Solving Permutation Flow Shop Scheduling Problem with Sequence-Independent Setup Time

In this paper, we study the resolution of a permutation flow shop problem with sequence-independent setup time.'e objective is to minimize the maximum of job completion time, also called the makespan. In this contribution, we propose three methods of resolution, a mixed-integer linear programming (MILP) model; two heuristics, the first based on Johnson’s rule and the second based on the NEH algorithm; and finally twometaheuristics, the iterative local search algorithm and the iterated greedy algorithm. A set of test problems is simulated numerically to validate the effectiveness of our resolution approaches. For relatively small-size problems, it has been revealed that the adapted NEH heuristic has the best performance than that of the Johnson-based heuristic. For the relatively medium and large problems, the comparative study between the two metaheuristics based on the exploration of the neighborhood shows that the iterated greedy algorithm records the best performances.


Introduction
e problem of scheduling in the manufacturing industry is characterized by the allocation of jobs on machines and establishing the correct sequence of those jobs in order to optimize an imposed criterion. Each job is identified by a set of tasks that form the path to be followed in the production cycle in the workshop. Every task has a running time on each machine when launching jobs in the industrial process. Several classifications are distinguished: one by machine grouping, one by job grouping, and one by mixed group. In this study, we are interested in a family where the set of jobs follows the same path on a set of machines arranged in series.
is type of problem is recognized in the scheduling literature as the flow shop problem (see Figure 1). In this type of problem, several constraints may be involved, for example, in logistical means, one can find, transferring, assembly, and disassembly of parts between machines, as well as the adjustment of production tools. Solving the scheduling problem is considered as an NP-difficult issue given a certain size of the batch and number of machines. e approaches of resolution are diverse, and we cite exact methods, heuristics, metaheuristics, and other combined methods. In this paper, we propose these three classes for solving the flow shop with permutation problem (PFSP) under the constraint of sequence-independent setup time (SIST). In this case, the SIST constraint depends on the technology nature of the machine, as well as the means used to prepare it for the execution of a new job. is preparation time is considered unproductive, and its introduction into the scheduling is of great use for the control of the process. We note that the PFSPs are widely studied in the last 50 years under various constraints and objectives. Research is currently knowing significant advances by introducing new constraints and new trends of multiobjective optimization. e scheduling problem PFSP is known in the literature as NP-complete optimization problem [1] for more than two machines. In general, since the number of jobs and the number of machines can be high, it is difficult to find the right solution with an exact method. Hence, metaheuristics are usually used to look for the right solution or at least approach it. e mainly used metaheuristics are based on an improvement of an initial solution by research in its neighborhood by one of the disruption procedures of the current solution. In the literature, we find simulated annealing (SA) [2][3][4], the genetic algorithm (GA) [5][6][7][8], and the combined metaheuristics SA and GA [9,10] that are used to solve flow shop scheduling problem. We find also other resolution methods such as tabu search [11][12][13][14] and the greedy randomized adaptive search procedure [15][16][17][18].
Other nature-inspired metaheuristics are applied for solving flow shop scheduling problem, the artificial bee colony algorithm [19][20][21], ant colony optimization algorithm [22,23], and water wave optimization algorithm [24,25]. e efficiency of these metaheuristics is observed during the resolutions of the flow shop scheduling problems with several jobs and machines. Sometimes, a machine has to be prepared in some way before it can process a certain job. It may need retooling, adjusting, and so on. e time required for such preparation is called setup time. e need to consider this new constraint, in simulation models, is pushing researchers to develop new models. Resolving flow shop problems with permutation and setup time is part of this new problem category. is topic has attracted a wide attention and has been subject of recent works for many researchers (see, for example, [26][27][28][29]).
Several works dealing with scheduling issues are presented in the optimization. Recently, a number of studies have been carried out on permutation flow shop scheduling problem under the constraint of setup time. We distinguish two types of setup time: the first depends on the sequence of the jobs on the same machine, called the sequence-dependent setup time (SDST), and the second is qualified as a setup time independent of the sequence of jobs in the same machine, called the sequence-independent setup time (SIST). For the first category of problem which is designated as the permutation flow shop scheduling problem with sequence-dependent setup time (PFSP-SDST), a large number of works have been carried out. is setup time depends on each job to be processed; for example, a big job requires more time of preparation than a small one. Often additional constraints are added such as no-waiting, blocking, and other new PFSP constraints that concertize industrial cases. e resolution of PFSP-SDST problem is presented in [30] using the migratory bird algorithm as well as a set of heuristics to minimize the makespan. In [31], the authors propose the water wave optimization algorithm to minimize the makespan with a blocking constraint for a PFSP-SDST problem. Similarly, [32] proposes an MILP model for the case of no-wait criterion for the same problem. In the same context, [33] proposes a case study of the PFSP-SDST problem under no-idle constraint to minimize the total flow time. Recently, PFSP-SDST problem is studied by using three metaheuristics with a local search neighborhood [29], and it was revealed that the iterated greedy gives good results comparing with the two other used methods. For the second category of problem, i.e., PFSP-SIST, we note that only few works has been tackled, which may be a way to be explored for future research. is second class of problem is very interesting to study when a given machine requires the same setup time for all jobs. For example, the machine can have an automatic fixed setup time to process every job; such a setup time can be interpreted in same industrial cases as cleaning, parts assembly-disassembly, or maintenance of a certain machine. We note that [34,35] propose a set of MILP models and lower bounds for specific cases of the PFSP-SIST problem. In the same context, [36] propose an MILP model and a set of constructive heuristics to solve F m |prmu, SIST|C max . We are continuing the investigation in PFSP-SIST issues by solving this problem for larger size of instances and testing two metaheuristics. To this end, we propose a study with three methods: MILP, heuristics, and metaheuristics. e two implemented metaheuristics for resolution are among the most powerful in the neighborhood exploration, the iterative local search (ILS) algorithm, and the iterated greedy (IG) algorithm. We note that our case study is particular because of the presence of the SIST constraint. In fact, this setup time may be the subject of several cases such as assembly and disassembly of parts or tools in the machine, cleaning, evacuation of production means, and so on. ese activities can be carried out by robots or manipulators independent of the processed jobs in the machine. In this perspective, we present a study to contribute to the resolution of this problem which forms an essential manufacturing case to be encountered in the new industrial technology. We continue in this line of research and propose a model in MILP and two metaheuristics for solving the PFSP-SIST problem.
Our paper is organized as follows. In Section 2, a description of the problem is presented; in Section 3, we develop all the proposed approaches to resolution. In Section 4. a comparative study between the different methods of resolution is presented. Section 5 summarizes our resolution approaches.

Description of the PFSP-SIST Problem
In the PFSP-SIST scheduling problem, a set of jobs J � J 1 , J 2 , . . . , J n constituting a production batch is launched in the production process consisting of a set of machines M � M 1 , M 2 , . . . , M m arranged in series. All jobs have the same path and m operations on all machines, and each job J j starts with the first machine and ends its cycle with the last machine. We consider that p i,j the processing time of the job J j on the machine M i , and each machine has an extra time of preparation s i when processing a new job. Denote a sequence π � π 1 , π 2 , . . . , π n constituting a possible permutation when processed by the machines. We note C i,π j the completion time of the job π j of the sequence on the M i machine. e goal is to find the right sequence among the n! possible sequence to minimize the production time of the batch of n jobs and determine the start and completion time of each job on each machine. e assumptions used are those of the classic case, that is to say the interruption is not allowed, and each machine processes only one job at time and job can process only once by the same machines. All jobs and machines are available at the initial moment. e aim is to minimize the maximum completion time called makespan C max under the selected assumptions as well as the availability constraints of the machines during scheduling in the real time of the entire batch. We give here a set of equations to determine the completion time of each job on each machine: Equations (1) and (2) allow to determine the end date of the first job and the other jobs on the first machine, respectively. Equations (3) and (4) allow to determine, respectively, the end date of the first job and the other jobs on all the other machines. Equation (5) calculates the makespan C max . In this model, the objective is to determine an optimal sequence π * in the set permutation Π, such as C max (π * ) ≤ C max (π), π ∈ Π.

Numerical Example.
We give here an numerical illustration with small instance represented by three jobs and three machines. e processing time of all jobs and the setup time for all machines of are given in Table 1. We represent the Gantt chart for the sequence π � (3, 1, 2), the makespan of this instance is 48 unite of time (see Figure 2). From this numerical example, it can be easily seen that jobs will be processed on the first machine continuously. On the contrary, when scheduling jobs in other machines, some machines remain on hold, which leads to a waste of time. In conclusion, the sequence in the first machine is a determining factor in the search for the right solution.

Resolution of PFSP-SIST Problem
e resolution of the scheduling problems during the last decades has undergone a great evolution and a great based on power of the height technology computers. To solve our problem F m |prmu, SIST|C max , we propose three approaches, a MILP, two heuristics among the most powerful for the scheduling problem and finally two metaheuristics algorithm ILS and IG. In the last resolution category, we rely on the exploration structures of the neighboring of a sequence π based on the permutation or insertion of a job in the sequence to obtain a new sequence in the space search for solution.

Mixed-Integer Linear Programming.
Modeling optimization problems in scheduling and in particular in the PFSP is one of the most important areas of research in operational research. In fact, these models propose a function to optimize under constraints solved by one of the solvers dedicated to this type of model. e models are numerous and diverse, often characterizing a practical case for industrial need or a theoretical case for academic research. Several models in MILP are proposed for different case study models of the PFSP; a comparative study is described in [37], and a new model is presented [38]. Furthermore, more models are presented in [39] for the PFSP-SDST problem. We present here a model in MILP for our problem, and we define a set of parameters and variables necessary for the realization of the proposed model.
e objective is to subject to Knowing that (7) defines the objective function to minimize. Constraints (8) and (9) make it possible to guarantee that each position in the sequence will be occupied by a single job, and each job can have only one position. Constraint (10) is used to determine the completion time of the job occupying the first position in the first machine. Constraint (11) calculates the completion time of the job at the position k, k ≥ 2 in the first machine. Constraint (12) guarantees that for a job at the k position, the completion time in the machine i is greater than or equal to the completion time in the previous machine i − 1 increasing its processing time on the i machine. Constraint (13) ensures that, for two jobs of successive positions, the completion time in the machine of the one in position k is greater than or equal to that of the one in position k − 1 increasing in processing and setup times. Constraints (14) and (15) ensure that the completion time is positive and the assignment variables x k,j are binary, respectively. To solve our mixed-integer linear programming model dealing with PFSP-SIST problem, we will use LINGO 17.3 software.

Heuristics.
e PFSP scheduling problems under various constraints can be solved through different heuristics. In fact, for the industrialists, the search of the solutions by heuristics constitutes simple and less expensive issue to ensure the production and to satisfy their customers. e heuristics are usually based on priority rules based on job characteristics such as processing time, setup time, and other priorities related to the problem being addressed. In this work, we present two heuristics, the heuristic Johnson's rule [40] and the one based on the NEH [41] algorithm. ese two basic heuristics are among the most promising approaches in finding the right solution for most PFSP scheduling problems.

An Heuristic Based on Johnson's Rule.
e PFSP problem with two machines is the basic problem of any flow shop scheduling problems. e studies that have followed since the first study refer to this problem. e extensions of these problems are numerous, and the cases studied often fall under the particular constraints like the preparation of the machines or the preparation of the jobs to schedule all the jobs in the workshop. e main objective of optimization is to minimize the makespan from which we can deduce many other intrinsic objectives. We were interested in this approach by proposing a heuristic based on Johnson's rule, which we adopted to solve our case study of F m |prmu, SIST|C max . e following algorithm presents our heuristic of resolution by highlighting the characteristics our problem.
In this investigation, the Johnson algorithm is adapted to fit more than two machines flow shop problems. Algorithm 1 shows the template of an heuristic based on Johnson's rule (HBJR) with construction of two virtual machines by assembling the first 1 < k < m − 1 machines into one virtual machine and the k − m remaining machines into the second virtual one. e processing time of a job π j on a virtual machine (p i,π j ′ ) is the sum of the processing times p i,π j of job π j on the machines that compose the virtual machine.

NEH Heuristic.
e NEH algorithm constitutes a heuristic pillar for solving the PFSP problem with n jobs and m machines. More and more researchers are developing more adopted versions of the case and following the constraints of each problem. We do several research work as in [42,43] and more recently in [44]. Indeed, Ruiz and Maroto compared NEH with more modern and more complex heuristics, such as those of [45,46] showing NEH algorithm realized the better performance. Here, we propose a version of the NEH algorithm for the resolution of PFSP-SIST, the Algorithm 2 presents the necessary steps to establish the sequence of NEH.
In order to adapt the NEH algorithm to the studied problem, the setup time is added to the sum of processing times of each job in the first step of the algorithm T j � m i�1 p i,π j + s i .
Input: processing time p j1 , . . . , p jm and the setup time s i in m machines.
Evaluate the total processing time and setup time of each job by: T(j) � m i�1 p ji + s i Construct the sequence π � (π 1 , . . . , π n ) by sorting the jobs in descending order of T(j) Schedule the first two jobs π 1 , π 2 and choose the best sequence π of the first two jobs for j � 3 to n do Insert π j in the all positions of the current sequence built and select the best sequence with the minimum C max in the last machine and update π NEH of the optimal sequence. end for Output: π NEH ALGORITHM 2: NEH algorithm.
Journal of Applied Mathematics diversification of neighborhood exploration that will be given in detail as follows. For our considered instances, the number of machines m ∈ 2, . . . , 30 and the number of jobs n ∈ 5, . . . , 600; the processing times p ij are defined in [1,49] and the setup times s i are defined in [1,10].

ILS Algorithm.
e ILS algorithm is considered one of the powerful optimization algorithms for solving scheduling problems. We find the latest applications of this algorithm in [47][48][49] for solving the PFSP problem and its extensions under various implementations. In Algorithm 3, we describe the steps of the development of this algorithm. In the first phase, an initial solution is given by one of the initialization heuristics. In the second, while the stopping criterion not satisfied a search procedure in the neighborhood of the current solution is triggered, each time, an update of the best solution is performed. In this implementation model, we choose an adaptation of the simulated annealing criterion to better explore the neighborhood. e implementation of the ILS algorithm requires a set of parameter settings to better converge to the optimal solution depending on the size of the problem. e first parameter is the stop condition of the algorithm that we consider here in number of iterations, the second parameter is the acceptance condition of the simulated annealing model T emp , We adopt the following model: is expression is given according to the characteristic of the PFSP-SIST problem. e parameter T 0 makes possible to control the speed of convergence of our algorithm, we choose in this study T 0 ∈ 0.5, . . . , 0.98 { }. is model of temperature has been adopted in several research works concerning the PFSP problem (see, for example, [50]). e local search metaheuristics requires the definition of the neighborhood structure exploration, we propose four structures (N i , i � 1, . . ., 4). Starting from the current sequence of Figure 3 where the jobs in Input: π the initial sequence. π (best) ⟵ π while unsatisfied stopping criterion do π ′ ⟵ π for u � 1 to d do Extract a job π u at random from the sequence π′ and add the job π u to the Φ subset. end for for u � 1 to d do Extract the π u ′ job from Φ subset.
Test the job on all positions in the current sequence π′ and choose the best position giving the best completion time. end for Choose π″ from the neighborhood of N k (π′) if C max (π″) < C max (π) then π ⟵ π″ if C max (π) < C max (π (best) ) then π (best) ⟵ π end if else if random ≤ exp − (C max (π″) − C max (π))/T emp then π ⟵ π″ end if end if end while Output: π (best) ALGORITHM 4: Iterated greedy algorithm. positions 2 and 7, respectively, are used to generate the set of neighborhoods considered. Neighborhood N 1 is defined by the permutation of the position of two jobs. Neighborhood N 2 adopts the insertion and shift to the right, and Neighborhood N 3 considers the insertion and shift on the left. Finally, neighborhood N 4 is defined by job blocks inversion between two different positions in the sequence.

IG Algorithm.
e second heuristic implemented to solve our problem is the IG algorithm (see Algorithm 4).
is algorithm is based on three stages of resolution: the step of destruction of the current solution, the step of the reconstruction of the solution and the last step concerns the local search procedure. e first two stages constitute the first main phase of the principle of the algorithm IG, the second main phase is that of the neighborhood exploration. Recent work in applying the IG [51][52][53] algorithm shows the rising power of this algorithm in solving the optimization problem in the scheduling domain. e implementation of this algorithm requires also a set of setting parameters, the stop condition, the number d of jobs extracted in the sequence to build the new sequence, the parameter of the simulated annealing temperature, and finally the structure of the neighborhood used in the exploration phase. For the stop condition and the temperature parameter T emp , we adopt the same condition as that adopted for the ILS algorithm. As for the neighborhood structure, we consider the same structure as ILS. For the d parameter of extracted jobs, we consider that it also depends on the size of the problem knowing that d ∈ ⌊n/4⌋, ⌊n/3⌋ { }, for n ∈ 5, . . . , 50 { } and d ∈ ⌊n/3⌋, ⌊n/2⌋ { } for n > 50.
We present here an illustrative example of the procedure of destruction and reconstruction of the IG algorithm. Knowing that the current solution is π � 7, 8, 5, 4, 2, 6, 1, 3 { }, we randomly extract two jobs, for example, 7 and 6, the remaining sequence is π d � 8, 5, 4, 2, 1, 3 { }. Subsequently, we insert the first 7 job in all positions, and we select the best constructed sequence giving the value of C max , for example π � 8, 5, 4, 7, 2, 1, 3 { }. We repeat the procedure for the new built sequence and insert job 6 in all possible positions, and at the end, we get the best value from the makespan. Figure 4 gives a representation of the procedure of destruction and reconstruction of the IG algorithm.

Computational Results
To validate our approaches of resolution for the studied problem, we propose two comparative studies between the different algorithms. e first comparative study is based on the calculation of the relative percentage deviation (RPD).
e second concerns the convergence study of resolution algorithms for medium and large instances. For the first comparative study, we give the expression of RPD by the following formula: where H 1 stands for HBJR and H 2 represents NEH heuristic.
In this expression, we measure the deviation of the solution found from the optimal solution given by the MILP, for relatively small instances, where C H i max represents the solution found by the heuristic H i and C best max represents the value found by the MILP. Table 2 shows the results found for some tested instances. We limit ourselves to such number of problems; our objective is to highlight the effectiveness of resolution heuristics and their relevance to solve this problem. From the results, we find that the heuristic NEH is better compared with the heuristic based on the rule of Johnson. Table 2 shows the RPD H i obtained for different combinations m × n, where m is the number of machines and n stands for the number of jobs. It is observed that for m � 2, the obtained results by the heuristics are the same and equal to those given by the MILP. It is also shown that for a number of jobs n ≥ 10, the NEH heuristic gives better results than those calculated by the heuristic based on Johnson's rule.
For relatively large and medium instances, we measure the deviation from the best solution given by the ILS and IG algorithms. For this last class of problem, we vary the number of jobs and machines to reach high instances. We note that computation is given for an average of five instances for each algorithm. e result in this comparative study is given in Table 3; this table shows the clear superiority of the IG algorithm with respect to ILS for all tested instances. We report that the number of machines tested is m ∈ 5, 10, 20, 30 { }, and the number jobs varies in a way to cover different sizes and reaches a maximum of 600 jobs. We have tried to better concertize the real size of the jobs processed in industrial batches. Journal of Applied Mathematics e analysis concerning the numerical results shows the efficiency of our suggested metaheuristics in term of convergence towards their good solutions.
In Figure 5, we represent the evolution of the values of the objective function as a function of the number of iteration for two instances 40 × 10 and 40 × 20; the two algorithms converge to their stable values about after 80 th iteration, and IG saves an important gain of units time compared with ILS. For the second instances, the two ILS and IG algorithms converge to their stable bounds at about 280 th iterations and that IG is better compared to ILS.  In Figure 6, we represent the evolution of the objective function for the two instances 100 × 10 and 100 × 20, respectively. ese two instances can be considered relatively larger instances. is type of size is frequently encountered in modern industry, where customers ask for a fairly high batch. For the first instance, we can see that the two algorithms stabilize after the 180 th iteration; a significant gain in calendar scheduling time is recorded by IG compared with ILS algorithm. For the second instance, we can conclude that the IG gives best results comparing with the ILS algorithm after 90 th iteration. e four tested instances, in the convergence comparison study, have been implemented for the NEH initialization heuristic. Indeed, this initialization algorithm gives us better results in terms of RPD for relatively small instances and relatively large and medium instances.

Conclusion
In this paper, we have studied a flow shop problem with permutation and sequence-independent setup time in order to minimize the makespan. First, we have conducted a comparative study between the exact mixed-integer linear programming method and two heuristics, one is based on Johnson's rule, and the other is based on NEH algorithm. e two heuristics were compared with the results of the MILP for all instances. Simulations are performed for relatively small instances by varying the number of jobs and machines. It was observed that the NEH heuristic gives better results than those given by the heuristic based on Johnson's rule. On the contrary, two metaheuristics are used for medium to relatively large instances. We have concluded that IG algorithm gives good results comparing with ILS metaheuristic. In this paper, we have studied PFSP-SIST problem by using MILP model and different metaheuristics. However, adding an other constraint to this problem like unavailability or no-idle machines can be considered as a potential future issue.

Data Availability
e data used the support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.