An integrated approach for scheduling flexible jobshop using teaching–learningbased optimization method
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Abstract
In this paper, teaching–learningbased optimization (TLBO) is proposed to solve flexible job shop scheduling problem (FJSP) based on the integrated approach with an objective to minimize makespan. An FJSP is an extension of basic jobshop scheduling problem. There are two sub problems in FJSP. They are routing problem and sequencing problem. If both the sub problems are solved simultaneously, then the FJSP comes under integrated approach. Otherwise, it becomes a hierarchical approach. Very less research has been done in the past on FJSP problem as it is an NPhard (nondeterministic polynomial time hard) problem and very difficult to solve till date. Further, very less focus has been given to solve the FJSP using an integrated approach. So an attempt has been made to solve FJSP based on integrated approach using TLBO. Teaching–learningbased optimization is a metaheuristic algorithm which does not have any algorithmspecific parameters that are to be tuned in comparison to other metaheuristics. Therefore, it can be considered as an efficient algorithm. As best student of the class is considered as teacher, after few iterations all the students learn and reach the same knowledge level, due to which there is a loss in diversity in the population. So, like many metaheuristics, TLBO also has a tendency to get trapped at the local optimum. To avoid this limitation, a new local search technique followed by a mutation strategy (from genetic algorithm) is incorporated to TLBO to improve the quality of the solution and to maintain diversity, respectively, in the population. Tests have been carried out on all Kacem’s instances and Brandimarte's data instances to calculate makespan. Results show that TLBO outperformed many other algorithms and can be a competitive method for solving the FJSP.
Keywords
Flexible job shop scheduling Local search Makespan Metaheuristics Teaching–learningbased optimizationIntroduction
Problems of scheduling occur in many economic domains like airplane scheduling, train scheduling, time table scheduling and especially in the shop scheduling of manufacturing organizations. Scheduling is a crucial factor to improve the productivity of any organization and to meet the deadlines (due dates). Effective scheduling has become mandatory to the manufacturing organizations to maintain the good will among their customers and for survival in the market. Among all the shop scheduling problems, FJSP is the most difficult NPhard (nondeterministic polynomial time hard) problem till date as pointed out by Garey et al. (1976). The completion time of the last job that leaves the shop floor is defined as the makespan for a scheduling problem by Pinedo (2008). A NPhard problem is a kind of problem that even a small change in problem size results in exponential increase of computation time. An FJSP consists of all the complexities involved in solving a basic JSP like sequencing problem (i.e., arrangement of operations allotted to a given machine) and additionally, it has routing problem (i.e., each operation’s allocation to a machine from given set of machines). The method of solving routing problem first and then the sequencing problem is known as hierarchical approach. Most of the researchers have chosen hierarchical approach to solve FJSP. Because, encoding a metaheuristic to FJSP is easier in hierarchical approach than an integrated approach. But the integrated approach is more powerful as it reduces the computational burden because both the subproblems of FJSP are tackled simultaneously. So an effort is put in this paper to apply proposed teaching–learningbased optimization (TLBO) in an integrated approach. A recent work by Garmdare et al. (2017) best demonstrates the integrated approach in scheduling problem. TLBO is proposed by Rao et al. (2011) has been applied to different kinds of optimization problems in the past. TLBO is found to be one of the efficient algorithms to give good quality solutions in a reasonable computation time. TLBO has been applied to different scheduling problems like permutation flow shop scheduling by Xie et al. (2014), for job shop scheduling by Keesari and Rao (2014), for flow shop and job shop scheduling cases by Baykasoglu et al. (2014), for flexible job shop scheduling with fuzzy processing times by Xu et al. (2015), for reentrant flexible flow shop by Shen et al. (2016) and for different scheduling problems by Buddala and Mahapatra (2016, 2017) and it is found that TLBO is one of the efficient metaheuristic that can be applied to these scheduling problems. So, research is focused in this paper to apply TLBO to solve the NPhard FJSP.
An effort to solve FJSP is first done by Brucker and Schlie (1990). They solved an FJSP problem with two jobs using a polynomial algorithm. Due to its complexity of FJSP, no exact method has been proposed to solve all sizes of FJSP problems till date. It is not always possible to obtain near optimal solutions to such an NPhard problem in a reasonable computation time. Therefore, many heuristic and metaheuristic techniques have been proposed in the recent past to generate near optimal solutions. Brandimarte (1993), based on integrated approach, has applied dispatching rules to solve the routing problem and used tabu search (TS) to solve sequencing problem. The most used algorithm to solve FJSP is genetic algorithm (GA). Chen et al. (1999) have applied GA for the first time, in hierarchical approach using chromosomal representation. Pezzella et al. (2008) have applied GA by integrating several rules to generate initial population, selection and crossover parameters. Gao et al. (2008) have applied GA with advanced mutation and crossover operators and hybridized GA with variable neighborhood descent (VND) technique to increase the search ability of FJSP. Zhang et al. (2011) proposed an improved chromosome representation for GA with a specially designed global selection and local selection parameters to generate good quality solutions. Chang et al. (2015) have proposed a hybrid taguchi GA to solve FJSP. It is a method that encodes feasible solutions to the initial chromosomes developed with Taguchi method behind mating. This is to increase the effectiveness of GA and to increase solution quality. Gambardella and Mastrolilli (1996) have developed two neighborhood functions using local search techniques and proposed two hybrid TS algorithms to solve single objective and multi objective FJSP, respectively. Kacem et al. (2002a) have proposed a paretooptimal approach by hybridization of evolutionary algorithms and fuzzy logic to solve multi objective FJSP. Kacem et al. (2002b) have proposed two approaches called approach by localization and evolutionary approach to solve FJSP with makespan criterion in hierarchical approach. Based on the behavior of flying birds (the swarm intelligence), Xia and Wu (2005) have applied particle swarm optimization (PSO). By hybridizing PSO with the local search algorithm called simulated annealing (SA) to solve multiobjective FJSP. Singh and Mahapatra (2012) and Singh and Mahapatra (2016) have proposed PSO and quantum behaved particle swarm optimization (QPSO), respectively, using chaotic numbers generated from logistic mapping function instead of random numbers to solve FJSP. Fattahi et al. (2007) have proposed a mathematical model and used different heuristic techniques to solve FJSP. Six different hybrid heuristics were proposed and results show that hybrid algorithm combination of TS and SA outperformed other heuristic combinations. Garmsiri and Abassi (2012) and Liouane et al. (2007) have used a combination of ant system (AS) algorithm with TS algorithm. They proposed a hybrid ant colony optimization (HACO) to solve six FJSP problems where TS is used as a local search algorithm. Xing et al. (2010) have proposed a knowledgebased ant colony approach (KBACO) by integrating the knowledge model to ant colony optimization (ACO) to solve FJSP. Xing et al. (2009a) have proposed a new simulation model to solve multiobjective FJSP. Xing et al. (2009b) have proposed a new algorithm based on some empirical knowledge to solve multiobjective FJSP. Bagheri et al. (2010) have applied an artificial immune algorithm (AIA) to solve FJSP. Li et al. (2010) have proposed a hybrid tabu search algorithm (HTSA) to solve multiobjective FJSP where a variable neighborhood structure with two adaptive rules is used to hybridize TS to increase its local search ability. Based on the application of multiple independent searches that increase the exploration of search space, Yazdani et al. (2010) have proposed a parallel variable neighborhood search (PVNS) technique. This technique uses various neighborhood techniques to solve FJSP. Based on the natural phenomenon of bee colony, Wang et al. (2012) have proposed an artificial bee colony (ABC) algorithm to solve FJSP with makespan criterion. Li et al. (2014) have proposed a discrete artificial bee colony (DABC) to solve FJSP considering the preventive maintenance and nonpreventive maintenance conditions. Based on the migration strategy of animals from one place to another, Rahmati and Zandieh (2012) have proposed a biogeographybased optimization (BBO) to solve FJSP. Yuan et al. (2013) have proposed a hybrid harmony search (HHS) algorithm based on an integrated approach. Some local search technique is embedded in the harmony search algorithm to solve FJSP. Using some existing heuristics applied to harmony search (HS), Gao et al. (2016) have proposed a discrete harmony search (DHS) to solve multiobjective FJSP. Karthikeyan et al. (2015) have proposed a hybrid discrete firefly algorithm (HDFA) to solve multiobjective FJSP. MalekiDarounkolaei et al. (2012), NooriDarvish and TavakkoliMoghaddam (2012), Mirabi et al. (2014) and Kia et al. (2017) worked on sequencedependent scheduling problems. Nouri et al. (2017) proposed a holonic multiagent model to solve FJSP.
Problem formulation
As JSP itself is an NPhard, its extension FJSP is also an NPhard. To find an optimal solution to an NPhard problem in a reasonable computation time, is not always possible. Therefore, it is advisable to find a near optimal solution using metaheuristic techniques in a reasonable computation time than searching an optimal solution with great computational effort. In a flexible jobshop scheduling problem, n jobs are to be arranged on m machines such that all the operations are executed successfully. There are n jobs (i = 1, 2, 3, …, n) with each job i having a predetermined number of operations J (j = 1, 2, 3, …, J) in a sequence that are to be executed on m machines (k = 1, 2, 3, … m). Manne (1960) proposed FJSP in a mixed integer linear programming (MILP) formulation with the following assumptions. All machines are different from each other. All jobs are different from each other. Machine breakdown during the operations is not allowed (nonpreemption condition, i.e., an operation without any interruption must be completed). Any machine can perform at most one operation at an instant (resource constraint).
 \(m\)

Total number of machines
 \(n\)

Total number of jobs
 \(Oij\)

ith job’s jth operation
 \(Ji\)

ith job’s total number of operations
 \(Pijk\)

\(Oij\) operation’s processing time if performed on machine k
 \(B\)

A big number
 \(Mij\)

Number of machines available for processing operation \(Oij\)
 \(Sijk\)

Begin time of operation \(Oij\) on machine k
 \(Cijk\)

Completion time of operation \(Oij\) on machine k
 \(Ci\)

Final completion time of job i in shop floor
 Cm

Makespan
 \(i, h\)

Job index (1, 2, 3, …, n)
 \(j, g\)

Operation index (1, 2, 3, …, Ji)
 \(k\)

Machine index (1, 2, 3, …, m)
 \(Xijk\)

1, if Oij is executed on machine k
0, otherwise
 \(Zijhgk\)

1, if operation \(Ohg\) succeeds operation \(Oij\) on machine k
0, otherwise
Inequality (1) determines the objective makespan. Constraint (2) governs completion time of all the jobs. Constraints (3) and (4) restrict that the minimum time difference between starting and completion times must be processing time on that particular machine k. In the set \(Mij \cap Mhg\), constraints (5) and (6) assure that resource constraint is not violated (i.e., no two operations are allotted to same machine at a given time). Precedence relationships between them are ensured by the constraint (7), i.e., second operation of a job cannot be started until the first operation of the same job is completed. An operation can be executed on one and only one machine. This is assured by the constraint (8). If an operation is not assigned to a machine k, then the starting and completion times of that operation are zero on that machine k. This is ensured by constraint (9). Constraint (10) determines the makespan.
As it is not always possible to obtain near optimal solutions in a reasonable computation time, scheduling of jobs in FJSP is acknowledged as NPhard problem is explained by Lenstra et al. (1977). The exact methods like dynamic programming proposed by Potts and Van Wassenhove (1987), branch and bound proposed by Brucker et al. (1994) and Artigues and Feillet (2008) are capable of solving only small size problems due to large complexity involved in FJSP. Most of them fail to solve large size problems to obtain good results and they also require large computation time and huge memory. The real world FJSP problems of medium and largescale size are beyond the scope of these exact methods, in spite of their relative success rate to achieve optimal solution. So researchers focused on nonexact heuristic methods like dispatching rules. This research further continued to the application of metaheuristic techniques like GA, PSO, TS, SA, ABC, BBO, HS, etc. which take less time compared to heuristic techniques.
Teaching–learningbased optimization
TLBO is proposed by Rao et al. (2011), Rao and Patel (2012, 2013) with an inspiration from the general teaching–learning process that how a teacher influences the knowledge of students. Students and teacher are the two main objects of a class and the algorithm explains the two modes of learning, i.e., via teacher (teacher phase) and discussion among the fellow students (student phase). A group of students of the class constitute the population of the algorithm. In any iteration, the best student of the class becomes the teacher. Execution of the TLBO is explained in two phases: they are teacher phase and student phase.
Teacher phase
Student phase
Accept Z_{new i} if it gives a better function value.
Problem mapping and representation
Example problem
Job  Operation  Machine 1  Machine 2  Machine 3  Machine 4 

1  O11  5  3  1  2 
O12  2  4  6  3  
2  O21  3  5  7  2 
O22  1  2  3  4  
O23  3  2  4  6  
3  O31  5  1  4  2 
O32  6  4  5  3  
O33  5  2  6  4  
O34  4  3  2  1 
Order of priority
Job  Operation  Priority 1  Priority 2  Priority 3  Priority 4 

1  O11  3  4  2  1 
O12  1  4  2  3  
2  O21  4  1  2  3 
O22  1  2  3  4  
O23  2  1  3  4  
3  O31  2  4  3  1 
O32  4  2  3  1  
O33  2  4  1  3  
O34  4  3  2  1 
Stochastic representation of subject value
Operation  O11  O12  O21  O22  O23  O31  O32  O33  O34 

Value of subject  2.8430  1.4356  1.1724  2.5219  3.2456  1.7951  3.5842  1.3216  4.2427 
Priority  2  1  1  2  3  1  3  1  4 
Machine assigned  4  1  4  2  3  2  3  2  1 
Initial sequence of operations before optimization
Machine 1  O34  O12  
Machine 2  O33  O22  O31 
Machine 3  O23  O32  
Machine 4  O21  O11 
Proposed algorithm
 1.
Input all the problem data like number of jobs, number of operations of each job, number of machines available for each job and the corresponding processing times.
 2.
Initialize students of the class. Generate initial subject values of the students randomly within the range. Sij = Sij min + (Sij max–Sij min) × r where Sij is the subject value of jth operation of job i. Sij min = 1, Sij max = (1 + tma) and r is a random number between (0, 1).
 3.
Generate the schedule using encoding scheme as proposed in problem mapping representation and the method to eliminate infeasible solution.
 4.
Evaluate each student’s knowledge (makespan).
 5.
Now evaluate the mean knowledge of all the students in the class.
 6.
Update the knowledge of all the students with teacher phase and student phase (Eqs. 11, 12 and 13) of TLBO.
 7.
Apply the local search as explained in the local search section.
 8.
Identify the best student of the class and replace the earlier teacher with best student if best student fitness is better than teacher.
 9.
Apply the mutation technique for every five iterations by randomly replacing 3% of the population.
 10.
Repeat the cycle to step 3 until the termination criterion is met.
 11.
End.
Method to eliminate infeasible solution
Because of the complexity of FJSP, there is very good chance for infeasible solution generation by the population of an algorithm during the run time. In this process too for example, on machine 2 in Table 4, operation O33 is sequenced before O31 which is an infeasible solution. In this paper, we propose a new method to eliminate infeasible solution when a real number encoding system is used. If we observe clearly, this problem arises when two or more operations of a job are allotted to a same machine and later operations of that job have low fractional value than the preceding operations. Here, O33 has a fractional value 0.3216 which is less than O31 fractional value 0.7951. Due to this reason, O33 is allotted before O31.
Elimination of infeasible solutions
Operation  O11  O12  O21  O22  O23  O31  O32  O33  O34 

Value of subject  2.8430  1.4356  1.1724  2.5219  3.2456  1.7951  3.5842  1.3216  4.2427 
Fractional value  0.8430  0.4356  0.1724  0.5219  0.2456  0.7951  0.5842  0.3216  0.2427 
Final fractional value  0.4356  0.8430  0.1724  0.2456  0.5219  0.2427  0.3216  0.5842  0.7951 
Final value of subject  2.4356  1.8430  1.1724  2.2456  3.5219  1.2427  3.3216  1.5842  4.7951 
Local search (LS)
Like many metaheuristics, TLBO also requires a local search technique to improve the solution exploration capacity of the algorithm while solving FJSP. A new local search technique is proposed in this paper. This local search technique can be applied to any metaheuristic technique in the future. Only solutions with best makespan are considered for local search at every iteration. This method is explained in two steps. (1) Sequence swap (2) Machine swap. Before going to these two steps, all the critical operations of the solution are to be found. To improve the solution quality of FJSP, a promising critical path concept in operation scheduling phase (Zhang et al. 2007) is used as explained below.
Sequence swap
Out of these critical operations, only operations allotted to the same machine such that two or more operations which are adjacent to each other are collected and pairwise swapping (pair exchange method used by Xia and Wu 2005) is done to these collected operations and makespan is evaluated for each swap. If the swapping gives a better makespan value, then the old solution is replaced with the current new solution. For example, in Fig. 2, operations marked with a black line are critical operations. Operations O11 and O21 on machine 4 and operations O22 and O33 on machine 2 come under this category. These operations are swapped and checked for new makespan.
Machine swap
Final sequence of operations before optimization
Machine 1  O12  O34  
Machine 2  O31  O22  O33 
Machine 3  O32  O23  
Machine 4  O11  O21 
Mutation strategy (MS)
A sudden change in the gene of the offspring compared to parent gene during the natural evolution process is called mutation. Even though exploration capability of TLBO improved with the local search technique, we observe that TLBO gets trapped at the local optimum. Observing the makespan values of all the students, it is clear that almost all the students reach the same knowledge level (local optimum) after several iterations and there is no further improvement. To eliminate this drawback and to maintain diversity in the population, thus increasing the balance between exploration and exploitation, mutation technique from the genetic algorithm is incorporated to the algorithm. The total population of the class considered here is 100. Out of the total population, randomly two percent of population is made to undergo mutation for every five iteration. It means that two percent population is replaced with a new random solution using the equation in step 2 of proposed algorithm. The reasons to implement mutation strategy are (1) there is an improvement in the quality of solutions obtained and (2) this does not increase the computation burden much.
Results and discussion
Final sequence of operations of optimized solution
Machine 1  O21  O22  O12 
Machine 2  O31  O33  O23 
Machine 3  O11  
Machine 4  O32  O34 
Percentage improvement in solutions
SI. no.  Problem size  TLBO  TLBO + LS  PI  TLBO + LS + MS (proposed TLBO)  PI 

1  4 × 5  11  11  0  11  0 
2  8 × 8  14  14  0  14  0 
3  10 × 7  11  11  0  11  0 
4  10 × 10  7  7  0  7  0 
5  15 × 10  14  13  7.142  12  14.286 
6 MK01  10 × 6  45  42  6.667  40  11.111 
7 MK02  10 × 6  33  30  9.090  28  15.152 
8 MK03  15 × 8  285  222  22.105  204  28.421 
9 MK04  15 × 8  97  74  23.711  63  35.052 
10 MK05  15 × 4  187  176  5.882  172  8.0214 
11 MK06  10 × 15  94  80  14.893  65  30.851 
12 MK07  20 × 5  166  150  9.638  144  13.253 
13 MK08  20 × 10  578  532  7.958  523  9.515 
14 MK09  20 × 10  394  332  15.736  311  21.066 
15 MK10  20 × 15  337  257  23.738  214  36.499 
Average percentage improvement (API)  9.771  14.882 
As best student of the class is considered as teacher in TLBO, after few iterations all the students learn and reach the same knowledge level. Due to this reason, it is observed that there is a loss of diversity in the population. So, like many metaheuristics, TLBO also has a tendency to get trapped at the local optimum. To avoid this limitation, mutation strategy from genetic algorithm is incorporated to improve the quality of the solution and to maintain diversity. Once again tests have been carried out to find the makespan value of all the problems. Now, we observed not only a very good improvement in the quality of solutions but also four more best solutions are obtained, i.e., for the problems of MK01, MK03, MK05 and MK08. For Kacem’s fifth problem and MK02 problems, the second best solutions are obtained. The percentage improvement of proposed algorithm is tabulated in the last column of Table 8. The API value is 14.882 which is considerably a large improvement.
Results comparison of 15 benchmark problems of FJSP
SI. no.  Size  GA Chen et al. (1999)  PSO + SA Xia and Wu (2005)  HACO Liouane et al. (2007)  Xing’s algorithm Xing et al. (2009b)  AIA Bagheri et al. (2010)  HTSA Li et al. (2010)  ABC Wang et al. (2012)  BBO Rahmati and Zandieh (2012)  HHS Yuan et al. (2013)  DABC Li et al. (2014)  HDFA Karthikeyan et al. (2015)  DHS Gao et al. (2016)  Proposed TLBO 

1  4 × 5  NA  NA  11  11  NA  11  11  11  NA  NA  11  NA  11 
2  8 × 8  16  15  NA  14  14  14  14  14  14  NA  14  NA  14 
3  10 × 7  NA  NA  11  11  NA  11  11  NA  NA  NA  11  NA  11 
4  10 × 10  7  7  7  7  7  7  7  7  7  NA  7  NA  7 
5  15 × 10  NA  12  12  11  11  11  11  12  11  NA  11  NA  12 
6 MK01  10 × 6  40  NA  NA  42  40  40  40  40  40  40  NA  40  40 
7 MK02  10 × 6  29  NA  28  28  26  26  26  28  26  26  NA  28  28 
8 MK03  15 × 8  204  NA  NA  204  204  204  204  204  204  204  NA  204  204 
9 MK04  15 × 8  63  NA  NA  68  60  61  60  64  60  60  NA  60  63 
10 MK05  15 × 4  181  NA  NA  177  173  172  172  173  172  172  NA  172  172 
11 MK06  10 × 15  60  NA  68  75  63  65  60  66  58  NA  NA  67  65 
12 MK07  20 × 5  148  NA  NA  150  140  140  139  144  139  139  NA  143  144 
13 MK08  20 × 10  523  NA  NA  523  523  523  523  523  523  523  NA  523  523 
14 MK09  20 × 10  308  NA  NA  311  312  310  307  310  307  NA  NA  309  311 
15 MK10  20 × 15  212  NA  NA  227  214  214  208  230  205  NA  NA  212  214 
Conclusion
In this paper, one of the most difficult NPhard flexible job shop scheduling problems with makespan as criterion is considered and an efficient and effective teaching–learningbased optimization is used to generate near optimal schedules for fifteen benchmark problems. A new local search procedure has been proposed and it is found to be effective. A new technique to successfully overcome the infeasible solutions that are generated during the run time is proposed using the real number encoding system. The mutation technique from the genetic algorithm is incorporated to algorithm to maintain the diversity in the population. Results show that the proposed TLBO is found to be one of the good problem solving approaches for solving FJSP, as it out performed many other algorithms from the literature. It gave the best results to 8 problems out of 15. As tuning the parameters of metaheuristics itself is a herculean task, this paper stands as a basement for future research to concentrate or develop tuning parameter less algorithms to solve FJSP. The proposed TLBO avoids this drawback and, thus, reduces the computational burden too. The future work can be extended to hybridize TLBO with other local search techniques. Also, a hierarchical approachbased TLBO can be experimented to solve FJSP. The work can be further extended by considering the various uncertainties that are encountered in a reallife FJSP problem. Also, a multiobjective optimization study of FJSP can be carried out in the future.
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