An integrated approach for scheduling flexible job-shop using teaching–learning-based optimization method

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 NP-hard (non-deterministic 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 NP-hard 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 meta-heuristic 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 sub-problems of FJSP are tackled simultaneously. So an effort is put in this paper to apply proposed teaching–learning-based 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 re-entrant 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 meta-heuristic that can be applied to these scheduling problems. So, research is focused in this paper to apply TLBO to solve the NP-hard 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 NP-hard problem in a reasonable computation time. Therefore, many heuristic and meta-heuristic 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 pareto-optimal 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 multi-objective 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 knowledge-based 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 multi-objective FJSP. Xing et al. (2009b) have proposed a new algorithm based on some empirical knowledge to solve multi-objective 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 multi-objective 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 non-preventive maintenance conditions. Based on the migration strategy of animals from one place to another, Rahmati and Zandieh (2012) have proposed a bio-geography-based 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 multi-objective FJSP. Karthikeyan et al. (2015) have proposed a hybrid discrete firefly algorithm (HDFA) to solve multi-objective FJSP. Maleki-Darounkolaei et al. (2012), Noori-Darvish and Tavakkoli-Moghaddam (2012), Mirabi et al. (2014) and Kia et al. (2017) worked on sequence-dependent scheduling problems. Nouri et al. (2017) proposed a holonic multiagent model to solve FJSP.

As JSP itself is an NP-hard, its extension FJSP is also an NP-hard. To find an optimal solution to an NP-hard problem in a reasonable computation time, is not always possible. Therefore, it is advisable to find a near optimal solution using meta-heuristic techniques in a reasonable computation time than searching an optimal solution with great computational effort. In a flexible job-shop 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 (non-preemption condition, i.e., an operation without any interruption must be completed). Any machine can perform at most one operation at an instant (resource constraint).

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 NP-hard 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 large-scale size are beyond the scope of these exact methods, in spite of their relative success rate to achieve optimal solution. So researchers focused on non-exact heuristic methods like dispatching rules. This research further continued to the application of meta-heuristic techniques like GA, PSO, TS, SA, ABC, BBO, HS, etc. which take less time compared to heuristic techniques.

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.

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.

To eliminate this problem, all the fractional values of each job are first arranged in an ascending order as prescribed in Table 5. For example, job 1 operations O11 and O12 have fractional values 0.8430 and 0.4356, respectively. After the ascending order arrangement, the final fractional values of O11 and O12 are 0.4356 and 0.8430, respectively. In this way, we can eliminate the possibility of infeasible solution generation. The pseudo-code for the same is given in Fig. 1. The initial gantt chart solution to the example problem before the application of TLBO is given in Fig. 2.

Table 5

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

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Like many meta-heuristics, 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 meta-heuristic 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.

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.

The computational experiments aim to find the performance of TLBO to solve FJSP with makespan as the objective. Figure 3 shows the final solution obtained for the example problem in gantt chart and Table 7 shows the final sequence of operations for obtained optimized solution of the example problem using the proposed algorithm. We can observe that solution improved from a makespan value of 14 time units to eight time units. The experiments are conducted using MATLAB software on an i7 processor running at 3.40 GHz on the Windows 7 operating system. Tests have been carried out on all the instances by Brandimarte (1993) and Kacem et al. (2002a, b). There are five Kacem’s instances with problem size ranging from 4 × 5 to 15 × 10. There are a set of 10 Brandimarte’s problems with size ranging from 10 × 6 to 20 × 15. These are the only two data sets that are widely solved and available for comparison from the literature.

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Table 7

Final sequence of operations of optimized solution

Machine 1	O21	O22	O12
Machine 2	O31	O33	O23
Machine 3	O11
Machine 4	O32	O34

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 meta-heuristics, 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.

Figures 4 and 5 show the rate of convergence of TLBO towards the optimal solution. After conducting experiments for different mutation percentage from the range one to five, it is found that most of the problems gave good results for two percent mutation. So mutation percent is fixed to two. Figure 6 shows the variation of results (makespan) of the problems for different mutation percentages.

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In this paper, one of the most difficult NP-hard flexible job shop scheduling problems with makespan as criterion is considered and an efficient and effective teaching–learning-based 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 meta-heuristics 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 approach-based TLBO can be experimented to solve FJSP. The work can be further extended by considering the various uncertainties that are encountered in a real-life FJSP problem. Also, a multi-objective optimization study of FJSP can be carried out in the future.