An Efficient Multiobjective Backtracking Search Algorithm for Single Machine Scheduling with Controllable Processing Times

The scheduling problem with controllable processing times (CPT) is one of the most important research topics in the scheduling field due to its widespread application. Because of the complexity of this problem, a majority of research mainly addressed single-objective small scale problems. However, most practical problems are multiobjective and large scale issues. Multiobjective metaheuristics are very efficient in solving such problems. This paper studies a single machine scheduling problem with CPT for minimizing total tardiness and compression cost simultaneously. We aim to develop a new multiobjective discrete backtracking search algorithm (MODBSA) to solve this problem. To accommodate the characteristic of the problem, a solution representation is constructed by a permutation vector and an amount vector of compression processing times. Furthermore, twomajor improvement strategies named adaptive selection scheme and total cost reduction strategy are developed. The adaptive selection scheme is used to select a suitable population to enhance the search efficiency of MODBSA, and the total cost reduction strategy is developed to further improve the quality of solutions. For the assessment of MODBSA, MODBSA is compared with other algorithms including NSGA-II, SPEA2, and PAES. Experimental results demonstrate that the proposed MODBSA is a promising algorithm for such scheduling problem.


Introduction
The scheduling problem with controllable processing times (CPT) has received increasing attention in manufacturing fields.The CPT denotes that operation duration of a job can be compressed or expanded by adjusting the available resources like fuel, equipment, manpower, and so on [1][2][3][4].Most classical scheduling problems assume that the job processing times are constant values.However, this assumption sometimes violates practical production.We can observe that job processing times are controllable in some cases.For example, in the chemical industry, the processing times of a chemical substance can be compressed by catalyzer or expanded by inhibitor, which requires additional costs [5].In the CNC manufacturing industry, the job processing times can be controlled by adjusting the cutting speed or the feed rate, which also entails more costs [6,7].Therefore, the consideration of CPT in scheduling problems may be more applicable for some manufacturing systems.This paper studies a single machine scheduling problem with CPT (SSPWCPT) for the following reasons: it fills the gap where a multiobjective evolutionary approach for the large scale SSPWCPT with multiple criteria has been rarely reported.Most existing methods to deal with singleobjective SSPWCPT can be classified into two categories, namely, exact and approximate approaches.Exact approaches like branch and bound algorithm have been successfully applied to small scale SSPWCPT.Unfortunately, the exact methods are incapable of solving large scale SSPWCPT.On the contrary, the approximate approaches can solve large scale SSPWCPT within an acceptable time.It is therefore important to conduct a study on efficiency and effectiveness of approximate algorithms for this studied problem.However, purely single-objective SSPWCPT cannot fully reflect the requirements of the real-world scheduling applications.Thus, the problem considers two important criteria, namely, total tardiness and total compression cost.The two objectives are widely accepted in single machine with CPT because they can affect satisfaction of customer and profits of enterprise.One straightforward strategy for addressing the multiobjective optimization problem (MOP) is to combine multiple objectives into a scalar function by giving fixed weights to each objective function [8].Nevertheless, in most practical scheduling problems, multiple criteria are usually in conflict with each other [9][10][11].In addition, the objective weight is difficult to determine due to different objective scales.Therefore, it is better to handle multiple objectives with knowledge about Pareto dominance.Pareto-based multiobjective evolutionary algorithm (MOEA) is suitable for solving multiobjective scheduling problems since it can yield nondominated solutions in a single run [4,12,13].
Recently, backtracking search algorithm (BSA) [14] is a promising method for solving single-objective scheduling problem due to its high convergence speed and ease of implementation.The core idea of BSA is that dual population is utilized to search for an optimal solution during search process.Based on the effectiveness of BSA and characteristics of the MOP, a novel multiobjective discrete backtracking search algorithm (MODBSA) is proposed to solve this multiobjective SSPWCPT.Although multiobjective backtracking search algorithm has been developed, it is mainly used to address continuous optimization problems [15].The main reason for adopting this MODBSA is that the problem under study is NP-hard [16] and BSA has been demonstrated to be an effective approach for solving this category of problem [17,18].Furthermore, dual population in BSA can be utilized to store information of different population for keeping diversity of population.In addition, to the best of the authors' knowledge, there exists no research about multiobjective BSA in the field of scheduling problem in literature.These reasons drive us to develop an efficient multiobjective algorithm based on BSA for this discrete optimization problem.
To achieve good performance of the proposed algorithm on SSPWCPT, several strategies of this method are developed.According to the characteristic of the addressed problem, the proposed MODBSA uses a two-part encoding scheme.The first part represents the job permutation and the second one denotes the amount of compression processing times of job.Moreover, an adaptive scheme is proposed to select a suitable population for enhancing search efficiency.Meanwhile, a total cost reduction strategy is proposed to improve convergence toward the efficient solution.
The remainder of the paper is organized as follows.In Section 2, some relevant work is described.In Section 3, a definition of the studied problem is stated.In Section 4, the proposed MODBSA for the SSPWCPT is elaborated.The experimental results of the proposed MODBSA are presented and analyzed in Section 5. Conclusions and future work are given in Section 6.

Literature Review
The SSPWCPT has been extensively studied since Vickson [19] initiated the CPT in a single machine scheduling problem.However, most single-objective methods or weighted sum methods have been used to solve SSPWCPT in previous literature.For example, Janiak and Kovalyov [20] addressed the SSPWCPT with deadlines and processing time which was a linear decreasing function of the amount of compression.They proposed an ( log ) algorithm for small scale cases with the objective to minimize the total resource consumption.Shabtay and Kaspi [21] considered a single machine scheduling problem for minimizing total weighted completion time.They presented and analyzed some special cases that were solvable by using polynomial time algorithms.They also gave some heuristic algorithms and a dynamic programming for the general case.Kayan and Akturk [6] studied a bicriteria scheduling problem on a single CNC machine.They proposed an exact algorithm and four heuristic methods to find a set of discrete points to approximate the continuous trade-off curve on small scale problem.Cheng et al. [22] considered a single machine scheduling problem where both job processing times and release dates were controllable.They proposed an ( 2 ) algorithm to solve this problem for minimizing the sum of makespan and the total compression cost.Yin and Wang [23] presented a heuristic to address a single machine scheduling problem with CPT and learning effect.The objective of this problem is to minimize a cost function including makespan, total completion, and total absolute differences in completion times.Xu et al. [24] proposed a polynomial time algorithm of ( 2 ) for this small scale problem with the objective of minimizing the total tardiness.Tseng et al. [25] proposed a net benefit compression (NBC) heuristic to minimize total tardiness plus compression cost on a SSPWCPT.Kayvanfar et al. [26,27] extended the work of Tseng et al. [25] by designing a net benefit compression-net benefit expansion (NBC-NBE) algorithm.Yedidsion et al. [28] proved the complexity of a single machine scheduling problem with CPT for several related criteria.Yin et al. [29] addressed a single machine batch delivery scheduling problem with an assignable common due date and CPT.They provided an ( 5 ) dynamic programming algorithm and developed an ( log ) algorithm to find the optimal solution for minimizing a cost function consisting of earliness, tardiness, job holding, and due date assignment.Nearchou [8] considered this SSPWCPT and the objective of this problem was to minimize total weighted completion time plus the compression cost.Several singleobjective population-based metaheuristics were adopted to solve this problem.Giglio [30] considered a class of single machine family scheduling problems, characterized by unreliable behavior of the machine, CPT, sequence-dependent setups, and due dates.The objective of this problem is to minimize the sum of the total weighted tardiness, the total weighted consumption cost, and the total setup cost.He proposed a dynamic programming approach to solve this problem.A survey on scheduling with CPT was provided by Shabtay and Steiner [31].In this paper, we only consider the single machine scheduling problem with CPT; thus, the scheduling problems in other environment are not reviewed.
As stated previously, the main criterion of most research is a single-objective or combined objective with the weighted sum approach.However, multiobjective scheduling problem should be the trend for the real-life scheduling production in the future.The majority of the previous research is focused on heuristics for addressing small scale problems.However, the research on addressing the studied problem using metaheuristics is relatively scarce.In fact, metaheuristics are very efficient in solving such type of large scale scheduling problems.To the best of authors' knowledge, efficient MOEAs based on BSA have not yet been applied to SSPWCPT in the previous literature.Thus, the motivation of this study from a theoretical perspective is to develop an efficient MOEA for minimizing both total tardiness and total compression cost on SSPWCPT.Nevertheless, the no free lunch theory implies that algorithm's performance is sensitive to the problem considered.Therefore, we should develop a more efficient MOEA than other high-performing metaheuristics if some characteristics of the problem as well as some techniques are considered.

Problem Formulation
The scheduling problem under study can be described as follows.A set of  independent jobs is available at time zero on a single machine.One machine can process at most one job at a time and job preemption is not allowed.Each job has a normal discrete processing time.The normal job processing times can be compressed by allocating additional resources, which entails compression cost.Assume that resources can only be assigned in discrete quantities and the job with normal processing time incurs no extra cost.
The parameters and decision variables used throughout the paper are as shown in the Notations.
Equations ( 1) define the objective to minimize the total tardiness and the total compression cost.Constraints (2) and (3) guarantee the precedence relationship and define that only one job can be processed at any instance in time.Constraints (4) impose the tardiness of each job in a sequence.Constraints (5) limit the amount of compression processing time for all jobs.Constraints (6) show nonnegativity of variables.This studied problem is NP-hard since the single machine total tardiness problem is already NP-hard [16].It is still valuable for us to research and explore such a problem.

The Proposed MODBSA for SSPWCPT
In this section, firstly we give a basic background on multiobjective optimization and then describe the original BSA and present a framework of the proposed MODBSA.Finally, the main improvement strategies of the MODBSA for optimizing SSPWCPT are elaborated.

Background on Multiobjective Optimization.
To better understand the proposed MODBSA for solving the above problem, we begin with a brief introduction of the basic concept of MOEA.Without loss of generality, a multiobjective optimization problem (MOP) can be formally defined as follows: where   (x) indicates the th subobjective function, x is a vector of the solution, which should satisfy the above constraints,   is the decision variable space, and   (x) and ℎ  (x) are inequality constraint and equality constraint, respectively. and  denote the total number of inequality constraints and equality constraints, respectively.
Let a and b ∈   , and a vector a = [ 1 ,  2 , . . .,   ]  dominates another vector b = [ 1 ,  2 , . . .,   ]  (denoted by a ≺ b) if a is not inferior to b for any of  objectives and a is superior to b for at least one objective.A solution x * ∈   is a Pareto optimal vector if there is not any solution x ∈   that dominates x * .The corresponding objective function in the objective space forms the Pareto optimal front point (x * ).For a Pareto optimal solution, the improvement in any objective will incur the deterioration of at least another objective.A set of all the Pareto optimal solutions is called Pareto optimal set (PS * ), while the set of all Pareto optimal front vectors is called the Pareto optimal front (PF * ).The main goal of multiobjective optimization is to find PF * .However, in general, a Pareto front consists of a large number of points.Therefore, a good Pareto front contains a limited number of points which should be as close as possible to the PF * and uniformly spread as well.

Brief Introduction of Backtracking Search Algorithm (BSA)
. The original BSA is described before presenting the MODBSA, and its main steps include five processes: Input: Mutant, mixrate,  and .Output: : Trial population.

Initialization.
Randomly initialize a population , which can be formulated as follows: where  and  are the population size and the number of decision variables, respectively, random is a real number value uniformly distributed in [0, 1],   and   denote the lower and upper bound for the th decision variable of the th solution.

(9)
BSA has a choice to generate the old  at the beginning of each generation through the following form: where fl is the initialization operation;  and  are real number values in the range [0, 1].The above equation defines that the BSA's historical population (old ) is from either the previous population or the old  itself.Once old  is generated, a permutation function is adopted to randomly alter the position of the individuals in old .

Mutation.
The form of a trial population  (i.e., offspring pupation) by the mutation operation can be written as follows: where  is a scale factor which controls the amplitude of the search-direction matrix (old −).The value  = 3⋅random, where random is a real number uniformly distributed in [0, 1].Since the historical population is employed in the calculation of the search-direction matrix, BSA takes some advantages of previous generations to obtain a trial population.line 4).The rand parameter indicates a random real number in [0, 1].The mixrate parameter controls the number of elements of individuals that will mutate in a trial individual by using ⌈mixrate ⋅ rand()⌉.Two strategies are employed to define BSA's map.In the first strategy, mixrate is used to define map (in lines 3-5).In the second strategy, only one randomly selected element is going to mutate (in lines 7-9).

Selection II.
In the stage of BSA's Selection II,   is used to update the   if it is better than   in terms of the fitness value.If the best solution  best outperforms the global optimal solution found so far, the global optimal solution is replaced by  best [14].Step 1 (initialization).Generate the initial and historical population according to the proposed solution representation stated in Section 4.4.2.
Step 2 (stopping condition).If stopping criterion is met, then stop and output PS and PF.Otherwise, go to Step 3.
Step 3 (Selection I).Calculate the selection probability value (denoted as ) of the population selected based on adaptive selection mechanism.This adaptive mechanism can select a suitable population as a historical population.Section 4.4.3gives details of the adaptive selection mechanism.
Step 4 (update).Perform update operation on the population.This update operator is described in Section 4.4.4.
Step 5 (total cost reduction strategy).The total cost reduction strategy can improve exploitation of the algorithm.This strategy is presented in Section 4.4.5.
Step 6 (Selection II).This stage of the MODBSA is different from that of the basic BSA.The offspring solutions are evaluated with regard to both fitness values (i.e., total tardiness and total compression cost).First, all the individuals in the population are sorted according to a nondominated sorting technique.This fast nondominated sorting method can be described as follows.First, for each solution we compute two entities: (1) domination count   , the number of solutions that dominate the solution , and (2)   , a set of solutions that the solution  dominates.The domination count of all solutions in the first nondominated level is equal to zero.Now, for each solution  with   = 0, we visit each individual  from its set   and reduce its domination count by one.When for any individual  the domination count becomes zero, it will be put into a separate set  where the individuals belong to the second nondominated level.This procedure continues until all nondominated levels are identified.That is, each individual has a rank equal to its nondominance level.Then, within each front or rank a crowding distance strategy The first part: 휋 The second part: Total tardiness: 6 Total compression cost: 13 is used to define an ordering among individuals.To achieve wide spread Pareto fronts (solutions differently balancing total tardiness and total compression cost), individuals with a large crowding distance are better than ones with a smaller crowding distance when they are the same nondominated level.The external archive is used to store nondominated solutions found so far.This external archive has a maximum size.To obtain PF with a uniform spread, the crowding distance technique is also employed to remove solutions with the worst crowding distance from the archive when this archive is full.Although Nearchou [8] presented a two-part encoding for this problem, he adopted subrange keys based on random key encoding scheme, which generates information redundancy.We used this proposed scheme to effectively avoid information redundancy.

The Main
To illustrate this solution representation, Figure 2 gives an example of a solution representation for a 5-job instance.This solution has two parts: (1) the first part: the job permutation, namely,  = (2, 1, 4, 3, 5); (2) the second one: the amount of compression processing times of job, namely, x = (1, 2, 1, 3, 0).There exists a corresponding relationship between the two parts.Each job has a corresponding amount of compression processing time.For the () job denoted by   , its corresponding amount of compression processing time is   .For example, the 1st job (i.e., (1)) in the sequence is job  2 , the amount of compression of  2 is  2 (i.e., 2).Similarly, the 2nd job (i.e., (2)) in the sequence is  1 and the corresponding amount of compression of  1 is  1 (i.e., 1).Therefore, the corresponding amount vector of compression is (2, 1, 3, 1, 0) for a sequence .In this manner, a feasible schedule is easily generated.In addition, this encoding scheme has two advantages below: (1) Solution structure is simple as it contains two parts, namely, job sequence and amount of compression of job processing times.
(2) In general, this discrete encoding scheme can reduce information redundancy compared with previous random keys encoding scheme like subrange keys [8].This view has been further certified in Section 5.4.

Initialization.
The MODBSA begins with a population of  initial individuals.As stated previously, each solution is constructed by a permutation vector and a vector of compression processing times.To ensure the high quality and good diversity of solutions, the job permutation of one individual is based on nondescending order of due time [33].
In detail, this solution is based on the property that if we have job  and job  that satisfy   <   and   <   (,  = 1, 2, . . ., ) then there exists an optimal processing sequence where job  precedes job .The amount of compression processing times of another individual is set to zero in order to minimize total compression cost.The other initial individuals are randomly generated in the feasible range as follows: , = (  (, 1) ,  (, 2) , . . .,  (, )  (, 1) ,  (, 2) , . . .,  (, ) ) , where (, ) denotes the th job in the th individual to be processed on the machine and (, ) presents the amount of compression for job  in the th individual.

Selection I Based on the Adaptive Mechanism.
The dual population scheme is a core idea of the BSA.The dual population is based on a random selection scheme, which can assist algorithm to maintain population diversity.However, this strategy cannot ensure a good convergence toward the optimal solutions, since too much emphasis on diversity would cause pure random search [15].To improve convergence performance, an adaptive selection scheme is presented.This adaptive selection scheme can select a suitable population as the current historical population.To simplify the calculation of the selection probability of population  as the current historical population, we record the population  that is chosen to participate in update operation.After the initial population and historical population are generated, a population is selected as the current historical population based on (10) so that each population has an equal selection probability.Afterwards, the current historical population is updated by an adaptive selection scheme during the search process.Let  ( is not a random number at this time) represent the probability of updating the historical population old  by replacing it with the current population .The adaptive selection scheme can be stated as follows.
Step 1.After population  is updated, calculate the update probability of historical population old ; namely,  = /||, where  is the number of nondominated solutions and || represents the total number of the nondominated solutions in the external archive at the current iteration.The updating of old  is executed by simply replacing the old  with the current population .
Step 2. Use the roulette-wheel approach to select a population from each population.
This selection strategy is simple yet efficient for improving performance of MODBSA.It implies that the selection probability is proportional to the number of nondominated solutions from the population .To avoid the situation where all solutions are obtained from the same population throughout the iterations, the population  has a minimum selection probability of  min .That is, after the calculation of the selection probability of population, if  <  min , then set  =  min .In this work,  min = 0.2.
Before updating population, select two parents in which one is from the current old  (note that the current old  is either from old  or from  according to the adaptive mechanism) and the other is from .In addition, the efficiency of this adaptive scheme has been proven in Section 5.5.

Update Operator.
It is evident that crossover and mutation operators of original BSA are not suitable for solving the SSPWCPT.To overcome this problem, we change the crossover and mutate operators to the traditional crossover and mutation operators.
Crossover can explore unknown areas of solution search space.For the first part (i.e., ), partially mapped crossover (PMX) [34] is adopted to update the permutation part and has been widely used in the scheduling field.For the second part, two-point crossover is used to update amount of compression processing times in this paper.The detailed steps of the crossover are as follows.
For the first part (see Figure 3), consider the following.
Step 1. Select the substring: randomly generate two crossover points and define the substring between two points as matching area (i.e., yellow area).
Step 2. Exchange the substrings: generate temporary offspring by swapping the matching area of two parents.Note that only the elements in the matching area (yellow area) are exchanged.We can find that temporary offspring is unfeasible when exchanging the substrings.For instance, job 5 appears twice in temporary offspring 1.Similarly, job 1 appears twice in temporary offspring 2.
Step 3. Mapped relationship: determine mapped relationship of the elements in conflict.When the same job sequence is assigned more than once, the mapped relationship of the sequence in crossover segments is defined.
Step 4. Legalize the offspring: make the job permutation part feasible by using the information from the mapped relationship without any changes to the substring (keep the substring unchanged).
For the second part (see Figure 4), consider the following.
Step 1. Select the substring: randomly generate two points and define the substring between two points as exchange area.
Step 2. Exchange the substring: exchange the substrings between the two points.Step (1) Step ( 2) Step ( 3) Step (4) Map relationship in offspring 2  Step ( 1) Step ( 2) Mutation operator assists the algorithm to escape from local optima.In this research, the mutation operator is composed of two mutation techniques with the probability of 0.5, respectively.That is, when this mutation operation is performed, one of the two techniques is executed.Therefore, the algorithm performs either the first technique or the second technique.
The first technique called swap mutation is only applied in the permutation part (i.e., the first part).The second one is only applied in the amount of resource compression part (i.e., the second part).To explain this mutation operator, an example is illustrated in Figure 5.For the first mutation technique as presented in Figure 5(a), the original sequence of jobs is  = (2, 1, 4, 3, 5).When the two positions are randomly selected (e.g., the 2nd and 4th positions), their corresponding job 1 and job 3 are swapped while the second part remains unchanged.That is, the sequence of jobs becomes  = (2, 3, 4, 1, 5) after performing the first technique.For the second mutation technique as shown in Figure 5(b), firstly two positions are randomly selected (e.g., the 2nd and 4th positions), and then the corresponding new feasible integer numbers are generated to replace the original values.That is, the amount value of compression of job 2 in the 2nd position is updated to a new value 1, while the amount value of job 4 in 4th position is replaced by new value 2. These new values are randomly generated in their own ranges.Therefore, the offspring after update operator is still feasible.

Total Cost Reduction Strategy.
Metaheuristics are usually combined with local search approaches, which may assist in searching for good solutions since they introduce an idea "greediness" within the metaheuristic [35].In this paper, however, any local search strategies are not applied in the proposed algorithm.Instead, a release cost procedure is developed to further improve quality of solutions.This procedure does not change the job processing sequence but reduces the total compression cost while keeping the same total tardiness.According to the characteristic of the problem, it can be observed that by adjusting amount of compression processing time of job the compression cost can be further reduced without affecting total tardiness.Thus, it can improve the quality of a solution for a given job sequence to some extent.The computational complexity of this heuristic is ().The main steps of the proposed total reduction strategy are below.

Amount of compression
Amount of compression The first part: 휋 The second part: x (a) The first mutation technique

Amount of compression
Amount of compression The first part: 휋 The second part: The second mutation technique  Step 4. Find the job with the maximum value Cost from   if   is not an empty set.This job is denoted by  () .Update its amount of compression processing time  () =  () − min{ () ,  () } and the completion time of jobs following the () job.
To explain this total reduction strategy, an example is given below.
Example 1.Consider a 5-job instance in Table 1 and Figure 6 with a given job sequence  = (2, 3, 4, 1, 5) and amount vector of compression processing time x = (4, 1, 1, 1, 0).The corresponding total tardiness and total compression cost are 2 and 2.6, respectively.Perform the total reduction strategy in the following steps.
Step 3. Conduct the following operation.
x (i) x (i) Step 4. Select the job with the maximum cost value from   , namely, the (4) job.Update the amount of processing time compressed:  (4) =  (4) − min{ (4) ,  (4) } = 4 − min{1, 4} = 3.Meanwhile, the corresponding total tardiness and total compression cost are updated to 2 and 2.1, respectively.Note that the total tardiness is fixed but the total compression cost is reduced.

Experimental Study
This section is devoted to assessing the performance of the proposed algorithm MODBSA on SSPWCPT.The experimental studies include the following four aspects: In the following subsections, performance metrics, test function, and parameter setting are described at first, and then the experimental studies are further investigated step by step.

Performance Metrics.
As mentioned previously, the final result is usually not a single optimal value rather than a set of optimal solutions for MOPs.To explain this problem, the parameters of a specific instance are provided in Table 2.
The PF obtained by MODBSA on this instance is shown in Figure 7.Some nondominated solutions and the corresponding objective values are summarized in Table 3.It can be observed that the obtained results consist of some trade-off solutions.It is also an interesting observation that there are two nondominated solutions that correspond to the same front point (i.e., solution 2).Unlike single-objective problem, the high quality results of MOP not only have good convergence but also evenly distribution along PF * .Therefore, how to evaluate results found by MOEAs is important for users.
To evaluate the results obtained by MOEAs, some metrics including the Spread [36], GD, and IGD [37] should be adopted as follows.(1) Spread (Δ).It is a diversity performance index that assesses the distribution of the obtained solutions in the front.This metric can be formulated as follows: where   is the Euclidean distance of each point in PF to its closest point in PF,  represents the mean value of all   ,    denotes the Euclidean distance between the extreme solutions in the th objective and the boundary solutions of the PF * , |PF| is the number of PF, and  is the number of objectives.
If the spread value is zero, then all the members of Pareto optimal front are evenly spaced.Lower values indicate better distribution and diversity.
(2) Generational Distance (GD).It is a convergence indicator, which represents how far the obtained PF is from PF * .It can be formulated as follows: where |PF| means the number of PF points and   is the Euclidean distance between the th member of PF obtained and the nearest member of PF * .A low GD value represents a good convergence performance.A normalization method is used in this metric.
(3) Inverse Generational Distance (IGD).It is a variant of the GD but represents a combined or comprehensive indicator.It measures the average distance between each solution consisting of the optimal Pareto front and obtained front.IGD can be defined as follows: where |PF * | is the number of the optimal Pareto fronts; dist(, PF) is the Euclidean distance between  and the nearest member of the approximation.Fronts with a lower IGD value are desirable.This metric uses a normalization method.
It should be mentioned that the true PF * of the studied problem may be unknown; therefore, the nondominated solutions obtained by different MOEAs on each instance in  all the independent runs are regarded as PF * on that instance [38].

Description of Test Function.
The instances generated are defined as shown in Table 4.There are six different numbers of jobs ( = 10, 30, 50, 80, 100, 200), where the normal processing times and the crash processing times are drawn from the discrete uniform distributions DU(20, 100) and DU(0.5 ⋅   ,   ), respectively.In due date calculation,  = ∑  =1   , and  is the discrete value from 0.2 to 1.0 with the step size of 0.2.The unit cost of compression is generated from a uniform distribution ranging between 0.5 and 2.5.Each instance can be labelled in the form of "n_r".For example, "10_02" represents the fact that the problem is featured by 10 jobs and  equal to 0.2.

Experimental Settings.
All algorithms are coded in Java on the platform jMetal [39].Experimental tests are implemented on a computer with Intel Core i5, 2.39 GHz, 4 GB RAM, with a Windows 8 operating system.
Parameter settings can affect the performance of the algorithm.The pilot experiments demonstrated that the population size and archive size were sensitive to the problem scale.Therefore, for various variants of MODBSA in Sections 5.4-5.6, the maximum number of function evaluations (NFEs) is 25,000 for 10-job and 30-job instances, 35,000 for 50-job and 80-job instances, and 45,000 for 100-job and 200-job instances.The population size and the external archive size are set to 50 for 10-job and 30-job instances, 80 for 50-job and 80-job instances, and 100 for 100-job and 200-job instances.The historical population size is equal to the population size.In Section 5.7, the parameter The optimal results are highlighted with bold in Tables 5, 7, 9, 12, 13, and 14.Due to the stochastic characteristic of all candidate MOEAs, the statistical analysis is necessary to provide confidential comparisons.A Wilcoxon sign rank test [38,40] is used to test the significant difference between the results obtained by different algorithms.The confidence level for all tests is set to 95% (corresponding to  = 0.05).The sign "+" indicates that our proposed MODBSA algorithm performs significantly better than the second best algorithm on average.While "−" represents the fact that the MODBSA algorithm is significantly worse than the best algorithm, the "=" sign denotes that there is no significant difference between MODBSA and the best or second best MOEA. + represents the sum of ranks for the problem where the MODBSA performs better than its competitor. − denotes the sum of ranks for the opposite.

Efficiency of Solution Representation.
To test the efficiency of the proposed solution representation, it is compared with subrange keys for MODBSA on medium and large scale problems.In this study, MODBSA sk represents MODBSA based on subrange keys.More detailed information on subrange keys can be found in Nearchou [8].Since subrange key is a real-coded scheme, the operators in MODBSA sk are different from that in MODBSA.The update operators of MODBSA sk are as follows: the simulated binary crossover (SBX) and polynomial mutation are used.The distribution indexes in both SBX and the polynomial mutation are set to 20.The crossover rate is 0.9, and mutation rate is 0.2.In addition, the proposed total cost reduction strategy is also adopted in MODBSA sk , but MODBSA sk requires converting real-coded scheme to discrete-coded scheme in this phase.MODBSA includes the proposed solution representation.The crossover rate is 0.9 and mutation rate is 0.2.Table 5 shows the mean and standard deviation metrics on these algorithms over 30 independent runs.Table 6 reports the significant test results over 30 runs.
Table 5 reveals that MODBSA obtains the optimal results on 13, 5, and 20 out of 20 test instances for GD, Spread, and IGD metrics, while MODBSA sk achieves the best values on 7, 15, and 0 problems, respectively.Table 6 records the  values of the Wilcoxon signed rank test.We can clearly observe from Table 6 that MODBSA has higher "+" counts than its compared algorithm in terms of GD and IGD.It means MODBSA is significantly better than MODBSA sk for GD and IGD metrics.This may be because the proposed discrete  on most instances.However, such advantage will no longer exist when only Spread metric is considered.The poor distribution performance of MODBSA may be associated with the characteristic of the problem.More specifically, this type of scheduling problem may be a multimodal optimization issue which contains several optimal solutions corresponding to the same objective value (i.e., the second nondominated solution on the case in Table 3).Therefore, the distribution distance between Pareto fronts is very crowded.MODBSA can obtain better results than MODBSA rs in terms of GD metric.In summary, the proposed MODBSA based on adaptive selection mechanism is superior to MODBSA rs on most instances.This means that adaptive selection can enhance search efficiency.Besides, the results computed by the proposed MODBSA are more stable, which indicates that the adaptive selection strategy can strengthen the stability of the MODBSA.

Efficiency of Total Cost Reduction in Proposed Algorithm.
To prove the efficiency of the MODBSA with the total cost reduction strategy, it is compared with MODBSA without the total cost reduction.In this experiment, MODBSA ntcr denotes the MODBSA without the total cost reduction strategy.MODBSA itself includes the total cost reduction strategy.The parameter settings of both MOEAs are the same as the above experiments.Table 9 records the statistical metrics on different algorithms over 30 independent runs.Table 10 shows the test results based on the best metrics for each instance with 30 independent runs.Table 9 presents that the MODBSA is superior or competitive to MODBSA ntcr in terms of all metrics on most instances.From Table 10, it can be clearly observed that the MODBSA with the total cost reduction strategy has a significant better performance than the one without this strategy on most instances.It means that MODBSA using the total cost reduction strategy has good convergence and coverage performance compared with MODBSA without total cost reduction.It also implies that the exploitation ability can be improved by the adoption of the total cost reduction technique in MODBSA for solving SSPWCPT.

Comparison MODBSA with Other Algorithms.
To further assess the performance of the MODBSA on these scheduling problems, MODBSA is compared with well-known MOEAs: NSGA-II [36], PAES [41], and SPEA2 [42].To fit the characteristic of the addressed problem and make a fair comparison, we modified these considered MOEAs.All MOEAs use the same population size and the NFEs as stated in Section 5.3.Moreover, the initial population is generated based on the proposed encoding scheme and strategy for all MOEAs.All MOEAs adopt the same operators including crossover, show the statistical results of GD, Spread, and IGD.From these tables, we can observe that the proposed MODBSA outperforms its counterparts for most instances.Especially on the comprehensive metric IGD and convergence metric GD, the outperformance of the MODBSA is overwhelming except for several problems.MODBSA is also competitive to NSGA-II with regard to Spread metric.
In addition, from Table 15, MODBSA shows a significant improvement over the other MOEAs with a level of significance  = 0.05 in terms of IGD metric.The major reasons for the good performance of the MODBSA can be explained as follows.First, the adoption of the dual population strategy can improve the diversity of population since different population may have different search directions, and thus the MODBSA can maintain a good diversity in search space.Second, to boost convergence performance, the adaptive selection mechanism can select an appropriate population as parent population for generating new candidate solutions according to different search environment and thus improve search efficiency.Third, to further enhance convergence, the total cost reduction strategy is proposed to improve solution quality.Therefore, we can draw a conclusion that these strategies have a positive effect on the behavior of the algorithm.Figure 8 presents the PF approximations found in the run with the best IGD value of each MOEA for three level instances with small, medium, and large scale.It is evident from Figure 8(a) that although all MOEAs can find some approximations with regard to convergence for the small scale problem, MODBSA is capable of covering more areas than other MOEAs.As is depicted in Figure 8(b), MODBSA can show better convergence and coverage performances for the medium scale problem, while the other MOEAs tend to fall into local optima.The outperformance of MODBSA can be attributed to the adaptive mechanism, by which MODBSA can search preferable solutions in different directions to enhance search diversity.We can also observe from Figure 8(c) that the MODBSA has good convergence performance compared with its MOEAs.The good performance of MODBSA on medium and large scale problems The statistical results are plotted as boxplots in Figure 9.The vertical axis of each subfigure represents the IGD value and the horizontal axis represents the different MOEAs.The lower position of box denotes better performance.The narrower the shape of the box is, the more stable the corresponding algorithm is.Clearly, the MODBSA is overwhelming without any exception in terms of IGD metric for three above level instances.It is consistent with previous numerical analysis and our view that the MODBSA outperforms other MOEAs considered for the SSPWCPT.The reasons behind the good performance of MODBSA are as follows.First, dual population scheme makes MODBSA have a better exploration ability as that it has a greater choice to search for different unknown areas of the search space.Second, total cost reduction strategy can improve the convergence of MODBSA since the quality of solution can be improved when total cost criterion is reduced while the tardiness criterion remains unchanged.

Conclusions and Future Work
In this paper, a multiobjective single machine scheduling problem with CPT is studied.The objective of this study Input (i) A stopping condition (ii) : the population size (iii) Input other parameters Output.Pareto front (PF) and Pareto solutions (PS) are stored in an external archive.

Figure 2 :
Figure 2: An example of solution representation.

Figure 3 :
Figure 3: PMX crossover in the first part.

Figure 4 :
Figure 4: Two-point crossover in the second part.

Figure 6 :
Figure 6: Gantt chart of Example 1 before (a) and after (b) using the total cost reduction strategy.

( 1 )
Evaluate efficiency of the proposed solution representation.(2) Evaluate efficiency of the adaptive selection scheme in the MODBSA.(3) Test performance on the total cost reduction strategy of the MODBSA.(4) Compare the MODBSA with other MOEAs on the instances.

Figure 7 :
Figure 7: Pareto front obtained by MODBSA for this scheduling instance.

Table 1 :
Data of Example 1.

Table 2 :
Parameter setting of the instance.

Table 3 :
The corresponding results for the instance in Figure7.

Table 4 :
Data set distribution.

Table 5 :
Mean and standard deviation value of all metrics between MODBSA sk and MODBSA.

Table 6 :
Wilcoxon signed rank test results based on the best metrics for each instance with 30 independent runs (a level of significant  = 0.05).

Table 7 :
Mean and standard deviation value of all metrics between MODBSA rs and MODBSA.
based on subrange keys may have a great choice to search for different areas of the search space and thus improve search diversity, although it can lead to information redundancy.5.5.Efficiency of Adaptive Strategy.To test the efficiency of the adaptive mechanism in MODBSA, we compare the MOEAs are the same for a fair comparison.Table7reports the statistical metrics on two strategies over 30 independent runs.Table8summarizes the  values of Wilcoxon signed rank test.From Tables7 and 8, it can be observed that MODBSA completely dominates MODBSA rs in terms of IGD metric

Table 8 :
Wilcoxon signed rank test results based on the best metrics for each instance with 30 independent runs (a level of significant  = 0.05).

Table 9 :
Mean and standard deviation value of all metrics between MODBSA ntcr and MODBSA.

Table 10 :
Wilcoxon signed rank test results based on the best metrics for each instance with 30 independent runs (a level of significant  = 0.05).

Table 12 :
Mean and standard deviation of GD obtained by NSGA-II, SPEA2, PAES, and MODBSA.

Table 13 :
Mean and standard deviation of Spread obtained by NSGA-II, SPEA2, PAES, and MODBSA.
(1) A multiobjective mathematical model of scheduling problem with CPT is constructed.A new multiobjective backtracking search algorithm is developed to solve this scheduling problem.

Table 15 :
Wilcoxon signed rank test results based on the best metrics for each instance with 30 independent runs (a level of significant  = 0.05).