Evolutionary heuristic for makespan minimization in no-idle flow shop production systems

This paper deals with no-idle flow shop scheduling problem with the objective of minimizing makespan. A new hybrid metaheuristic is proposed for the scheduling problem solution. The proposed method is compared with the best method reported in the literature. Experimental results show that the new method provides better solutions regarding the solution quality to set of problems evaluated.


Introduction
The flow shop scheduling problem is a problem of work scheduling in which n tasks must be scheduled, in the same sequence, in a set of m distinct machines.A particular case of flow shop scheduling, called permutational, occurs when in each machine is maintained the same processing order of tasks.
The solution for the problem is to determine among the (n!) possible sequences of tasks, the one that optimizes some measure of scheduling performance, being that the most common consist of minimizing the total scheduling duration (makespan), or minimize the sum of the tasks flow times (total flowtime).The first relates to an efficient use of resources (machines) while the second aims to minimize the stock in processing.
This scheduling problem has been intensely studied in literature, since the first study reported by Johnson (1954) in obtaining the optimal solution for the problem with two machines.
The restrictions commonly considered in the flow shop scheduling problem restrain but do not exclude their practical applications in fact (DUDEK et al., 1992).One of the restrictions is that the machines that compose the flow shop do not suffer interruption once initiated.A representation for this problem is presented in the Figure 1.This type of problem is known as no-idle flow shop (NIFS), and occurs when the time for preparation or the cost of using the machine are relatively high, considering that to turn it on or to prepare it more times than the necessary provokes a costly process.Clear examples are the equipments used in the production of integrated circuits, by means of photolithography, a technique that creates the desired mould on the semiconductor slide.
Other examples are the ovens of ceramic cylinder that consume a large amount of natural gas when in operation, due to the thermal inertia that makes impossible to stop or restart their functioning.In all cases, the best alternative is to determine a sequencing that considers the restriction of non-idleness in the machines.This same problem can be denoted by F/prmu,no-idle/C max , and its computational complexity was considered as a NP-hard type by Tanaev et al. (1994).
Given the above, it is verified the importance and the difficulties found to solve the NIFS problem.This article aimed to present a new evolutionary heuristic method called HE-NIFS, and compare it with the most recent method proposed currently.
Heuristic methods for the NIFS problem Adiri and Pohoryles (1982) were the first to address the no-idle problem.In this study, they also addressed the no-wait flow shop (NWFS) problem.The main contribution was the development of an algorithm which optimally solves the problem F2/prmu,no-idle/∑C j .They also showed results for the problem with more than two machines, however, in special cases of dominance among the machines.
The heuristic methods for the problem of mmachines were initially addressed by Woollam (1986) with the objective to minimize the makespan.Basically, several methods were selected from the literature, including the known method NEH.In the study, five methods were adapted, and the computational experimentation was performed with problems up to 25 tasks and 25 machines.However, for such problems, the adapted NEH method produced the best results.Narain and Bagga (2003) studied the problem F3/prmu,no-idle/C max .They presented an integer linear programming method and an algorithm Branch & Bound with limited computational results.The same problem with three machines was studied by Saadani et al. (2003).They proposed a lower limiting (lower bound) and an efficient heuristic method.The heuristic was compared to a previous method presented in Saadani et al. (2005) and presented better results.An important note is that this study was subsequently published in Saadani et al. (2005).Kamburowski (2004) proposed a network representation that provided a better representation of the problem, indentifying some paradoxes resulting from the no-idle condition, and conducting some relationships of dominance among the machines in which the problem becomes effectively resolved.Saadani et al. (2005) presented heuristics based on Travelling Salesman Problem (TSP) for F/prmu,noidle/C max .Basically, the authors modeled as distances the makespan of pairs of tasks.Based on the minimum distance, the heuristic applies search engines in the neighborhood, inserting the tasks, one by one, in every possible position.The heuristics possesses a complexity O(n 3 ) and it is easily implementable.The authors tested the proposed heuristic comparing its solutions to an integer linear programming model for problems up to 17 tasks and 30 machines.Kalczynski and Kamburowski (2005) proposed a heuristic for F/prmu,no-idle/C max with computational complexity of O(n 2 m).The heuristic was compared with Saadani et al. (2005), and presented the best results in most the evaluated problems.The authors presented also an adaptation of the NEH heuristic for the no-idle problem, and the results showed that the proposed method also exceeds this heuristic.
Most recently, the NIFS problem was once again studied by Kalczynski and Kamburowski (2007), whereby were presented special situations between the no-wait and no-idle problems.The authors presented a network representation in which the longest path lengths in the network represent the makespan.The networks revealed the duality between the two problems, and a graphical explanation about the condition no-wait and no-idle for makespan was discussed.Baraz and Mosheiov (2008) proposed a heuristic method of two phases for F/prmu,no−idle/C max .In the first phase, the tasks were added, one by one at the end of the current sequence, and the task that resulted in the shortest makespan was sequenced.This phase was performed until all the tasks had been sequenced.In the second phase, all the tasks were interchanged within the sequence obtained in the first phase.The authors concluded that the execution time of their heuristic was O(n 2 ).However, it is worth highlighting that the authors did not consider the additional complexity of the makespan calculation in each stage.As this calculation has a computational complexity O(nm), they showed the superiority of their proposed heuristic in comparison to the results obtained by Saadani et al. (2005), however they did not present a comparison to the heuristic of Kalczynski and Kamburowski (2005).
Pan and Wang (2008a and b) in two similar studies, proposed a procedure that accelerates the search in the neighborhood of insertion and reduces the computational complexity from O(n 3 m) to O(n 2 m).This acceleration is based on procedures presented in Taillard (1990) for the same search in the neighborhood, but for the traditional flow shop problem.Both algorithms used a method of search called Iterated Greedy (RUIZ; STÜTZLE, 2007).
The authors used the problems of Taillard (1993) of the classical flow shop for the no-idle problem.In the two studies were compared the proposed methods to the heuristics of Baraz and Mosheiov (2008) and Kalczynski and Kamburowski (2005).The results indicated the superiority of the proposed methods; however the results were not compared among them.
Finally, an extensive computational experimentation was carried out by Ruiz et al. (2009) for the no-idle flow shop problem.The authors performed two computational experiments: the first was the evaluation of deterministic heuristic methods; and the second was the evaluation of metaheuristic methods.As for the first experiment, were evaluated nine heuristic methods adapted to the NIFS problem as listed below: -  and 12 (FRB4 4 and FRB4 12 ), with computational complexity of O(kn 2 m) or O(n 2 m).
In the second computational experimentation were evaluated four metaheuristic methods adapted to the NIFS problem: - The results obtained by the authors in the first computational experimentation indicated that the methods FRB3 and GH_BM2 had the best performances.The first regarding the relative average deviation, and the second regarding the results quality and the computing time.
The results obtained in the second experimentation pointed out that the IG LS had results significantly better than the HDPSO.
Finalizing according to the literature review performed and reported in this study, it can be concluded that the adapted IG method, as well as the improved proposed method GH_BM2, together with the heuristics of Rad et al. (2009) are the best existing methods for the NIFS problem with the makespan minimization criterion.

New evolutionary heuristic
The evolutionary heuristic proposed in this research is a hybridization of the Cluster Search (CS) presented by Ribeiro Filho et al. ( 2007) with the IG LS method proposed by Ruiz and Stützle (2007).
In this evolutionary process the stopping criterion is determined by the computing time (elapsed time) of the problem (MORAES; NAGANO, 2012).The time limit will be adopted as being n(m/2)T milliseconds, where T is a regulatory parameter of the stopping criterion.With a stopping criterion proposed in that way the computational time will be proportional to the number of tasks and machines.Thus, the greater the (n; m) problem, the greater will be the time available for the iterations.
The quality of the solution obtained by the CS is mostly dependent on the initialization processes present in its structure.Among these initializations exists the initial population initialization and the clusters initialization.
The initial population is formed by a vector of k positions, where each position is a solution.The size of this vector is defined as being the number of individuals possible to be generated in a time limit, being 500 the maximum value for k.Defining this time limit to generate the population as being 10% of the computing time (elapsed time) for the problem already defined, the initial population will be in the maximum of 500 individuals or a limited value by the 10% of the elapsed time.In this way it is understood that in the maximum of 10% of the computing time for the problem will be spent to initialize the population.
In order to ensure the quality of the individuals generated in the population, it is proposed a combination of the heuristic NEH of Nawaz et al. (1983) and the method Iterated Greedy (IG) (RUIZ; STÜTZLE, 2007), in which are generated solutions (sequences) through two phases: destruction and construction.In the phase of destruction some elements of the sequence are removed to be inserted subsequently, reconstructing the sequence.The motivation for using a combination of methods in the evolutionary heuristic (HE-NIFS) proposed, was the search for a greater variety of solutions with better quality in the process of initialization.
In the population initialization, the first individual inserted in the population is generated by the procedure NEH.All the other individuals are generated by the method Iterated Greedy (IG), applying the phases of destruction and construction to an individual already created.Thus, the first individual is generated by the procedure NEH, the second will be generated by the destruction and subsequent construction of the first individual.The third will be found by the process of destruction and construction of the second, and so on.Following this process, the forming of an individual requires other already formed, that is, the process is recursive, except the first individual generated by the heuristic NEH.
This procedure aims to provide a greater diversity of solutions in the process of initialization, without having to apply a simply random process, which would lead to the loss of quality of the generated solutions in the initial population.
The evaluation of the generated individuals is made directly by the optimization criterion for the makespan, and the evaluation is responsible for maintaining the population ordered, with the best individual (that individual with the shortest makespan) occupying the first position.This same procedure does not allow the insertion of repeated individuals in the population.
A procedure of initialization of clusters is created to seize the good individuals from the initial population.In the same way as the population, the clusters created will be submitted to the process of evaluation to order them.Once completed its initialization, the clusters evaluated as having their makespan among the best 1/3, within the all generated, will pass a process of local search (LS1) in view of a better selection of the space of solutions to be worked by the method Cluster Search.
The clusters initialization scans the population from the best to the worst individual, generating new clusters or assimilating the individuals into the clusters already created.New clusters are created when the individual in question is not found within the radius (r = 0.85n) of any clusters already existing, where n is the number of tasks of the sequence.The assimilation of an individual is made in the cluster whose center this individual is found closer.
The measurement of distance between the permutations of the individuals p i and the centers of the clusters c j was adopted as being the amount of necessary exchanges to transform p i into c j .The process of generation of these initial clusters ends when the entire population is scanned or when generated the maximum number of clusters, in the present study it was determined as being 200.
A Cluster Search characteristic is the evolutionary behavior, in which after a start with several clusters created, its number is slowly reduced, prevailing the clusters, in the regions of the space, with the best solutions.
The assimilation process of an individual in a cluster has as basis the Path Relinking process (GLOVER, 1996).From an individual p i , successive exchanges of a pair of tasks will be made until its sequence becomes identical to the sequence of the cluster center.At every exchange a new sequence is generated and evaluated.The pair of tasks chosen to be exchanged is that pair that produces the sequence with the best evaluation at every step.In the end, if the own assimilated individual p i , or the best sequence found in the successive exchanges, has the best evaluation than the cluster center, this becomes the new center of this cluster.
At every attempt to generate a new individual two others are selected in the population, one of them among the best 10%, called base individual, and other among all the individuals in the population, called guide.A crossover process, or recombination, known as BOX (Block Order Crossover) proposed by Syswerda (NAGANO et al., 2012), is used to generate the new individuals.In this technique, the parent individuals are combined through random copy of blocks of genes of both parents, which results in the generation of a single child containing part of the parent's heritage.In this study, the child was generated with 75% of genes from the parent 1 (base) and 25% from the parent 2 (guide).
After recombination, the new individual has 60% of probability to go through a process of improvement in the form of a local search.This improvement can be done by two types of local search, LS1, with probability of 40% to occur, or LS2, with 20%.
Both the LS1 and the LS2 are hybrid processes that use two types of neighborhood, the permutation and the insertion.Both are detailed in Figures 2 and 3, being the Figure 2 about the LS1 and the Figure 3 about the LS2.Another relevant characteristic in these searches is the random character, because both are made through the random choice and without repeating the tasks to be worked in the neighborhood.Such a procedure was proposed, once again, to guarantee a greater diversity of solutions.In the neighborhood of permutation all the possible pairs of tasks of the sequence are exchanged, thereby generating n(n-1)/2 new sequences.In the neighborhood of insertion each task is removed from its position and inserted in all other possible positions, being laterally moved to fill the position left by it, thereby generating (n-1) 2 new sequences.

Local
Every new individual, most times improved by one of the two local searches, and which is not identical to any individual already belonging to the population, is inserted into the population in a position relating to its evaluation, thereby causing the removal of the worst individual present in the population up to the moment.Thus, the evolutionary process eventually updates the population at every new generated individual.evaluation criterion for the makespan.This second local search will be performed when the computing time of the method HE-NIFS exceeds 50% of the computing time proposed by the stopping criterion (elapsed time).This local search will be performed only once, not being repeated until the proposed method ends.During all the process, the best cluster, i.e., that whose center has the best evaluation will be maintained safe.Being this the solution presented by the Cluster Search.
For a better understanding of the Cluster Search, the Figure 4 presents the procedure in detail.

Cluster Search(π)
tls2 := 0; while time < 10% of the total processing time; for i ← 1 up to n;

Results and discussion
Aiming to evaluate the performance of the method HE-NIFS, the method was compared with the best method reported in the literature, known as IG LS proposed by Ruiz et al. (2009).All methods were coded into language C and processed in a microcomputer Intel Xeon 3GHz and 32GB of RAM.
It was used the Taillard database made up by 10 problems for each class, totaling 120 test problems.
The statistics used to evaluate the methods performance were the Percentage of Success (PS) and the Relative Deviation (RD).
The first statistics is defined by the quotient between the total number of problems for which the method obtained the best solution, and the total number of problems resolved.The second quantifies the relative deviation (RD h ) that the method h obtains in relation to the best solution for a same problem, being calculated by: where: M h : makespan obtained by the method h; M .: best makespan obtained by the methods for a determined problem.
The Table 1 presents the summarized results for the values of percentage of success and percentage of relative average deviation for the methods IG LS and HE-NIFS.The same results are presented individually in the Figures 5, 6 and 7.The Figure 5 shows the percentage of success for the methods IG LS and HE-NIFS for the total of problems evaluated according to the variation of the parameter T.  It was observed that none of the evaluated methods presented predominant trend.Both methods oscillated according to the variation of the parameter T.
The Figure 6 presents the percentage of relative average deviation for the methods IG LS and HE-NIFS for the total of problems evaluated according to the variation of the parameter T. In the Figure 6 it was observed a gradual trend for reduction of the relative average deviation for the method HE-NIFS as the value of the parameter T increased.For this reason, according to the Figure 6, it can be concluded that the method HE-NIFS obtained solutions with better quality in comparison to the method IG LS .
The Figure 7 presents the statistical significance of the difference between the heuristics IG LS and HE-NIFS for the values of T referent to the relative deviations.In the figure it was calculated the mean value of heuristic pairs corresponding to 95% confidence intervals.
The Figure 7 pointed out that the quality of the solution between the methods IG LS and HE-NIFS presented significant differences as the values of T were evaluated.The method HE-NIFS has proved to be more stable in relation to the quality of the obtained solutions, on the other hand, the IG LS method presented a growing increase in the confidence interval variation, reflecting in solutions of lower quality according to increasing values of T. The results for the computational experimentation in the present study are motivating, despite the percentage of success obtained by the method HE-NIFS not being far superior when compared to the method IG LS , the results indicate that the method HE-NIFS obtains solutions of better quality presenting little variation when compared to IG LS .Thus, it was verified the superiority of the new method HE-NIFS in relation to the achievement of solutions of better quality comparing the set of evaluated problems.

Conclusion
The experimental results showed that the heuristic method HE-NIFS presented equal performance considering the percentage of success, and superior regarding the relative deviation (quality of the obtained solutions) in comparison to the method IG LS , which is considered the best method currently available.Thus, it can be affirmed that the method HE-NIFS is an alternative method with highest quality for the no-idle flow shop scheduling problem with criterion of minimizing makespan.
The fact that HE-NIFS presented solution of better quality is guaranteed by the great diversity of initial solutions allowed by the Cluster Search that provides a better selection of space of final solutions.
Although the initial results presented had already been considered satisfactory, the method HE-NIFS can still be improved by finding optimal parameters, through an exhausting computational experimentation, in which can be evaluated the parameters variations and their respective provided solutions.This process will constitute a continuity
NEH of Nawaz et al. (1983) with acceleration of Pan and Wang (2008a and b), with computational complexity of O(n 2 m); -The original NEH without acceleration, referred to as NEH na , with computational complexity of O(n 3 m); -SGM of Saadani et al. (2005), as computational complexity of O(n 3 ); -KK of Kalczynski and Kamburowski (2005), with computational complexity of O(n 3 m); -GH_BM of Baraz and Mosheiov (2008), with computational complexity of O(n 3 m); -The new GH_BM2 proposed method with acceleration, with computational complexity of O(n 2 m); -GH_BM2 without acceleration referred to as GH_BM2 na , with computational complexity of O(n 3 m); -FRB3 of Rad et al. (2009), with computational complexity of O(n 3 m); -FRB4 k of Rad et al. (2009) with k values of 4 HDPSO of Pan and Wang (2008a); -DDELS of Pan and Wang (2008b); -FRB5 of Rad et al. (2009); -IG LS of Ruiz and Stützle (2007).

Figure 3 .
Figure 3. Local Search LS2.Aiming to select even more the space of solutions to be worked by the Cluster Search a new local search (LS2) will be performed in the clusters classified as belonging to the best 1/3 portion, by the

Figure 5 .
Figure 5. Percentage of success for the methods IG LS and HE-NIFS.

Figure 6 .
Figure 6.Percentage of relative average deviation for the methods IG LS and HE-NIFS.

Figure 7 .
Figure 7. Mean and confidence intervals (95%) for the pairs of algorithms evaluated.
. 35, n. 2, p. 271-278, Apr.-June, 2013 of the present study with the goal of applying the technique of design of experiments (Design of Experiments -DOE).

Table 1 .
Percentage of success and percentage of relative average deviation for the methods IG LS and HE-NIFS.
*Percentage of success; ** Percentage of relative average deviation.