A Hybrid Shuffled Frog Leaping Algorithm for solving No_Idle Permutation Flow Shop Scheduling Problems

: This paper presents a novel hybrid shuffled frog-leaping algorithm (HSFLA) for solving the no_idle permutation flow shop scheduling problems (NIFS) with the criterion to minimize the maximum completion time (makespan). First, the algorithm employs insert-neighborhood-based local search to enhance the searching ability. Second, it adopts roulette wheel selection operator to generate the global best frog in the early stage of the evolution which can expand the searching solution space. The experimental results show that the proposed algorithm is effective and efficient for different scale benchmarks of NIFS.


Introduction
No-idle permutation flow shop scheduling problems as a typical manufacturing challenge have gained wide attention in academic fields, which occurs in the scheduling environment where the setup time or cost of a machine is so high that shutting down the machine after its initial setup is not cost-effective.In a no-idle permutation flow shop scheduling problem, the processing of each machine has to continuously work.That is, each machine must process jobs without any interruption from the start of processing the first job to the completion of processing the last job.Therefore, it is important both in theory and in engineering applications to develop effective and efficient novel approaches for the no-idle permutation flow shop scheduling problems [1].
However, the no-idle permutation flow shop scheduling problems have not been researched enough so far [2,3].According to the research work [4], the no-idle permutation flow shop scheduling problem to minimize makespan with three machines is strongly NP-hard.Baptiste and Lee [4] and Saadani, Baptiste and Moalla [5] developed a branch and bound algorithm for problem.Kalczynski and Kamburowski [2] developed a constructive heuristic for minimizing the makespan in NIFS problems.Baraz and Mosheiov [3] introduced an efficient improved greedy algorithm consisting of a simple greedy heuristic and an improvement step.With the rapid development of computer technology, some meta-heuristic algorithms are presented to solve NIFS problems.Such as discrete differential evolution (DDE) [6], discrete harmony search algorithms [7], discrete particle swarm optimization (DPSO) [1] and discrete shuffled frog leaping algorithm (DSFLA) [8], etc.
The SFLA has been designed as a meta-heuristic to perform an informed heuristic search using a heuristic function (any mathematical function) to seek a solution of combinatorial optimization problem [9].In essence, it combines the benefits of the genetic-based MAs and the social behavior-based PSO algorithms.But there are a few papers that addressed the shuffled frog leaping algorithm [9][10][11][12][13][14][15].This paper develops a novel hybrid shuffled frog leaping algorithm(HSFLA) for solving the no-idle permutation flow shop scheduling problems with makespan criterion.In the proposed HSFLA, the balance between global exploration and local exploitation is well stressed by employing roulette wheel selection to effectively perform exploration and by applying an effective insert neighborhood search algorithm to perform exploitation.

The no-idle permutation flow shop scheduling problem
The no-idle permutation flow shop scheduling problem can be described as follows: Each of n jobs, available at time zero, from set J = {1, 2,..., n} will be sequenced through m machines (k =1, 2,..., m).Job j∈J has a sequence of m operations (o j1 , o j2 ,..., o jm ).Operation o jk corresponds to the processing of job j on machine k during an uninterrupted processing time p(j, k), where setup time is included into the job processing time.At any time, each machine can process at most one job and each job can be processed on at most one machine [16].To follow the no-idle restriction, each machine must process jobs without any interruption from the start of processing the first job to the completion of processing the last job.In other words, there must be no idle time between the processing of any consecutive operations on each machine.The problem is then to find a schedule such that the processing order of jobs is the same on each machine and the maximum completion time (makespan) is minimized.Paper [1] proposed a new way of calculating makespan whose computational complexity is O(mn) and a speed-up method to evaluate the whole insert neighborhood of a job permutation with (n−1) 2 neighbors in time O(mn 2 ).

Shuffled frog-leaping algorithm (SFLA)
The SFLA is inspired by the behavior of a group of frogs seeking food.It is a combination of deterministic and random approaches.The deterministic strategy allows the algorithm to use response surface information effectively to guide the heuristic search as in PSO.The random elements ensure the flexibility and robustness of the search [6].In the SFLA, the population consists of a set of frogs (solutions) that is partitioned into subsets referred to as memeplexes.The different memeplexes are considered as different cultures of frogs, each performing a local search.Within each memeplex, the individual frogs hold ideas, that can be influenced by the ideas of other frogs, and evolve through a process of memetic evolution.The frogs with the best and the worst fitness are identified as P b and P w , respectively in each memeplex, Also, the frog with the global best fitness is identified as P g , Then, a process similar to PSO is applied to improve only the frog with the worst fitness (not all frogs) in each cycle.Accordingly, the position of the frog with the worst fitness is adjusted as follows: where rand( ) is a random number between 0 and 1, and D max is the maximum allowed change in a frog's position.If the new P w is better than the old one, calculations in Eq.1 and Eq.2 are repeated but with respect to the global best frog (i.e.P g replace P b ).If no improvement becomes possible in this case, then the spread of defective meme is stopped by randomly generating a new frog at a feasible location to replace the frog whose new position was not favorable to progress.After a defined number of memetic evolution steps, ideas are passed among memeplexes in a shuffling process [8].The local search and the shuffling processes continue until defined convergence criteria are satisfied [7].

Solution representation
It is important to design solution representation in a meta-heuristic algorithm.One of the most widely used representation for sequencing problems including permutation flow shop sequencing problem is job-to-position representation.In this kind of representation, a single row array of the size equal to the number of the jobs to be scheduled is considered.The value of the first element of the array shows which job is scheduled first.The second number shows which job is scheduled second, and so on.Table 1 shows how this representation is depicted.

Initial population
In order to guarantee the initial population with certain diversity in solution space, the frogs in the initial population are generated as follows: Step 1 i=1; Advanced Engineering Forum Vol. 1 111 Step 2 Generate a permutation corresponding to the ith frog, in which and j π was generated in random.

Step 3 i ++;
Step 4 If i is not more than the number of frogs in the population, go to step 2 and continue, otherwise, stop.

Update individual frogs
Eq.1 and Eq.2 show that the essence of updating the worst frog P w is a process of P w studying the local best frog P b or the global best frog P g .Where rand() is a random number between 0 and 1, and it represents the degree of the frog P w inherit the genes of the best frog.So the process of above can be replaced by crossover.When rand() is 1, P w is replaced by the best frog P b or P g .When rand() is 0, P w stays the same.The crossover is implemented detailed as follows: Step 1 Select a part of P b randomly as crossover string.
Step This process shows that the update of individuals is easy to implement and the new frog can inherit the effective memes from its parents.

Insert-neighborhood-based local search
The population depends extremely on the local best result and the group best result.Based on this character, the speed-up algorithm was proposed employs the simple neighborhood search [1], and it can improve the convergence rate of SFLA.In SFLA every memeplex evolves through memetic algorithm (MA).All frogs are reordered and classified, so the information can flow and is shared among all memeplexs.This can speed up the convergence rate of population.In genetic algorithm (GA) evolutional information is obtained from the entire population in probability.Otherwise frogs can get updated information from the local best solutions and the global best solutions.Compare of GA the information flows in one way, and the purpose of flow is more efficient in SFLA.The search was deeply affected by the best individuals.So strengthening the detection capacity of the local search can improve SFLA's performance.Based on this analysis, the improved algorithm was proposed employs the simple neighborhood search on the best results.But the scope of precise search is only limited to the nearest neighborhood of the best results.It is likely to make the solutions "wandering" around the old state for a long time, which make the algorithm more easily trapped into local minima.In order to enhance the algorithm's ability to jump out of local optimum, we can implement a random insert move after a neighborhood search and repeat the process several times.If a better solution is found, the old solution is replaced by the new one.Through the precise search the population can move to the optimal solution more quickly, and it can improve the convergence rate of SFLA.Insert or swap neighborhood structure can be used in simple neighborhood search, and the process depends on the problems.

Roulette wheel selection the global best solution
The best solution P g is the one whose performance is best in the population, and the search was largely infected by the global best solution P g .So in the early iterations, the evolution tends to be controlled by the super-frog who has the absolute advantage.If the objective function is multimodal, in the long run, the frog is perhaps not able to guide the search to the global optimum, so it is prone to premature convergence.In order to other "less good" frogs have the opportunity to guide the search, roulette wheel selection operator is used in SFLA.This measure can reduce the control function of the super-frogs and avoid the premature phenomenon to some extent.Detailed implementation method of roulette wheel selection P g from all the local best frogs P b is as follows:

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Steps of the hybrid shuffled frog-leaping algorithm
The HSFLA meta-heuristic strategy is summarized in the following steps.

Global exploration
Step 0 Initialize.Select M and N, where M is the number of memeplexes and N is the number of frogs in each memeplex.Therefore, the total sample size F in the swamp is given by F = M×N.Select Tr, where Tr is the time of applying roulette wheel selection operator to get the best frog P g.
Step 1 Generate a virtual population.Sample F virtual frogs U(1),U(2), . . .,U(F ) in the feasible space , where n is the number of jobs.The ith frog is represented as a solution of NIFS problems . Compute the makespan as f (i) for each frog U(i).So the smaller of f (i), the better of U(i).
Step 2 Rank frogs.Sort the F frogs in order of decreasing performance value.Store them in an array X = {U(i), f(i), i = 1,… , F }, so that i = 1 represents the frog with the best performance value.
Step 3 Partition frogs into memeplexes.Partition array Sort X in order of decreasing performance value.Update the population best frog's position P g .Step 9 Check convergence.If the convergence criteria are satisfied, stop.Otherwise, return to step 3.
Typically, the decision on when to stop is made by a prespecified number of consecutive time loops when at least one frog carries the "best memetic pattern" without change.Alternatively, a maximum total number of function evaluations can be defined.

Local exploration: frog-leaping algorithm
In step 7 of the global search, evolution of each memeplex continues independently iN times.After the memeplexes have been evolved, the algorithm returns to the global exploration for shuffling.Below are details of the local search for each memeplex.
Step 0 Set im = 0 where im counts the number of memeplexes and will be changed from zero to the total number M of memeplexes.Set iN = 0 where iN counts the number of evolutionary steps and will be changed from zero to the maximum number N max of steps to be completed within each memeplex.
Advanced Engineering Forum Vol. 1 113 Step 3 Improve the worst frog's position.A order crossover is applied to the worst frog P w and the best local frog P b.
Step 4 If this process produces a better frog (solution), it replaces the worst frog.Otherwise, the order crossover is repeated with respect to the global best frog P g .
Step 5 If no improvement becomes possible in this latter case, then a new solution is randomly generated to replace the worst frog with another frog having any arbitrary fitness.
Step 6 If iN < N ma , go to step 2.
Step 7 If im < M, go to step 1. Otherwise return to the global search to shuffle memeplexes.

Experimental results
To test the performance of the proposed HSFLA , computational simulations were carried out with the flow shop benchmark set of Taillard [17], which is composed of 12 sub-sets of given problems with the size ranging from 20 jobs and five machines to 500 jobs and 20 machines, and each sub-set consists of ten instances.Paper [8] proposed DSFLA and a speed-up algorithm which we referred as DSFLA*.For each instance of each sub-set, DPSO, DSFLA, DSFLA* and HSFLA algorithms independently carried out five replications with T=30mn millisecond, and each replication was compared to the solution produced by NEH [18] when modified for the no idle permutation flow shop scheduling problems.The parameters were set as follows: M=20, N=5, F=100, Ts=T/10 millisecond.The average percentage relative deviation (APRD) and the standard deviation (SD) were calculated as the statistics for the performance measures.The algorithm was coded in C++ and run on an Intel PIV 2.0 GHz PC with 2.0 GB memory.And the results are reported in Table 2. From Table 2, it is shown that the statistical results generated by the HSFLA are significantly better than those by DSFLA and DPSO algorithm, which indicates that by insert neighborhood search the algorithm convergences more quickly and by roulette wheel selection it's searching space is more expanded.The results of APRA and SD carried by HSLA are less than DSFLA* show that HSFLA is super to DSFLA*.This indicates that roulette wheel selection the global best solution in early time can bring down the dependence on the global best individual, and can avoid trapping into local optimum to a certain extend.The experimental results indicate that the proposed HSFLA outperforms the DSFLA with respect to these measures and is able to improve the quality of the obtained solutions, especially for the large-size problems.Thus, it is concluded that HSFLA is more robust, effective and efficient than other algorithms of above.

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Conclusion
A hybrid shuffled frog leaping algorithm (HSFLA) has been proposed in this paper for solving no_idle permutation flow shop scheduling problems with the criterion to minimize the maximum completion time( makespan).Insert-neighborhood-based local search improves the quality of the meme of the best individuals which guide the population towards a goal more quickly.Roulette wheel selection the global best individual can make the frogs with better memes(ideas) contribute more to the development of new ideas than frogs with poor ideas, which ensure the fast convergence.Also can it make the infection process to be competive and bring down the algorithm's independence on someone excessively, which can avoid the search trapping into the local optimum to a certain degree.Experimental results indicate that the algorithm proposed in this paper can balance the exploration and exploitation, and it is an effective and efficient algorithm for solving no_idle permutation flow shop scheduling problems.

*
The work was supported by a grant from the Research Development Program of Shandong Education Committee (No. J10LG25).

Table 1
Individual vector and corresponding job permutations 2Combine P b and P w by put the crossover string before (or behind) P b and delete the element which has appeared in crossover string.
Application insert-neighborhood-based local searches to all the local best frogs.Step 5 Selection the best frog P g.If the execution time is less than Ts, get the best frog's position P Step 6 Application insert-neighborhood-based local search to the global best frogs P g .Step 7 Memetic evolution within each memeplex.Evolve each memeplex Y k , k = 1, . . ., M, according to the frog-leaping algorithm outlined below.Step 8 Shuffle memeplexes.After a defined number of memetic evolutionary steps within each memeplex, replace Y 1 N}, k=1,2,…,M e.g., for m = 3, rank 1 goes to memeplex 1, rank 2 goes to memeplex 2, rank 3 goes to memeplex 3, rank 4 goes to memeplex 1, and so on.Record the local best frog's position P b of every memeplex, such that P bk = U(k), k=1,2,…,M.Step 4 g by roulette wheel selection from all the local best frogs P b.Otherwise, record the best frog's position P g in the entire population (F frogs) (where P g = U(1)).

Table 2
Comparison of different algorithms