests on Brandimarte instances We compare our algorithm CRODFJSP on benchmark instances of Brandimarte [28] adapted for the DFJSP such as N is the number of job and M correspond to the number of machines with the approach of Chang and Liu [10] called Hybrid Genetic Algorithm (HGA) and the results show that the two works give the same results in two instances but HGA surpasses our algorithm in the last instance, see table 6.

4.2. Tests on DFJSP instances We test our algorithm CRODFJSP in two instances of Chan et al. [11] such as N correspond to the number of job and M correspond to the number of machines and the results show that our algorithm is better in term of makespan than the approach of Chan et al. [11] called a genetic algorithm with dominant genes for DFJSP (GADG) in 50% of instances and gives the same results with the HGA and IGA approaches in the first instance, see table 7.


1432 Table 5. Hurink instances with 4 factories


Table 6. Brandimarte Instances

Table 7. DFJSP Instances

Instance DFJSP 1 DFJSP 2

N 10 10 10 10 10 15 15 15 15 15 20 20 20 20 20 10 10 10 10 10

M 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 10 10 10 10 10

LB 413 394 349 369 380 413 376 369 382 443 413 408 382 443 378 717 646 663 617 756

Instance MK1 MK2 MK3

N 20 20 30

M 6 6 8

N 10 10

M 13 13

F 2 2

IGA 413 394 349 369 380 413 376 369 382 443 413 408 382 443 397 717 646 663 617 756

F 2 2 2

GADG 42 49

GA-JS 413 394 349 369 380 413 376 369 382 443 413 408 382 443 378 717 646 663 617 756

HGA 51 39 210

HGA 37 37

CRODFJSP 413* 394* 349* 369* 380* 413* 376* 387 409 443* 413* 424 394 463 407 717* 646* 663* 617* 756*

CRODFJSP 51 39 227

IGA 37 37

CRODFJSP 37 49

5. Conclusion and perspectives In this paper, we study the Distributed and Flexible Job shop Scheduling Problem and we propose, to solve this problem, an algorithm named ”Chemical Reaction Optimization for the Distributed and Flexible Job shop Scheduling Problem” (CRODFJSP) in order to minimize the maximum completion time (Makespan). An initial solution is created firstly which consists in affect firstly the different jobs to all factories using a heuristic which consists in scheduling the current job in a each factory and we choose the best scheduling. Then, an optimization phase based on chemical reaction optimization metaheuristic is launched. In the optimization phase, four basic functions are performed, two to establish the diversification phase which are ”decomposition” and ”synthesis” and two to establish the intensification phase which are ”on wall-inneffective collision” and ”inter inneffective collision”. To evaluate the performance of our algorithm, we tested our algorithm on two groups of instances, the first group consists of instances of classical FJSP from the literature adapted for the Distributed and Flexible Job shop Scheduling Problem, and the second group consists of instances of DFJSP and the results show that our algorithm gives promising results. As a perspective, we work to solve the DFJSP by using other metaheuristics such as Ant Colony, Particle Swarm Optimization (PSO), etc., or using the multi agent system knowing that the DFJSP is naturally decentralized.


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