Solving the flexible job shop problem by hybrid metaheuristicsbased multiagent model
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Abstract
The flexible job shop scheduling problem (FJSP) is a generalization of the classical job shop scheduling problem that allows to process operations on one machine out of a set of alternative machines. The FJSP is an NPhard problem consisting of two subproblems, which are the assignment and the scheduling problems. In this paper, we propose how to solve the FJSP by hybrid metaheuristicsbased clustered holonic multiagent model. First, a neighborhoodbased genetic algorithm (NGA) is applied by a scheduler agent for a global exploration of the search space. Second, a local search technique is used by a set of cluster agents to guide the research in promising regions of the search space and to improve the quality of the NGA final population. The efficiency of our approach is explained by the flexible selection of the promising parts of the search space by the clustering operator after the genetic algorithm process, and by applying the intensification technique of the tabu search allowing to restart the search from a set of elite solutions to attain new dominant scheduling solutions. Computational results are presented using four sets of wellknown benchmark literature instances. New upper bounds are found, showing the effectiveness of the presented approach.
Keywords
Scheduling Flexible job shop Genetic algorithm Local search Holonic multiagent Hybrid metaheuristicsIntroduction
Scheduling is a field of investigation which has known a significant growth these last years. The scheduling problems appear in all the economic areas, from computer engineering to industrial production and manufacturing. The job shop scheduling problem (JSP), which is among the hardest combinatorial optimization problems (Sonmez and Baykasoglu 1998), is a branch of the industrial production scheduling problems.
The flexible job shop scheduling problem (FJSP) is a generalization of the classical JSP that allows to process operations on one machine out of a set of alternative machines. Hence, the FJSP is more computationally difficult than the JSP. Furthermore, the operation scheduling problem, the FJSP presents an additional difficulty caused by the operation assignment problem to a set of available machines. This problem is known to be strongly NPHard even if each job has at most three operations and there are two machines (Garey et al. 1976).
To solve this problem, Pinedo (2002) developed a set of exact algorithms limited for instances with 20 jobs and 10 machines. Birgin et al. (2014) presented a mixed integer linear programming (MILP) model, but it took a very large time to generate a scheduling solution. Shafigh et al. (2015) developed a mathematical model integrating layout configuration and production planning in the design of dynamic distributed layouts. Their model incorporated different manufacturing attributes such as demand fluctuation, system reconfiguration, lot splitting, work load balancing, alternative routings, machine capability, and tooling requirements. On the other hand, a community of researchers used the metaheuristics to find nearoptimal solutions for the FJSP with acceptable computational time. Brandimarte (1993) proposed a hierarchical algorithm based on tabu search metaheuristic for routing and scheduling with some known dispatching rules to solve the FJSP. Hurink et al. (1994) developed a Tabu Search procedure for the job shop problem with multipurpose machines. DauzèrePérès and Paulli (1997) presented a new neighborhood structure for the problem, and a list of Tabu moves was used to prevent the local search from cycling. Mastrolilli and Gambardella (2000) used tabu search techniques and presented two neighborhood functions allowing an approximate resolution for the FJSP. Bozejko et al. (2010a) presented a tabu search approach based on a new golf neighborhood for the FJSP, and in the same year, Bozejko et al. (2010b) proposed another new model of a distributed tabu search algorithm for the FJSP, using a cluster architecture consisting of nodes equipped with the GPU units (multiGPU) with distributed memory. A novel hybrid tabu search algorithm with a fast Public Critical Block neighborhood structure (TSPCB) was proposed by Li et al. (2011) to solve the FJSP. For the genetic algorithm, it was adopted by Chen et al. (1999), where their chromosome representation of solutions for the problem was divided into two parts. The first part defined the routing policy and the second part took the sequence of operations on each machine. Kacem et al. (2002a) used a genetic algorithm with an approach of localization to solve jointly the assignment and job shop scheduling problems with partial and total flexibility, and a second hybridization of this evolutionary algorithm with the fuzzy logic was presented in Kacem et al. (2002b). Jia et al. (2003) proposed a modified genetic algorithm for the FJSP, where various scheduling objectives can be achieved such as minimizing makespan, cost, and weighted multiple criteria. Ho et al. (2007) developed a new architecture named LEarnable Genetic Architecture (LEGA) for learning and evolving solutions for the FJSP, allowing to provide an integration between evolution and learning in an efficient manner within a random search process. Gao et al. (2008) adapted a hybrid genetic algorithm (G.A) and a variable neighborhood descent (V.N.D) for FJSP. The G.A used two vectors to represent a solution and the disjunctive graph to calculate it. Then, a V.N.D was applied to improve the G.A final individuals. Zhang et al. (2014) presented a model of lowcarbon scheduling in the FJSP considering three factors, the makespan, the machine workload for production, and the carbon emission for the environmental influence. A metaheuristic hybridization algorithm was proposed combining the original Nondominated Sorting Genetic Algorithm II (NSGAII) with a local search algorithm based on a neighborhood search technique. Kar et al. (2015) presented a productioninventory model for deteriorating items with stockdependent demand under inflation in a random planning horizon. This model is formulated as profit maximization problem with respect to the retailer and solved by two metaheuristics, which are the genetic algorithm and the particleswarm optimization. Kia et al. (2017) treated the dynamic flexible flow line problem with sequencedependent setup times. A set of composite dispatching rulebased genetic programming are proposed to solve this problem by minimizing the mean flow time and the mean tardiness objectives. Moreover, the particleswarm optimization was implemented by Xia and Wu (2005) in a metaheuristic hybridization approach with the simulated annealing for the multiobjective FJSP. A combined particleswarm optimization and a tabu search algorithm were proposed by Zhang et al. (2009) to solve the multiobjective FJSP. Moslehi and Mahnam (2011) presented a metaheuristic approach based on a hybridization of the particleswarm optimization and local search algorithm to solve the multiobjective FJSP. In addition, other types of metaheuristics were developed in this last few years, such as (Yazdani et al. 2010) implementing a parallel variable neighborhood search (PVNS) algorithm to solve the FJSP using various neighborhood structures. A new biogeographybased optimization (BBO) technique is developed by Rahmati and Zandieh (2012) allowing to search a solution area for the FJSP and to find the optimum or nearoptimum scheduling to this problem. Shahriari et al. (2016) studied the just in time single machine scheduling problem with a periodic preventive maintenance. A multiobjective version of the particleswarm optimization algorithm is implemented to minimize the total earliness–tardiness and the makespan simultaneously. In addition, it is noted that metaheuristics based on constraint programming (CP) techniques have been used for the FJSP. Hmida et al. (2010) proposed a variant of the climbing discrepancy search approach (C.D.S) for solving the FJSP, where they presented various neighborhood structures related to assignment and sequencing problems. Pacino and Hentenryck (2011) considered a constraintbased scheduling approach to the flexible job shop problem. They studied both the large neighborhood search (LNS) and the adaptive randomized decomposition (ARD) schemes, using random, temporal, and machine decompositions. Oddi et al. (2011) adapted an iterative flattening search (IFS) algorithm for solving the flexible job shop scheduling problem (FJSSP). This algorithm applied two steps, a first relaxation step, in which a subset of scheduling decisions was randomly retracted from the current solution, and a second solving step, in which a new solution was incrementally recomputed from this partial schedule. Moreover, a new heuristic was developed by Ziaee (2014) for the FJSP. This heuristic is based on a constructive procedure considering simultaneously many factors having a great effect on the solution quality. Furthermore, distributed artificial intelligence techniques were used for this problem, such as the multiagent model proposed by Ennigrou and Ghédira (2004) composed by three classes of agents, job agents, resource agents, and an interface agent. This model is based on a local search method which is the tabu search to solve the FJSP. In addition, this model was improved in Ennigrou and Ghédira (2008), where the optimization role of the interface agent was distributed among the resource agents. Henchiri and Ennigrou (2013) proposed a multiagent model based on a hybridization of two metaheuristics, a local optimization process using the tabu search to get a good exploitation of the good areas and a global optimization process integrating the particleswarm optimization (PSO) to diversify the search towards unexplored areas. Rezki et al. (2016) proposed a multiagent system combining many intelligent techniques such as: multivariate control charts, neural networks, bayesian networks, and expert systems, for complex process monitoring tasks that are: detection, diagnosis, identification, and reconfiguration.
In this paper, we present how to solve the flexible job shop scheduling problem by a hybridization of two metaheuristics within a holonic multiagent model. This new approach follows two principal steps. In the first step, a genetic algorithm is applied by a scheduler agent for a global exploration of the search space. Then, in the second step, a local search technique is used by a set of cluster agents to improve the quality of the final population. Numerical tests were made to evaluate the performance of our approach based on four data sets of Kacem et al. (2002b), Brandimarte (1993), Hurink et al. (1994), and Barnes and Chambers (1996) for the FJSP, where the experimental results show its efficiency in comparison with other approaches.
The rest of the paper is organized as follows. In the next section, we define the formulation of the FJSP with its objective function and a simple problem instance followed by which we detail the proposed hybrid metaheuristic algorithm with its clustered holonic multiagent levels. The experimental and comparison results are provided in the subsequent section. The final section rounds up the paper with a conclusion.
Problem formulation

The operations assignment subproblem assigns each operation to an appropriate machine.

The operations sequencing subproblem determines a sequence of operations on all the machines.

All the machines are available at time zero.

All jobs are ready for processing at time zero.

The order of operations for each job is predefined and cannot be modified.

There are no precedence constraints among operations of different jobs.

The processing time of operations on each machine is defined in advance.

Each machine can process only one operation at a time.

Operations belonging to different jobs can be processed in parallel.

Each job could be processed more than once on the same machine.

The interruption during the process of an operation on a machine is negligible.
Simple instance of the FJSP
Job  Operation  \({M}_{1}\)  \({M}_{2}\)  \({M}_{3}\)  \({M}_{4}\)  \({M}_{5}\) 

\({J}_{1}\)  \({O}_{1,1}\)  2  9  4  5  1 
\({O}_{1,2}\)  –  6  –  4  –  
\({J}_{2}\)  \({O}_{2,1}\)  1  –  5  –  6 
\({O}_{2,2}\)  3  8  6  –  –  
\({O}_{2,3}\)  –  5  9  3  9  
\({J}_{3}\)  \({O}_{3,1}\)  –  6  6  –  – 
\({O}_{3,2}\)  3  –  –  5  4 
A metaheuristic hybridization within a holonic multiagent model
Glover et al. (1995) elaborated a study about the nature of connections between the genetic algorithm and tabu search metaheuristics, searching to show the existing opportunities for creating a hybrid approach with these two standard methods to take advantage of their complementary features and to solve difficult optimization problems. After this pertinent study, the combination of these two metaheuristics has become more well known in the literature, which has motivated many researchers to try the hybridization of these two methods for the resolution of different complex problems in several areas.
Ferber (1999) defined a multiagent system as an artificial system composed of a population of autonomous agents, which cooperate with each other to reach common objectives, while simultaneously each agent pursues individual objectives. Furthermore, a multiagent system is a computational system where two or more agents interact (cooperate or compete, or a combination of them) to achieve some individual or collective goals. The achievement of these goals is beyond the individual capabilities and individual knowledge of each agent (Botti and Giret 2008).
Koestler (1967) gave the first definition of the term “holon” in the literature, by combining the two Greek words “hol” meaning whole and “on” meaning particle or part. He said that almost everything is both a whole and a part at the same time. In fact, a holon is recursively decomposed at a lower granularity level into a community of other holons to produce a holarchy (Calabrese 2011). Moreover, a holon may be viewed as a sort of recursive agent, which is a superagent composed by a subagents set, where each subagent has its own behavior as a complementary part of the whole behavior of the superagent. Holons are agents able to show an architectural recursiveness (Giret and Botti 2004).
In fact, the choice of this new metaheuristic hybridization is justified by that the standard metaheuristic methods use generally the diversification techniques to generate and to improve many different solutions distributed in the search space, or using local search techniques to generate a more improved set of neighbourhood solutions from an initial solution. However, they did not guarantee to attain promising areas with good fitness converging to the global optimum despite the repetition of many iterations; that is why, they need to be more optimized. Therefore, the novelty of our approach is to launch a genetic algorithm based on a diversification technique to only explore the search space and to select the best promising regions by the clustering operator. Then, applying the intensification technique of the tabu search allowing to relaunch the search from an elite solution of each cluster autonomously to attain more dominant solutions of the search space.
Scheduler agent
The scheduler agent (SA) is responsible to process the first step of the hybrid algorithm using a genetic algorithm called neighborhoodbased genetic algorithm (NGA) to identify areas with high average fitness in the search space during a fixed number of iterations MaxIter, see Fig. 2. In fact, the goal of using the NGA is only to explore the search space, but not to find the global solution of the problem. Then, a clustering operator is integrated to divide the best identified areas by the NGA in the search space to different parts, where each part is a cluster \({\rm {CL}}_{i} \in {\rm {CL}}\) the set of clusters, where \({\rm {CL}} = \lbrace {\rm {CL}}_{1},{\rm {CL}}_{2},\dots ,{\rm {CL}}_{N}\rbrace.\) In addition, this agent plays the role of an interface between the user and the system (initial parameter inputs and final result outputs). According to the number of clusters N obtained after the integration of the clustering operator, the SA creates N cluster agents (CAs) preparing the passage to the next step of the global algorithm. After that, the SA remains in a waiting state until the reception of the best solutions found by the CA for each cluster. Finally, it finishes the process by displaying the final solution of the problem.
Individual’s solution presentation
Population initialization
Selection operator
 The fitnessneighborhood total for the population:$$\begin{aligned} {\rm{FN}} = \sum _{k=1}^P [1/(C_{\rm {max}}[k] \times {\text {Neighborhood}}[i][k])]. \end{aligned}$$(5)
 The selection probability \({\text {sp}}_{k}\) for each individual \({I}_{k}\):$$\begin{aligned} {\text {sp}}_{k} = \frac{1/(C_{\rm {max}}[k] \times {\text {Neighborhood}}[i][k])}{\text {FN}}. \end{aligned}$$(6)
 The cumulative probability \({\text {cp}}_{k}\) for each individual \({I}_{k}\):$$\begin{aligned} {\text {cp}}_{k} = \sum _{h=1}^k {\text {sp}}_{h}. \end{aligned}$$(7)
Crossover operator
The crossover operator has an important role in the global process, allowing to combine in each case the chromosomes of two parents to obtain new individuals and to attain new better parts in the search space. In this work, this operator is applied with two different techniques successively for the parent’s chromosome vectors MA and OS.
Mutation operator
The mutation operator is integrated to promote the children generation diversity. In fact, this operator is applied on the chromosome of the new children generated by the crossover operation. In addition, each part of a child chromosome MA and OS has separately its own mutation technique.
Replacement operator
Clustering operator
Cluster agents
Experimental results
Experimental setup

Kacem data (Kacem et al. 2002b): The data set consists of five problems considering a number of jobs ranging from 4 to 15 with a number of operations for each job ranging from 2 to 4, which will be processed on a number of machines ranging from 5 to 10.

Brandimarte data (Brandimarte 1993): The data set consists of ten problems considering a number of jobs ranging from 10 to 20 with a number of operations for each job ranging from 5 to 15, which will be processed on a number of machines ranging from 4 to 15.

Hurink edata (Hurink et al. 1994): The data set consists of 40 problems (la01–la40) inspired from the classical job shop instances of Lawrence (1984), where three test problems are generated: rdata, vdata, and edata which are used in this paper.

Barnes data (Barnes and Chambers 1996): The data set consists of 21 problems considering a number of jobs ranging from 10 to 15 with a number of operations for each job ranging from 10 to 15, which will be processed on a number of machines ranging from 11 to 18.

Crossover probability 1.0.

Mutation probability 0.5.

Maximum number of iterations 1000.

The population size ranged from 15 to 400 depending on the complexity of the problem.

The fixed threshold Distfix represents 50% of the maximal dissimilarity distance Distmax.
Results of the Kacem instances (part 1)
Instance  Problem \(n\times m\)  AL\(+\)CGA  LEGA  MOPSO\(+\)LS  BBO  

Best  Dev (%)  Best  Dev (%)  Best  Dev (%)  Best  Dev (%)  
Case 1  4 \(\times\) 5  16  13.250  11  0  16  31.25  11  0 
Case 2  8 \(\times\) 8  15  6.666  N/A  –  14  0  14  0 
Case 3  10 \(\times\) 7  15  26.666  11  0  15  26.666  N/A  – 
Case 4  10 \(\times\) 10  7  0  7  0  7  0  7  0 
Case 5  15 \(\times\) 10  23  52.173  12  8.333  11  0  12  8.333 
Results of the Kacem instances (part 2)
Hybrid NSGAII  Heuristic  GATS\(+\)HM  

Best  Dev (%)  Best  Dev (%)  Best  Avg \(C_{\rm {max}}\)  Avg CPU (in s) 
11  0  11  0  11  11.00  0.05 
15  6.666  15  6.666  14  14.20  0.36 
N/A  –  13  15.384  11  11.40  0.72 
7  0  7  0  7  7.60  1.51 
11  0  12  8.333  11  11.60  29.71 
Results of the Brandimarte instances (part 1)
Instance  Problem \(n\times m\)  TS  LEGA  MATSLO\(+\)  TS3  

Best  Dev (%)  Best  Dev (%)  Best  Dev (%)  Best  Dev (%)  
MK01  10 \(\times\) 6  42  4.761  40  0  40  0  40  0 
MK02  10 \(\times\) 6  32  15.625  29  6.896  32  15.625  29  6.896 
MK03  15 \(\times\) 8  211  3.317  N/A  –  207  1.449  204  0 
MK04  15 \(\times\) 8  81  20.987  67  4.477  67  4.477  65  1.538 
MK05  15 \(\times\) 4  186  6.989  176  1.704  188  7.978  173  0 
MK06  10 \(\times\) 15  86  24.418  67  2.985  85  23.529  68  4.411 
MK07  20 \(\times\) 5  157  8.280  147  2.040  154  6.493  144  0 
MK08  20 \(\times\) 10  523  0  523  0  523  0  523  0 
MK09  20 \(\times\) 10  369  15.718  320  2.812  437  28.832  326  4.601 
MK10  20 \(\times\) 15  296  25  229  3.056  380  41.578  227  2.202 
Results of the Brandimarte instances (part 2)
BBO  MATSPSO  Heuristic  GATS\(+\)HM  

Best  Dev (%)  Best  Dev (%)  Best  Dev (%)  Best  Avg \(C_{\rm {max}}\)  Avg CPU (s) 
40  0  39  −2.564  42  4.761  40  40.80  0.93 
28  3.571  27  0  28  3.571  27  27.80  1.18 
204  0  207  1.449  204  0  204  204.00  1.55 
64  0  65  1.538  75  14.666  64  65.60  4.36 
173  0  174  0.574  179  3.351  173  174.80  8.02 
66  1.515  72  9.722  69  5.797  65  67.00  110.01 
144  0  154  6.493  149  3.355  144  144.00  19.73 
523  0  523  0  555  5.765  523  523.00  11.50 
310  −0.322  340  8.529  342  9.064  311  311.80  79.68 
230  3.478  299  25.752  242  8.264  222  224.80  185.64 
Results of the Hurink edata instances
Instance  Problem \(n\times m\)  LB  N11000  MATSLO\(+\)  GATS\(+\)HM  

Best  Dev (%)  Best  Dev (%)  Best  Dev (%)  Best  Avg \(C_{\rm {max}}\)  Avg CPU (in s)  
la01  10 \(\times\) 5  609  0  611  0.327  609  0  609  609.00  24.64 
la02  10 \(\times\) 5  655  0  655  0  655  0  655  655.00  4.65 
la03  10 \(\times\) 5  550  −3.091  573  1.047  575  1.391  567  567.40  10.67 
la04  10 \(\times\) 5  568  0  578  1.730  579  1.900  568  569.60  22.13 
la05  10 \(\times\) 5  503  0  503  0  503  0  503  503.00  10.22 
la16  10 \(\times\) 10  892  0  924  3.463  896  0.446  892  909.60  73.14 
la17  10 \(\times\) 10  707  0  757  6.605  708  0.141  707  709.60  116.58 
la18  10 \(\times\) 10  842  −0.119  864  2.431  845  0.237  843  848.60  34.98 
la19  10 \(\times\) 10  796  −1.005  850  5.412  813  1.107  804  813.40  36.88 
la20  10 \(\times\) 10  857  0  919  6.746  863  0.695  857  859.80  70.36 
Results of the Barnes data instances
Instance  Problem \(n\times m\)  BBO  GATS\(+\)HM  

Pop  Best  Avg \(C_{\rm {max}}\)  Dev (%)  Avg CPU (in s)  Pop  Best  Avg \(C_{\rm {max}}\)  Avg CPU (in s)  
mt10c1  10 \(\times\) 11  350  946  947.00  2.008  401  300  927  930.00  84.26 
mt10cc  10 \(\times\) 12  350  946  946.00  3.065  405  300  917  918.60  78.40 
mt10x  10 \(\times\) 11  350  955  961.00  3.350  416  300  923  931.40  82.56 
mt10xx  10 \(\times\) 12  350  939  945.00  2.236  480  300  918  924.40  81.73 
mt10xxx  10 \(\times\) 13  350  954  954.50  3.773  497  300  918  921.00  95.22 
mt10xy  10 \(\times\) 12  350  951  951.00  4.521  458  300  908  910.00  93.48 
mt10xyz  10 \(\times\) 13  350  858  858.00  −1.165  495  300  868  871.80  72.03 
setb4c9  15 \(\times\) 11  350  959  959.00  3.336  762  250  927  936.60  104.10 
setb4cc  15 \(\times\) 12  350  944  950.00  0.635  770  300  938  946.80  150.34 
setb4x  15 \(\times\) 11  350  942  951.00  −0.212  749  200  944  956.20  61.05 
setb4xx  15 \(\times\) 12  350  967  967.00  2.585  761  300  942  953.60  145.74 
setb4xxx  15 \(\times\) 13  350  991  991.00  4.238  797  300  949  958.60  133.19 
setb4xy  15 \(\times\) 12  350  978  982.00  4.805  778  250  931  941.80  118.25 
setb4xyz  15 \(\times\) 13  350  930  930.50  0.430  651  200  926  929.80  62.04 
Experimental comparisons
To show the efficiency of our GATS\(+\)HM algorithm, we compare its obtained results from the four previously cited data sets with other wellknown algorithms in the literature of the FJSP.

The TS of Brandimarte (1993), N11000 of Hurink et al. (1994) (with its literature lower bound LB), and the AL\(+\)CGA of Kacem et al. (2002b) obtained the first results in the literature for their proposed instances.

The LEGA of Ho et al. (2007), the BBO of Rahmati and Zandieh (2012), and the Heuristic of Ziaee (2014) are standard heuristic and metaheuristic methods.

The TS3 of Bozejko et al. (2010a) is the paper from which we inspired the computation method of the dissimilarity distance.

The MOPSO\(+\)LS of Moslehi and Mahnam (2011) and the Hybrid NSGAII of Zhang et al. (2014) are two recent hybrid metaheuristic algorithms.

The MATSLO\(+\) of Ennigrou and Ghédira (2008) and the MATSPSO of Henchiri and Ennigrou (2013) are two new hybrid metaheuristic algorithms distributed in a multiagent model.
Analysis of the comparative results
By analyzing Tables 2 and 3, it can be seen that our algorithm GATS\(+\)HM is the best one which solves the fives instances of Kacem. In fact, the GATS\(+\)HM outperforms the AL\(+\)CGA in four out of five instances; the Heuristic in three out of five instances; and the LEGA, the MOPSO\(+\)LS, BBO, and the Hybrid NSGAII in two out of five instances. In addition, by solving this first data set, our GATS\(+\)HM attains the same results obtained by the chosen approaches such as in the case 1 for LEGA, BBO, Hybrid NSGAII, and Heuristic; in the case 2 for MOPSO\(+\)LS and BBO; in the case 3 for LEGA; in the case 4 for all the algorithms; and in the case 5 for MOPSO\(+\)LS and Hybrid NSGAII.
From Tables 4 and 5, the comparison results show that the GATS\(+\)HM obtains eight out of ten best results for the Brandimarte instances. Indeed, our algorithm outperforms the TS in nine out of ten instances. Moreover, our GATS\(+\)HM outperforms the LEGA and the MATSLO\(+\) in eight out of ten instances. In addition, our hybrid approach outperforms the TS3 in five out of ten instances. For the comparison with the BBO, the GATS\(+\)HM obtains the best solutions for the MK02, MK06, and MK10 instances, but it gets slightly worse result for the MK09 instance. Furthermore, the MATSPSO attained the best result for the MK01 instance, but our algorithm obtains a set of solutions better than it for the remaining instances. In addition, our algorithm outperforms the Heuristic in all the Brandimarte instances. By solving this second data set, our GATS\(+\)HM attains the same results obtained by some approaches such as in the MK01 for LEGA, MATSLO\(+\) and TS3; in the MK02 for MATSPSO; in the MK03 for TS3, BBO and Heuristic; in the MK04 for BBO; in the MK05 for TS3 and BBO; in the MK07 for BBO and TS3; and in the MK08 for all the algorithms only it is not the case for the Heuristic.
From Table 7, the results for the Barnes instances demonstrate that our GATS\(+\)HM dominates the BBO algorithm in different criteria such as the \(C_{\rm {max}},\) the Avg \(C_{\rm {max}},\) the Avg CPU, the deviation, and the population size. In fact, for the \(C_{\rm {max}}\) criterion, our GATS\(+\)HM outperforms the BBO in 12 out of 14 instances, see Fig. 10, with deviations varying from 0.430 to 4.805%. In addition, we attain average values for the \(C_{\rm {max}}\) solutions dominating the BBO in 12 times. In addition, as shown in Fig. 11, the used population sizes for our algorithm are less than the BBO in all the 14 instances, which influenced on the CPU execution time for each solution, see Fig. 12.
By analyzing the computational time in seconds and the comparison results of our algorithm in terms of makespan, we can distinguish the efficiency of the new proposed GATS\(+\)HM relatively to the literature of the FJSP. This efficiency is explained by the flexible selection of the promising parts of the search space by the clustering operator after the genetic algorithm process and by applying the intensification technique of the tabu search allowing to start from an elite solution to attain new more dominant solutions.
Conclusion
In this paper, we present a new metaheuristic hybridization algorithmbased clustered holonic multiagent model, called GATS\(+\)HM, for the flexible job shop scheduling problem (FJSP). In this approach, a neighborhoodbased genetic algorithm is adapted by a scheduler agent (SA) for a global exploration of the search space. Then, a local search technique is applied by a set of cluster agents (CAs) to guide the research in promising regions of the search space and to improve the quality of the final population. To measure its performance, numerical tests are made using four wellknown data sets in the literature of the FJSP. The experimental results show that the proposed approach is efficient in comparison with others approaches. In the future works, we will search to treat other extensions of the FJSP, such as by integrating new transportation times in the shop process, where each operation must be transported by a moving robot to continue its treatment on its next machine. In addition, this problem can be improved by considering a nonunit transport capacity for the moving robots, where the problem becomes a flexible job shop scheduling problem with transportation times and nonunit transport capacity robots. Therefore, we will plan to make improvements to our approach to adapt it to this new transformation of the problem, and study its effects on the makespan.
Notes
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest.
Ethical approval
This paper does not contain any studies with human participants or animals performed by any of the authors.
Informed consent
Informed consent was obtained from all individual participants included in this paper.
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