An Enhanced Differential Evolution Algorithm with Fast Evaluating Strategies for TWT-NFSP with SSTs and RTs

The no-wait ﬂow-shop scheduling problem with sequence-dependent setup times and release times (i.e., the NFSP with SSTs and RTs) is a typical NP-hard problem. This paper proposes an enhanced diﬀerential evolution algorithm with several fast evaluating strategies, namely, DE_FES, to minimize the total weighted tardiness objective (TWT) for the NFSP with SSTs and RTs. In the proposed DE_FES, the DE-based search is adopted to perform global search for obtaining the promising regions or solutions in solution space, and a fast local search combined with three presented strategies is designed to execute exploitation from these obtained regions. Test results and comparisons with two eﬀective meta-heuristics show the eﬀectiveness and robustness of DE_FES.


Introduction
e no-wait flow-shop scheduling problem (NFSP) has been widely studied for more than 30 years [1,2]. Nevertheless, literature reviews on production scheduling with no-wait or setup constraints in [1,3] manifest that the NFSP with sequence-dependent setup times (SSTs) and release times (RTs) has not received the attention it needs. In fact, SSTs and RTs are two kinds of constraints widely existing in the real-life NFSPs. ese constraints can be found in pharmaceutical processing, metal processing, and chemical processing [4][5][6][7][8]. Moreover, in the current increasingly fierce global competition, many enterprises are trying to reduce the tardy jobs in order to maintain customer satisfaction and avoid the loss of customer orders. Hence, it is important to design an effective algorithm to minimize the total weighted tardiness objective (TWT) for the NFSP with SSTs and RTs (i.e., F m /no − wait, ST sd , r j / w j T j ).
Optimization algorithms have been successfully used to deal with a variety of important engineering problems [9][10][11][12][13][14]. In recent years, evolutionary algorithms have become an important class of optimization algorithms for addressing production scheduling problems [15][16][17][18][19]. Differential evolution (DE) is a very competitive evolutionary algorithm for solving continuous optimization problem [20][21][22]. It iteratively executes three key operators, i.e., mutation, crossover, and selection. Due to its easy implementation, simple mechanism, and quick convergence, DE has been applied to addressing various optimization problems in academia and industry. Nevertheless, due to its continuous nature, the studies on DE for combinatorial optimization problems are restricted. Tasgetiren et al. [23] developed a novel DE via introducing the interchange-based local search and the smallest position value (SPV) rule for the flow-shop scheduling problems (FSPs), whose criterion is to minimize makespan. Onwubolu et al. [24] presented a new approach based on DE for the FSPs. ey considered three criteria, i.e., mean flowtime, makespan, and total tardiness. Pan et al. [25] designed a discrete DE to address the FSPs effectively. Qian et al. [26] proposed a hybrid DE (HDE) for the NFSP with makespan criterion. By reasonably utilizing the corresponding problem's properties, HDE is of excellent search ability. Wang et al. [27] designed an effective discrete DE for the blocking FSPs. Hu et al. [28] designed an effective DE, namely, DE_NTJ, for the NFSP with SSTs and RTs. e criterion is to minimize the number of tardy jobs. Qian et al. [29] developed a DE with two speed-up methods (DE_TSM) to minimize the total completion time for the NFSP with SSTs and RTs. Tang et al. [30] presented an improved DE (IDE) to deal with a dynamic scheduling problem in steelmaking-continuous casting production. IDE can obtain better performance than the compared algorithms. Santucci et al. [31] proposed an algebraic DE for the FSPs with total flowtime criterion. Wu and Che [32] developed a memetic DE (MDE) to solve the biobjective unrelated parallel machine scheduling problem. In their tests, MDE outperforms two famous multiobjective algorithms. However, according to the existing literatures, there are no studies on F m /no − wait, ST sd , r j / w j T j . erefore, it is imperative to develop an efficient algorithm for the TWT-NFSP with SSTs and RTs. e developed algorithm can be easily extended for other kinds of flow-shop scheduling problems with no-wait constraint.
is paper proposes an efficient DE-based algorithm with fast evaluating strategies (DE_FES) to minimize the total weighted tardiness for the NFSP with SSTs and RTs. Compared with the existing research, the main contributions of this work are summarized as follows: (1) e permutation-based model for the TWT-NFSP with SSTs and RTs is formulated for the first time. Moreover, the NP-hardness of this problem is proved via a reduction chain starting from a known NP-hard problem. e remaining part of this paper is organized as follows. In Section 2, the TWT-NFSP with SSTs and STs is formulated. In Section 3, three fast solution evaluating strategies and some theoretical analyses are presented. In Section 4, DE_FES is proposed after introducing its main components. In Section 5, the test results are provided. Finally, conclusions and future research are given in Section 6.

TWT-NFSP with SSTs and RTs
e NFSP with SSTs and RTs is described as follows. ere are n jobs to be processed on m machines in the same order. Each machine can only process one job at a time. Each job must be processed without waiting time between consecutive operations. Setup times depend on the sequence of the jobs. If the job has not been released, the machine remains idle until it is released.

NFSP with SSTs and RTs.
Denote π � [j 1 , j 2 , . . . , j n ] is the permutation or schedule to be processed, p j i ,l is the processing time of job j i on machine l, sp j i is the total processing time of job j i on m machines, ML j i ,l is the minimum delay time between j i−1 and j i on the machine l, L j i−1 ,j i is the minimum delay time between j i−1 and j i on the first machine, s j i−1 ,j i ,l is the sequence-dependent setup time between j i−1 and j i on machine l, r j i is the arrival time of j i , St j i is the start processing time of j i on the first machine, and C j i is the completion time of j i on the last machine. Let p j 0 ,l � 0 for l � [1, . . . , m]. en, ML j i ,l can be calculated as follows: By using (1), L j i−1 ,j i can be calculated with the following formula: Obviously, St j i can be written as follows: us, C j i can be calculated as follows: Figure 1(a) gives an example of the NFSP with SSTs when n � 4 and m � 2. It can be seen that ML j i ,l and L j i−1 ,j i are determined by the model of the NFSP with SSTs. Figure 1(b) gives an example of the NFSP with SSTs and RTs when n � 4 and m � 2. It can be seen that St j i and C j i are determined by the model of the NFSP with SSTs and RTs.

TWT-NFSP with SSTs and RTs.
Denote d j i is the due date of j i , w j i is the weighted value of j i , and Π is the set of all permutations. en, TWT(π) can be calculated as follows: 2 Complexity e TWT-NFSP with SSTs and RTs is to find an optimal permutation π * in Π such that 2.3. Problem Complexity. e TWT-NFSP with SSTs and RTs can be described by a triplet F m /no − wait, ST sd , r j / w j T j . e NP-hardness of this problem can be determined by using a reduction chain starting from a known NP-hard problem, i.e., F m /no − wait, ST sd / C j [3,8]. Problem A reduces to problem B (denoted A ∝ B) means A is just a special case of B, and B is at least as difficult to solve as A [2].
Proof. Obviously, F m /no − wait, ST sd / C j reduces to F m /no − wait, ST sd / C j by setting r j � 0 for all j, and F m /no − wait, ST sd / C j reduces to F m /no − wait, ST sd / C j by setting w j � 0 and d j � 0 for all j. Since each of the above reductions is completed in a polynomial time, a polynomial Turing reduction chain can be established as follows: By using (8), it can be concluded that F m /no − wait, ST sd , r j / w j T j is NP-hard because F m /no − wait, ST sd / C j is NP-hard [3,8]. eorem 1 holds. e reductions between objective functions are given in Figure 2. It can be seen from Figure 2 that w j T j is one of the most difficult criteria. Based on the reduction concept in [2], all the other types of no-wait flow-shop scheduling problems reduce to the NFSP with SSTs and RTs and almost all the other objective functions reduce to TWT (see Figure 2), which means the TWT-NFSP with SSTs and RTs is a most difficult one and can be utilized to represent a wide range of real-life scheduling problems. us, the study on the TWT-NFSP with SSTs and RTs has general and theoretical meanings.

Three Fast Evaluating Strategies
e reasonable utilization of problem-dependent properties is very useful for designing effective algorithms [33]. So, this section analyses the properties of the considered problem Sj 0 , j 1 , 2 Sj 0 , j 1 , 1

Strategy One: Fast Computing Strategy.
In the model of the NFSP with SSTs and RTs, L j i−1 ,j i is only decided by j i−1 and j i . By utilizing this property, the computing complexity (CC) of TWT(π) can be reduced. at is, L j i−1 ,j i , sp j i , and n i�1 sp j i are calculated and recorded at DE_FES's initial stage, and then they are treated as constant values at DE_FES's evolution stage. is strategy reduces the CC of TWT(π) from O(nm) to O(n).

3.2.
e Interchange-Based Neighbourhood. According to [34], the diameter of interchange is n − 1. at is to say, one solution π can transit to any other solution by utilizing interchange at most n − 1 times. e diameter of interchange is one of the shortest among those most-used neighbourhoods.
at is, the neighbors or solutions generated by interchange are closer to each other. So, this subsection chooses to analyze and utilize the properties of interchange's neighbourhood.
e interchange-based neighbourhood of π can be given as where interchange(π, u, v) denotes the interchange of j u and j v . e size of N interchange (π) is n(n − 1)/2. Figure 3 shows a small example of π n,u,v interchange when n � 10, u � 4, and v � 7.

Strategy Two: Fast Scanning Strategy for N interchange (π).
A speed-up neighbourhood scanning strategy is presented in this subsection. is strategy is necessary for devising a fast local search.
Denote Find Best N interchange (π) is the method of obtaining the best neighbor with TWT criterion in N interchange (π) (i.e., the N interchange (π)-based neighbourhood scanning method). When u � 1,. . .,n − 1 and i � 1,. (5) and (6), it has It is clear from (10) that in Find Best N interchange (π), St j i and i ll�1 w j ll T j ll (i � 1, . . . , n − 1) can be computed and recorded before evaluating or scanning each neighbor in N interchange (π), and then they can be treated as constant values when evaluating can be directly replaced by St j u−1 and u−1 i�1 w j i T j i , respectively, and St j u ′ can be calculated from St j u−1 . When scanning each neighbor in N interchange (π), only n i�u w j i ′ T j i ′ needs to be calculated. So, by using strategies one and two, the CC of Find Best N interchange (π) can be decreased to some extent.

Strategy ree: Fast Nonimproving Neighbor Identification Strategy for N interchange (π).
According to the N interchange (π)-based neighbourhood properties, two lemmas are stated and then are used in the proof of eorem 2. By utilizing eorem 2, the nonimproving neighbors in N interchange (π) can be identified with lower CC, which means the CC of Find Best N interchange (π) can be further reduced.
Denote ΔSt j y (y � 1, 2, . . . , n − 1) and ΔSt j y ≥ 0) is a start processing time delay of j y on the first machine, St j y ′ is the start processing time of j y on the first machine when ΔSt j y is added to is the start processing time of job j y+ll on the first machine when St j y is set to St j y ′ , and T j y+ll ′ (ll � 1, . . . , n − y) is the tardiness of job j y+ll on the first machine when St j y is set to St j y ′ . 1 0 1 0 9 9 j 1 j 2 j 3 j 4 j 5 j 6 j 7 j 8 j 9 j 10 j′ 1 j′ 2 j′ 3 j′ 4 j′ 5 j′ 6 j′ 7 j′ 8 j′ 9 j′ 10 j u-1 j v+1 j′ u-1 j′ v+1 Figure 3: A small example of π n,u,v interchange when n � 10, u � 4, and v � 7. Proof. Let Δ y+ll � St j y+ll ′ − St j y+ll (ll � 1, . . . , n − y) denote the start processing time delay of j y+ll . Lemma 1 can be proved by using mathematical induction.
If St j y ′ > St j y , according to the value of d j y+ll , three cases are discussed as follows: Under this case, with (16) and (17), it has (16) and (17), it has Under this case, with (16) and (17), it has Based on the above analysis, Lemma 2 holds.
Based on the above analysis, eorem 2 holds. By utilizing eorem 2, after v i�u w j i ′ T j i ′ and St j v+1 ′ have been computed, the nonimproving neighbors can be detected without calculating n i�v+1 w j i ′ T j i ′ if the conditions in eorem 2 are satisfied. So, eorem 2 is helpful in reducing the CC of Find Best N interchange (π). Let S123 Find Best N interchange (π) denote Find Best N interchange (π) with the above three strategies. Note that S123 Find Best N interchange (π) is a key component in the local search of our proposed DE_FES.

DE_FES for TWT-NFSP with SSTs and RTs
In this section, DE_FES's main components designed for addressing the TWT-NFSP with SSTs and RTs are explained as follows.

Solution Representation.
As mentioned in Section 1, the original DE cannot be directly used to solve the TWT-NFSP with SSTs and RTs. Hence, a largest-order-value (LOV) rule in [35] en, π i can be obtained by using the following formula: It is worth mentioning that the LOV rule achieved better results than the famous random key rule in our previous tests.

DE-Based Global Search.
Since DE exhibited strong global search ability in many previous studies, DE_FES's global search is devised according to the DE/rand-to-best/1/ exp scheme [28,29,35,36], in which the base vector is set to the best of all individuals. is means the valuable information of the best individual can be shared among all individuals during the evolution process.

Fast Local Search.
Denote insert(π, u, v) is the operator of inserting j u in the v th dimension of π. e pseudocode of our fast local search is provided as follows (Algorithm 1).
In Algorithm 1, Step 2 is utilized to avoid plunging into local optima and it leads the search to a quite different region. In Step 3, three proposed strategies in Section 3 are adopted in S123 Find Best N interchange (π), which can help the local search reach more regions in the same running time. Hence, Step 3 performs a deep and efficient exploitation from the region found by Step 2. It is worth noting that S123 Find Best N interchange (π), designed according to the properties of the problem, is the key part that distinguishes the above local search from the local searches in the existing hybrid DE-based algorithms. is is the first time to analyze the lowbound property of the neighbors in N interchange (π) (see eorem 2 in Subsection 3.4) and utilize this property to reduce the CC of local search. By utilizing the low-bound property (corresponding to strategy three) to further reduce the CC of neighbor evaluation, all neighbors in N interchange (π) can be evaluated in a very short time. Obviously, searching the whole neighbourhood with low CC is an efficient and ideal search mode. erefore, S123 Find Best N interchange (π) adopts the "comprehensive neighbourhood search scheme" similar to that in [29], instead of the "first-improvement-move neighbourhood search scheme" in [28] and the "random variable neighbourhood search scheme" in [35]. Meanwhile, due to the use of the low-bound property, the CC of S123 Find Best N interchange (π) is smaller than that of the neighbourhood search in [29], which allows DE_FES to have a more efficient local search engine.

DE_FES. Based on Subsections 4.1-4.3, the pseudocode of DE_FES is given as follows (Algorithm 2).
It can be seen from Algorithm 2 that DE_FES remains the basic characteristic of the original DE. In DE_FES, DE's standard crossover and mutation are utilized to generate candidate individuals, and DE's competition scheme is still used to get new individuals. Meanwhile, DE_FES also adopts the LOV rule, fast computing strategy, and fast local search with three proposed strategies to make DE suitable for addressing the considered problem efficiently. Not only does DE_FES adopt DE's parallel searching scheme to find promising regions, but it also utilizes an efficient local search based on three problem-dependent strategies to execute deep exploitation in these promising regions. Because the global and local searches are well balanced, DE_FES is expected to acquire satisfactory solutions in a reasonable time.

Experimental Setup.
To test the performance of DE_FES, numerical tests are carried out by using a series of randomly generated instances. e n × m combinations are set as {20, 30, 50, 70} × {5, 10, 20}. Both the setup time s j i−1 ,j i ,l and the processing time p j i ,l are generated from a uniform Step 1: transform individual X i (t) to π i 0 via the LOV rule.
Step 5: transform π i 0 back to X i (t).

ALGORITHM 1: Fast local search.
Step 0: denote CR is the crossover probability, random (0, 1) is the random value in the interval (0, 1), t max is the maximum generation, t is a generation, Pop(t) is the population with size N p at t, , and tmp l is the lth variable of tmp. Each X i (t) is evaluated or calculated via strategy one. Step 1: input N, N p ≥ 3, CR ∈ [0, 1], and let bounds be lower(x i,l ) � 0 and upper(x i,l ) � 4, l � 1, . . . , N.
Step 4: set t � 1 and select an individual X best (0) with the minimum objective value from Pop(0) as best.
Step 7: execute DE's Mutation and Crossover.
Step 10: set i � i + 1. If i ≤ N p , then go to Step 6.
Step 11: execute fast local search on best.
Step 12: set t � t + 1. If t ≤ t max, then go to Step 5.
Step 13: output the current best with its objective value. Complexity distribution [1,100]. e release time r j i is randomly generated from (0, 150nα), in which α is a control parameter. Obviously, the value of r j i increases with the value of α. If r j i ≥ St j i−1 + L j i−1,j i and r j i ≥ d j i for i � 1, . . . , n, the optimal solution can be obtained when each job j i is processed at its release time. is means that a larger value of α may reduce the difficulty of solving the problem. Hence, the values of α are set to 0, 0.2, 0.4, 0.6, 0.8, 1, and 1.5, respectively. e weight value w j i is randomly generated in (0, 1). Let d p,j i be the due date of the job j i on the instance p, C p,j i the completion time of the job j i on the instance p, and random(−C p,j i , 0) a random value in (−C p,j i , 0). en, d p,j i is set as follows: Step 1: randomly generate a sequence (j 1 , j 2 , . . . , j n ) for each instance p.
Step 3: specify d p,j i by Obviously, the due date d p,j i is relatively tight, which helps maintain the company's competitiveness. e total number of test instance is 4 × 3 × 7 � 84. ese instances can be downloaded from https://pan.baidu.com/s/ 1mCdtt3MisBGf16W19xP2aA (password: cmka).
DE_FES's three main parameters are set as follows: the scaling factor F � 0.7, the crossover parameter CR � 0.1, and the population size popsize � 30. Furthermore, KK in DE_FES's local search is set to 3. In order to make a fair comparison, the maximum generation of a modified simulated annealing algorithm with first move strategy
To demonstrate the effectiveness of DE_FES for the TWT-NFSP with SSTs and RTs, DE_FES is tested against two effective scheduling algorithms, i.e., a modified simulated annealing algorithm with first move strategy (MSA_FMS) [37] and an iterated greedy algorithm (IGA) [38]. MSA_FMS outperforms a well-known simulated annealing algorithm presented by Osman and Potts [37,39]. Based on our previous tests, MSA_FMS also performs better than a hybrid genetic algorithm [40]. IGA is one of the best algorithms for solving the FSPs with SSTs [38,41]. In addition, to show the effectiveness of the proposed strategies two and three, DE_FES is also compared with its variant DE_FES_V1, which removes these two strategies from DE_FES. e test results are given in Tables 1 and 2. Table 1 shows the values of BIP, AIP, SD, and CT of each compared algorithm. Table 2 provides the details of each compared algorithm for addressing the instances 20 × 10, 30 × 10, and 70 × 10 under different α.
It is clear from Tables 1 and 2 that, in most instances, the values of AIP and BIP obtained by DE_FES are larger than those obtained by IGA and MSA_FMS and, in all instances, are larger than those obtained by DE_FES_V1. is confirms the superiority of DE_FES and clarifies the effect of utilizing the proposed strategies in DE. In addition, it can be seen that the CT values of DE_FES are obviously larger than those of IGA and MSA_FMS, and it increases quickly as the problem's scale increases. is means that the proposed speed-up strategies can significantly reduce the CC of evaluating solutions and neighbors. More precisely, with the help of the proposed strategies, DE_FES can search more regions or solutions under the same running time.
is greatly increases its probability of obtaining high-quality solutions. Besides, the SD values of DE_FES are relatively small, which indicates the robustness of DE_FES. In summary, DE_FES has powerful search ability to solve the considered problem.

Conclusions
As far as we know, this is the first paper on differential evolution (DE) for dealing with the TWT-NFSP with SSTs and RTs. In view of the complexity of the problem, a differential evolution algorithm with fast evaluating strategies (DE_FES) is developed to find satisfactory solutions of the considered problem. First, the LOV rule is adopted to ensure that DE is suitable for solving the flow-shop scheduling problems. Second, the DE-based global search is used to guide the search to enough promising regions distributed in solution space. ird, by investigating the problem's model structure and the neighbourhood properties, three fast evaluating strategies are proposed and then applied to design a fast local search, which is used to execute thorough and fast exploitation from the promising regions obtained via DEbased search. Since the DE-based parallel search and the problem-dependent local search in DE_FES are well balanced, it can effectively solve the TWT-NFSP with SSTs and RTs. Test results manifest the efficiency and robustness of the proposed DE_FES. Future research direction is to find more valuable properties and neighbourhoods for scheduling problems and extend the DE-based algorithm to uncertain scheduling problems.

Data Availability
Data were curated by the authors and are available upon request.

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