A Novel Differential Evolution Algorithm for TWET-NFSSP with SDSTs and RDs

A novel differential evolution (DE) algorithm, namely DE_TWET, is presented to deal with the no-wait flow-shop scheduling problem (NFSSP) with sequence-dependent setup times (SDSTs) and release dates (RDs). The criterion is to minimize a total weighted earliness/tardiness (TWET) cost function. The presented algorithm is a hybrid of DE, problem’s properties, and a special designed local search. In DE_TWET, DE is adopted to execute global search in the solution space, and the problem’s properties are utilized to give a speed-up evaluation method and construct the local search, and the special local search is designed to enhance the local search ability of DE. Experimental results and comparisons demonstrate the effectiveness and robustness of the presented algorithm.


Introduction
With the development of just-in-time (JIT) manufacturing systems, the study on the scheduling problems with both earliness and tardiness (E/T) costs is of greater significance. In this paper, a typical production scheduling problem with strong engineering background [1,2], the no-wait flow-shop scheduling problem (NFSSP) with sequence-dependent setup times (SDSTs) and release dates (RDs), is considered, whose criterion is to minimize a total weighted earliness/tardiness (TWET) cost function. In such a case, each job j must be processed through all machines without any interruption, and both the setup times and the release dates need to be explicitly treated, and an optimal schedule is the one that all jobs finish exactly on their due dates. This type of model is classified as , which can also be identified as it can be concluded that is NP-hard [3]. Moreover, literature review show that the researches on the scheduling problems with both sequence-dependent setup times and release dates are very limited [4]. Thus, it is meaningful and practical to develop an effective algorithm for the considered problem. Differential evolution (DE) algorithm, which was first designed for optimizing complex continuous problems [5], is one of the latest population-based evolutionary methods. Owing to its quick convergence and easy implementation, nowadays, the DE algorithm has gained many successful applications in different fields. However, due to DE's continuous nature, the applications of the DE-based algorithms to scheduling problems are still limited. Tasgetiren et al. [6] devised a DE-based algorithm for flow-shop scheduling problems (FSSPs) to minimize makespan. Onwubolu and Davendra [7] developed a DE-based approach for FSSPs, where makespan, mean flowtime, and total tardiness were considered. Qian et al. [8] designed an very efficient DE-based algorithm for NFSSPs with the makespan criterion. Wang et al. [9] proposed an efficient discrete differential evolution algorithm for FSSPs with blocking. Recently, Hu et al. [10] U  r  ST  wait  no  Fm  /  ,  ,  / , which is the current best approach for the problem considered. To the best of our knowledge, there has no promising results on , and there has no published work addressing it by using DE-based algorithm.
In the current paper, a novel DE algorithm (DE_TWET) is proposed to deal with TWET-NFSSP with SDSTs and RDs. In our DE_TWET, firstly, a largest-order-value (LOV) in [11] is utilized to map the real-valued vectors or individuals in DE to job permutations so as to make DE suitable for solving NFSSP; secondly, a speed-up evaluation method based on the property of the considered scheduling problem is given to calculate the cost function efficiently; thirdly, the DE-based search is adopted to perform global exploration in the solution space and guide the whole search to the promising regions/solutions, while a special local search based on problem's properties is developed to emphasize exploitation from those regions. Test results and comparisons demonstrate the efficiency and robustness of the proposed DE_TWET.
The remainder of this paper are partitioned into four sections. Section 2 introduces the mathematical model of TWET-NFSSP with SDSTs and RDs. Section 3 presents DE_TWET in details. Section 4 provides and discusses test results and comparisons. Finally, Section 5 gives some concluding remarks and suggestions of future research.

TWET-NFSSP with SDSTs and RDs
The NFSSP with SDSTs and RDs can be described as follows. There are n jobs and m machines. Each of n jobs will be sequentially processed on machine m ,..., 2 , 1 . The processing time of each job on each machine is deterministic. At any time, preemption is forbidden and each machine can process at most one job. To satisfy the no-wait restriction, each job must be processed without interruptions between consecutive machines. Thus, all jobs are processed in the same sequence on all machines. In a flow-shop with SDSTs, setup must be performed between the completion time of one job and the start time of another job on each machine, and setup time depends on both the current and the immediately preceding jobs at each machine. In a flow-shop with RDs, if a machine is ready to process a job but the job has not been released yet, it stays idle until the release date of the job. Then l j i ML , can be calculated as follows: (1) Accordingly, can be calculated by using the following formula: ( ( So, can be calculated as follows: The aim of this paper is to find a permutation * π in the set of all permutations Π such that

DE_TWET for TWET-NFSSP with SDSTs and RDs
In this section, we will propose DE_TWET for TWET-NFSSP with SDSTs and RDs after explaining the solution representation, speed-up evaluation method, DE-based global search, and special designed local search. Solution Representation. Owing to the continuous nature of the individuals in DE, the standard encoding scheme of DE cannot be directly applied to NTJ-NFSSP with SDSTs and RDs. In this paper, we adopt a largest-order-value (LOV) rule in [11] ]. According to LOV rule, are firstly ranked by descending order to get the sequence = ]. Then the job permutation i π is calculated by the following formula: According to our tests, LOV rule can achieve better results than the well-known random key representation.

Speed-up Evaluation Method.
Based on the mathematical model of TWET-NFSSPs with SDSTs and RDs in Section 2, the interchange of the job at the uth dimension (i.e., u j ) and the job at the vth dimension (i.e., v j ). The interchange-based neighborhood of π can be expressed as As for a fixed u , the subset of ) (π e interchang N can be written as Thus, it has . Then, based on Eq. 4-Eq. 7, when is given as follows: Step 1: Set 1 = u and π π = best ; Step 2: Advanced Engineering Forum Vols. 6-7 751 Step 3: If , can be given as follows: Step 1: Set KL u = and π π = best ; Step  N is.
Procedure of Local Search. Let ) , , ( v u insert π denote the insertion of u j in the vth dimension of π . The procedure of DE_TWET's local search is given as follows: Step

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In the above procedure, Step 2 is the perturbation phase, which can avoid cycling and overcome local optima, and Step 3 executes exploitation from the region obtained by Step 2. DE_TWET. According to the above solution representation, speed-up evaluation method, speed-up evaluation method, and special designed local search, the procedure of DE_TWET is presented as follows: Step 0: Let t denote a generation, the lth variable of tmp , CR the crossover probability, and ) 1 , 0 ( random the random value in the interval [0,1]. The objective value of each individual is calculated by using speed-up evaluation method.
Step 1: Input N , Step 4: Set t=1 and select an individual ) 0 ( best X from ) 0 ( Pop as best with the minimum objective value. Step 5: Evolution phase (Step 5 through Step 11). Set i =1.
Step 7: Perform DE's Mutation and Crossover.
Step 11: Apply special designed local search to best .
Step 12: Set t=t+1. If ≤ t max _ t (the maximum number of iteration), then go to Step 5.
Step 13: Output best and its objective value. It can be seen that DE_TWET not only applies the DE-based algorithm to obtain promising regions within the entire solution space, but also applies a special designed local search to enhance the search quality. Because both global and local search are well balanced, DE_TWET is expected to achieve good results. r is an integer that is randomly generated in ] 150 , 0 [ α n , where the parameter α is used to control the jobs' arrival speeds. The values of α are set to 0, 0.2, 0.4, 0.6, 0.8, 1 and 1.5, respectively. Moreover, the due date of each job is specified as follows:

Test Results and Comparisons
Step 1: For each problem p , randomly generate a permutation of the jobs.
Step 2: Calculate the completion time of each job in the permutation specified in Step 1.
Step 3: Specify the due date of each job by where i j p d , is the due date of job i j to problem p , For the purpose of evaluating the effectiveness of DE_TWET, we carry out simulations to compare our DE_TWET with an iterated greedy heuristic (IG) [13] and a hybrid DE (DE_NTJ) [10]. IG is the new state-of-the-art approach for solving FSSPs with SDSTs [13], and DE_NTJ is one of the current best algorithms for NFSSPs with SDSTs and RDs [10]. Moreover, we also compare DE_TWET with its five variants, whose abbreviations are as follows: (1) DE_TWET_S1: In DE_TWET's local search, only speed-up evaluation method is used.
(2) DE_TWET_S12: In DE_TWET's local search, only speed-up evaluation method and speed-up search method are used.
(4) DE_ TWET_0.3n: In Step 3 of DE_TWET's local search, KL is set to ) is a round function rounds a real-type value a to an integer-type value. the average value of )) ( the best value of )) (

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Comparisons of DE_TWET with Its Five Variants. In order to investigate DE_TWET's local search ability, we compare DE_TWET with its five variants. Statistical results can be found in Table  1. In Table 1, it can be seen from ARI metric that DE_TWET can obtain better results than its five variants for most instances, which manifest the importance of simultaneously adopting two speed-up methods and two strategies in local search and show the necessity of setting KL to the smallest value. From SD metric, it can be seen that the SD values of DE_TWET are smaller than those of five variants for almost all instances. This means DE_TWET is more robust than its five variants. Comparisons of DE_TWET, IG, and DE_NTJ. To show the effectiveness of DE_TWET, we carry out some comparisons with IG [13] and DE_NTJ [10]. The test results of the three algorithms are shown in Table 2. From Table 2, it can be concluded that the searching quality of DE_TWET is better than that of IG for almost all instances and is superior or comparable to that of DE_NTJ. Therefore, it is concluded that DE_TWET is an effective and robust algorithm for dealing with TWET-NFSSP with SDSTs and RDs.

Conclusion and Future Research
To the best of the current authors' knowledge, this is the first report on the application of differential evolution (DE) algorithm for NFSSP with SDSTs and RDs with the criterion to minimize the total weighted earliness/tardiness. In our presented algorithm, DE-based global search was used to perform exploration for promising regions within the entire solution space, while a special local search based on problem's properties was developed to stress exploitation in these regions. Due to the hybridization of DE and local search, DE_TWET's search behavior can be enriched and its search ability can be enhanced. Simulations and comparisons showed the effectiveness and robustness of DE_TWET. The future work is to develop some effective DE-based algorithms for dynamical scheduling and reentrant scheduling.