1 Introduction
This work aims at optimizing the computer control systems when these systems have two dedicated processors: the assignment of tasks to these processors is fixed. For this problem, we have three types of tasks. Some tasks must be processed only by the first processor, others by the second processor, and the remaining tasks need simultaneously both processors. This problem represents a practical issue in computer control systems, where a task is performed in several copies on different processors in order to ensure better safety of the system. In production management, we can cite the case where a task requires several operators for its execution.
The contribution of our work is to propose lower bounds for the three studied criteria (makespan, total tardiness and total completion time) and to develop genetic algorithms to solve this problem in the multiobjective case. The lower bounds allow us to assess the quality of the feasible solutions and the genetic algorithms incorporates the optimization part. We implemented our approach by considering aggregative, NSGAII and Pareto scenarios on a large set of instances. The results show the effectiveness of the implemented algorithms. The studied problem is according the standard ternary notation.
Few studies have dealt with this problem. The most important studies are mentioned in the following paragraphs.
Coffman et al.[9] studied the file transfer problem in the field of computer networks where each computer has a number of different ports for data exchange. File transfer uses a subset of ports, therefore a multiprocessor task on dedicated processors. The boot time of the transfers is also taken into account, then different transfer protocols are proposed, and performance results are demonstrated. Drozdowski[12] cited this paper to describe the actual applications of scheduling problems on dedicated processors.
Craig et al.[10]
studied the problem in testing integrated circuits VLSI (very largescale integration). To test a component of these circuits, several other electronic components are needed simultaneously. The authors addressed the problems in case when the processing times are unitary or arbitrary. A heuristic based on the maximum degree of incompatibility has been proposed to solve these two problems
and .Hoogeveen et al.[15] showed that the problem is NPhard in the strong sense. The preemption of tasks does not make the problem easier. Oguz and Ercan[35] proved that the problem is NPhard in the strong sense. Afrati et al[1] proposed a PTAS approximation scheme for the problem and a second PTAS approximation scheme proposed by Afrati and Milis[2].
Chu[8] proposed a lower bound for the minimization of total tardiness problem; the calculation involves the SRPT priority rule (Shortest Remaining processing Time) for a relaxed problem with preemption. The main idea is that each time the processor becomes available, an unfinished task available with the shortest remaining processing time is set. The execution of a task is interrupted when its remaining processing time is strictly greater than the length of processing task that becomes available.
Leung and Wang[28] proposed a genetic algorithm with multiple fitness functions to conduct research in order to solve a multiobjective problem. The authors applied an experimental design method called Uniform Design to select the weights used with the objective functions and diversify uniformly selected solutions.
Kacem[19] developed two lower bounds for tardiness minimization problem on a single machine with Family Setup Times. The first lower bound is based on Emmons theorem [13]
and the SPT rule, the second is achieved by sorting tasks by processing times and the idea of due dates exchange. Another idea of solving the linear programming problem, was also proposed.
Berrichi et al.[5] studied a biobjective model of parallel machine problem using reliability models to take into account the service side. Two genetic algorithms were developed to obtain an approximation of the Pareto front: One algorithm that uses the two objectives weighted and NSGAII algorithm.
Rebai et al.[33] introduced three lower bounds for minimization tardiness problem on one machine to schedule preventive maintenance tasks. The first lower bound is based on the Lagrangian relaxation of mathematical model. The second is obtained by the sum of M costs calculated for M tasks, and the third is an adaptation of the lower bound given by Li[26] for the problem of earliness tardiness minimization with a single due date for each task.
Manaa and Chu[29] proposed the method of separation and evaluation to minimize the makespan. In their article, the authors presented a lower bound that has been proven. This method can treat all instances generated up to 30 tasks for the most difficult cases in less than 15 minutes.
Vallada and Ruiz[37] studied the unrelated parallel machine scheduling problem. A genetic algorithm is developed to solve this problem. The proposed method includes a fast local search and a local search enhanced crossover operator. The computational and statistical analysis shows an excellent performance in a comprehensive benchmark set of instances.
Kacem and Dammak[22] studied the problem of biobjective scheduling of multiprocessor tasks on two dedicated processors. The authors adapted the genetic algorithm to solve the problem of minimizing the makespan and the total tardiness for the large size instances. The results found showed the effectiveness of the proposed genetic algorithms and the encouraging quality of the lower bounds constructed in [29, 20]. For that, we decided to study a new extension of this problem by adding additional objective.
In the next section, some notations are detailed and the proposed lower bounds for the makespan, total completion time and total tardiness are given. In Section 3, we present the solving approaches. Three methods to solve the considered problem are developed. The one is aggregative with Uniform Design, the second is Pareto and the third is the NSGAII. Section 4 deals with the generation of instances, the computational results and the qualitative and quantitative analysis. Finally concluding remarks are given in Section 5.
2 Lower bounds
We study three scheduling problems on two dedicated processors. To assess the quality of the results found by such a method, we use the following lower bounds.
2.1 Notation
The following fields denote:

: Two processors.

: Each task has one or two dedicated processors and the assignment of each task is fixed.

: Release date of task .

: Makespan.

: Tardiness of task ; with the completion time of task .

: Due date of task .
2.2 Lower bound for problem
Manaa and Chu [29] proposed two ideas to construct a lower bound for the considered problem:

The idea of dividing the problem into two subproblems on one processor by relaxing the studied problem.

The idea of Bianco et al.[6] an optimal solution to minimize the makespan for oneprocessor problem.
The relaxation of the studied problem allows us to obtain two simple problems:
a) Scheduling tasks that necessitate using simultaneously both processors and tasks that require the first processor.
b) Scheduling tasks that require employing simultaneously both processors and tasks that necessitate the second processor.
The optimal solutions of problems (a) and (b) can be found by scheduling tasks according to the order of their release dates.
The lower bound for the studied problem corresponds to the maximum value of the solutions of problems (a) and (b).
2.3 Lower bound for problem
In this study, we use and combine three ideas to build a lower bound:

The idea of reducing the problem into two subproblems on one processor by partitioning the biprocessor tasks.

The idea of dividing the mono processor tasks into two tasks.

The idea of underestimating the completion times of the tasks on a single processor (originally proposed by Chu
[8]).
The first of this lower bound is to partition the bi processor tasks into two monoprocessor tasks, each of them on one of the processors. We get two independent problems on each processor.
On the first processor, we consider the monoprocessor tasks with a weight , and the bi processor subtasks on processor a weight , with .
Similarly, we consider the second processor , the monoprocessor tasks with a weight . However, the biprocessor subtasks on processor a weight . Thus, we obtain a problem on each processor:
and .
We consider . The next step is to divide the monoprocessor tasks (with a weight ) in two tasks. We get for each divided task two sub tasks and with release date ; and processing time .
We divide the weight on for each sub tasks. We are getting if . From where,
, with is a penalty to be added according to Webster formula.
.
We apply the same principle for the problem . We are getting .
is then a lower bound for problem . The calculation of the lower bounds of completion times for the problem on each
processor uses the following theorem for .
Theorem 1
(Chu[8])
Let be the completion time of the task in the position of a feasible schedule .
is the completion time of the task in the position of a feasible schedule constructed by the SRPT (Shortest Remaining Processing Time) priority rule.
Chu proved that for every feasible schedule , we have:
By applying the theorem (Chu[8]), we compute a lower bound on the completion time of each job.
Example : Let us consider the instance in Table 1.
We apply the principle of calculation of the lower bound mentioned above and we get two subproblems on each processor. On the first processor , we schedule the tasks . We divide the tasks on into two, we get the following tasks: with the following parameters described in Table 2.
The sequence built by the SRPT rule with preemption gives the solution described in Figure 1.
The total of completion time, giving the following lower bound:
(1) 
Respectively, we calculate the lower bound for the problem .
Thus, we consider as a lower bound for the problem .
.
2.4 Lower bound for the problem
Kacem and Dammak[20] proposed an adapted lower bound for the problem of minimization of total tardiness on two dedicated processors. The authors exploited and combined three ideas to construct this lower bound:

The idea of reducing the problem in two subproblems on one processor by partitioning the biprocessor tasks.

The idea of underestimating the completion time of the tasks (initially suggested by Chu[8]).

The idea of calculating the lower bound by assigning the due dates to the reduced completion times (originally proposed in Rebai et al.[33] for another scheduling problem).
The first step of this lower bound is to divide the biprocessor tasks into two monoprocessor tasks; each of which is executed on one of the two processors. Consequently, we obtain two independent problems on each processor. On the first processor , we consider the monoprocessor tasks with a weight , and the biprocessor subtasks on processor having a weight with .
Similarly, we consider, on the second processor , the monoprocessor tasks with a weight . However, the biprocessor subtasks on the processor have a weight . Thus, we obtain a problem on each processor
and .
Using the idea of Chu (described in the previous section), we compute a lower bound on the completion time of each task.
The next step of computing the lower bound is based on the idea of assigning the weight and the due date of each task to completion times’ lower bounds. The total tardiness is minimized by the Hungarian algorithm.
Let be the cost of assigning a reduced to the task supposed to end at the position of the schedule. This cost can be calculated according to the following formula: .
This assignment technique, presented by Rebai et al.[33], allows us to elaborate a new lower bound. We apply the Hungarian algorithm to determine, from the assignment matrix , a lower bound () to solve the following problem .
(2) 
(3) 
(4) 
(5) 
Applying the same process, we calculate for the problem. Thus, we consider as a lower bound for the problem .
Optimization of the lower bound
To improve the constructed lower bound, we look for the weights , which maximize . We use the following method to optimize the bound .

For a biprocessor task , we calculate the gap between and tardiness of task according to the associated two subproblems on the two processors obtained by the Hungarian algorithm.

According to the gap, , we increase the value for a negative gap (and we reduce it respectively for a positive gap).

We apply the Hungarian algorithm to the new matrix and we calculate a new lower bound .

We repeat this procedure for all biprocessor tasks.
Next, we present the study of the problem .
3 Solving approaches
We adapt the genetic algorithm to the multiobjective case. We propose three methods to solve the considered problem. The first is aggregative, second is Pareto and the third is the NSGAII.
3.1 The genetic algorithm
To represent the data of the studied problem, we used a standard coding technique. This coding consists in representing an individual with a permutation containing distinct numbers that correspond to the set .
To form the diversified initial population, we used a random method to create a feasible sequence and to generate the other individuals of the initial population.
To assess the quality of individuals in a population, we have presented three methods to evaluate the studied problem in a given sequence.
The literature has several selection techniques such as proportional selection by tournament, by rank, random selection, etc (see Karasakal and Silav[23]). For our algorithm, we implemented three selection approaches: the aggregative approach, the Pareto one and NSGAII.
The process of crossover between two parents leads to the birth of two children. In this case, an exchange position is randomly determined (see Vallada and Ruiz[37]). The first part of the first child is directly obtained from the first parent. The second part is provided by respecting the order of the remaining tasks as they appear in the second parent tasks. The same process is applied to the second child by reversing the parents. For our algorithm, we implemented the onepoint crossover, which is a folklore (see Holland[18]).
Several methods of mutation exist in the literature such as the method of permutation, insertion and inversion. In our case, we used the permutation method of swapping two positions of the individual.
3.2 Aggregative approach
To adapt our genetic algorithm to the multiobjective case, we constructed an aggregative selection method that consists in generating weights for each sequence of a given population. To calculate the weights, we used an experimental design method called Uniform Design (Leung and Wang[28]). We choose a new population by a scaling method, which consists in calculating the weighted sum of normalized objective functions. Several combinations of weight are considered for the three objective functions (makespan, total tardiness and completion time). Each combination of these weights transforms the problem into a monoobjective case. Accordingly, the search directions are uniformly dispersed to the Pareto front in the objective space. With multiple fitness functions, we design a selection scheme to maintain the quality and the diversity of the population.
In what follows, we will describe the Uniform Design method used for calculating the weight and we will give the formula for the scaling method for the selection of a new population.
3.2.1 Calculation of weight with Uniform Design
The main objective of the Uniform Design is to sample a small set of points from a given set of points, so that the selected points are uniformly dispersed. This method is a branch of statistics that has been used to calculate the weight. As an illustration of the Uniform Design method, the reader could consult (Leung and Wang[28]).
We consider a unit hypercube in a dimensions space ( is the number of objectives) and a point in , Where
, such that .
For any item from the hypercube , we can create a hyperrectangle between the center and , with . This hyperrectangle is described by the following formula:
(6) 
We consider a set of points from , We can associate with each point , a subset of points that belongs to the hyperrectangle . Let be the cardinality of such a sub set and the fraction of the points included in the hypercube and is the fraction of volume value of the hyperrectangle . The uniform design is to determine points in such that the following discrepancy is minimized.
(7) 
The authors presented the points solution calculated using the uniform matrix given by Fang and Li [14].
With and is a parameter that depends on and .
Now, we consider our problem studied, which consists in optimizing three objectives. In our case, we have , we take , so see (Leung and Wang[28]). Using the formula given by Fang and Li[14], we get the following uniform matrix:
We consider the weighting vector
. The components of this vector are calculated by the following formula:(8) 
3.2.2 Scaling method
We use the weight components vector calculated by using the Uniform Design to build a scaling method that allows us to choose a new population by sorting individuals of the current population in ascending order according to the following formula:
(9) 
Such that,
(10) 
(11) 
(12) 
where are the vector components of the weight described in the above section, is a feasible solution from population and , , , are respectively the makespan, the total tardiness and total completion time of a solution .
By exploiting the uniform matrix, we obtain seven evaluation functions (fitness). The list of functions is given by the following formula:
(13) 
Each combination of these weights transforms the problem into a monoobjective case. For each combination, the genetic algorithm is applied and the population is stored. At the end of this process, such populations are merged and only the nondominated solutions are kept.
3.3 Pareto approach
We adapt classical genetic algorithm for multiobjective case using the Pareto approach [32]. For each generation, we transform the population by crossing the nondominated solutions and mutating the dominated solutions. Then, we concatenate the current population and the new individuals created by crossover and mutation(see Alberto and Mateo[3]). The new population is then obtained by keeping all nondominated solutions. In case the number of nondominated solutions is less than the population size, we complete the remaining population by the best individuals according to three fairly studied criteria: The onethird of the remaining population by the best individuals according to the makespan criterion and the onethird of the remaining population by the best individuals according to the total tardiness criterion. The best individuals according to the total completion time criterion will complement the rest of the population. In the last generation, only nondominated solutions are kept.
3.4 NSGAII algorithm
The NSGAII algorithm is based on the following principle (Deb et al.[11]):

With each generation , merging the population of parents of size with the population of children of the same size to build a new population of size .

Sort the results population according to the nondominance criterion. This makes it possible to distribute in several fronts . The first fronts contain the best individuals.

Building the new parent population by adding the fronts while the size of does not exceed . In the case where the size of the new population is less than , the crowding method is applied.
The calculation of the crowding distance of an individual is based on the following principles:

Repeat these steps for all objectives.

Sort the solutions of an objective in ascending order.

Assign infinite distance for the individuals having extreme values (the first and last according to the sorts).

For all other individuals, calculate the normalized difference of the two adjacent solutions. Add the value and calculate the distance of the current individual.
After calculating the crowding distance of front from , the list of solutions must be sorted in a descending order. The best solution is selected by using the crowded comparisonoperator ; between two different rank solutions, we choose the one with the smallest rank, if they have the same rank we choose the solution that has the greatest crowding distance.
4 Numerical results
In this section, we present some experimental results obtained on randomlygenerated instances. Then, we analyze these results and we provide some conclusions.
We implemented our genetic algorithm using a compiler on an Intel i3 4005U CPU 1.7 GHz, 1.7 GHz and 4 GB of RAM.
We randomly generated instances by taking into account five types of problems illustrated in Table 3 presented by Manaa and Chu[29]. The parameter is an integer (), and [x] corresponds to the integer part of . The variables , and respectively represent the number of , and .
For these five types of problems, Manaa and Chu[29] considered the distribution of the three types of tasks and the number of tasks on each processor (load on the processor).
For , the distribution of tasks is balanced () and the distribution of the load on each processor ( and ) is therefore balanced.
For , the number of tasks exceeds that of the two other types ( and ), while the processor is more loaded than . For , the number of tasks , which requires the use of the two processors, exceeds that of tasks of the other two types ( and ). But, the distribution of load on the processors is balanced.
For , the load on the processors is balanced, which is not the case for . The processing times are randomly generated from the set .
The values are randomly generated from the set , with equal to the integer part of: , where and and are respectively the totals of the processing time of , and .
The due dates are randomly generated from the set .
We consider that the group of instances represents the set of instances having the same parameters , and .
For the experimental results, 10 instances of each group are generated and the average values are provided. We fixed the number of generations to for each population where is the number of tasks to be processed. Some preliminary tests have motivated our choices.
4.1 Computation time
Tables 4 summarizes the numerical results in terms of average computation of time (second). These results show the importance of distinguishing not only the total number of tasks and the length of the release dates interval, but also the different types of problems. The results also show that instances corresponding to the problem of type 4 (with the largest number of tasks compared to other types and the tightest distribution of the release dates with ) are the most difficult to solve.
For the aggregative and NSGAII methods, our genetic algorithm requires an average of computation time equal to seconds for the type of problem (with ). For the Pareto method, the average of computation time is equal to seconds. The problem of remains the easiest to solve. The numerical results also reveal that the aggregative approach and NSGAII require an average of computation time more than the Pareto approach.
From Table 5, Our genetic algorithm with NSGAII approach requires an average computation time equal to seconds for the type of problem (with ). In Manaa and Chu[29] the branchandbound algorithm to minimize the makespan criterion, needs in average more than seconds to find the optimal solution. The problem of (having the smallest number of tasks compared to others) requires less computation time compared to other problems. This can be justified by the fact that the processors are loaded with less than the number of biprocessor tasks compared to other cases.
4.2 Solution quality
Table 6, provides the averages of makespan, total tardiness and total completion time from instances randomly generated. These instances satisfy the same parameters , and .
The first column indicates the types of problems , and that denote respectively the makespan, the total tardiness, the total completion time and the number of nondominated solutions.
Types  initial solution  Aggregative GA  Pareto GA  NSGAII  Lower bound  

210  929  2771  160  106  1588  25  175  134  1688  23  170  118  1632  43  144  83  1126  
309  1019  4365  226  109  2627  28  240  71  2520  29  243  103  2575  47  201  24  2168  
385  1285  5856  292  82  3350  28  297  79  3323  21  306  71  3300  29  284  10  2887  
263  1494  4288  196  253  2619  23  232  244  2752  22  202  242  2663  24  175  80  1740  
362  1605  6655  270  187  4220  32  302  111  4026  33  292  113  4046  26  243  37  3102  
523  1676  9484  414  315  6464  29  399  137  5654  32  402  164  5764  19  380  9  4789  
292  1747  4766  237  418  2986  23  264  489  3143  29  243  445  3043  35  210  194  1738  
402  2070  7325  291  258  4458  24  314  145  4252  33  306  153  4058  31  258  31  3203  
525  1971  930  416  364  6659  26  427  112  6249  52  436  201  6315  35  377  18  4967  
340  2474  6396  254  581  4127  25  265  414  3827  29  261  577  3939  37  213  299  2317  
477  2841  9754  340  557  6639  26  323  229  6049  37  342  313  6025  22  283  44  4530  
603  2648  13456  495  510  9050  25  489  367  8862  41  506  442  9120  26  430  32  6831  
253  1306  3315  194  259  1975  25  223  258  2008  31  200  229  1932  51  173  221  1233  
339  1373  4797  226  90  2663  23  233  78  2536  41  252  81  2678  36  208  28  2091  
409  1149  6051  299  125  3679  28  325  81  3433  29  333  58  3569  38  290  9  3072 
The results for the three approaches listed in Table 6 show that the aggregative selection technique is more effective for the problems of , with(). For the problems of and with (), Pareto is more efficient. For other cases of problems, each technique has an advantage on one or two criteria.
The quality of the solutions found by the NSGAII approach is good for the problem of with () and with (). The space of solutions found by NSGAII is the most diverse in many cases containing a significant number of nondominated solutions. This is justified by the fact that this approach ensures elitism by archiving nondominated solutions in the evolution from one generation to another.
The results found by the three approaches are close to the lower bounds for the makespan criterion. In some cases, the results of total tardiness are close to the lower bounds with NSGA and Pareto approach.
For the total completion time criterion, the results with the aggregative approach are close to the lower bounds in some cases and quite far from these lower bounds for the other cases. The solution space remains the least diversified with the aggregative approach, but it is most effective for the makespan criterion.
The graphical representation of the results obtained for the makespan criterion described in Figure 2 shows that the Pareto selection technique is less effective for such problems of and with () . For the problems of and with () NSGAII is more effective. In other cases, the three techniques are almost identical.
The results for the total tardiness criterion described in Figure 3, show that the Pareto selection technique is more effective for the problem of with () and with (). In other cases, both techniques NSGAII and Pareto are almost identical. For the problems of with (), and the aggregative approach is less efficient.
The three techniques are almost identical. For the problems of with () and , the aggregative approach is slightly less efficient.
Types  initial solution  Aggregative GA  Pareto GA  NSGAII  Lower bound  

459  4687  11971  365  1484  7996  23  376  1283  7766  32  355  1114  6729  20  301  342  4158  
673  6089  20427  493  1161  12675  19  473  521  11505  23  489  942  12644  25  409  73  8383  
819  5665  24914  617  1236  16554  17  618  849  15894  56  634  940  16403  18  582  21  11720  
543  6814  17757  414  2526  12400  25  412  1769  11813  53  431  2193  11919  27  319  759  6250  
734  6377  27034  583  2123  19932  22  573  1633  18492  52  569  1879  19212  28  482  133  12227  
1010  8200  39710  801  2697  29150  20  783  1858  27821  44  811  2066  27958  30  723  27  18463  
614  8128  19334  490  3833  14127  14  461  2656  13615  41  482  3465  13858  18  388  798  6411  
847  9783  31231  643  2736  20772  18  592  1619  19863  41  665  2023  19484  20  494  235  12200  
1025  9851  40043  811  3122  28309  24  833  1707  27331  59  803  2394  26554  16  701  36  17768  
688  10762  26394  547  4810  19011  27  545  4117  18261  17  541  4365  18544  24  403  1107  8650  
943  10051  40377  766  3427  31064  26  760  3228  32847  31  691  2634  25813  26  588  168  17036  
1220  14008  57961  949  4045  41358  22  975  3871  39914  61  998  3634  39887  32  837  89  25383  
518  5265  12921  404  1931  9003  24  403  1362  8366  22  429  1719  8939  20  311  683  4376  
662  5948  19576  488  1634  12887  10  440  744  11321  102  491  1157  12877  25  396  175  8096  
872  6214  26521  668  1529  18115  23  673  651  16114  66  664  783  16683  21  590  58  11928 
The results for the three criteria listed in the Table 7 show that the Pareto selection technique is more effective for problems of and with (), and with () and with ().
For the problems of with () and with () NSGAII method is more effective. For the problem of with () each technique (aggregative, Pareto or NSGAII) has an advantage over the other on a one criteria. In other cases, Pareto techniques and NSGAII are almost identical.
The graphic representation for the makespan criterion described in Figure 5 shows that NSGAII method is less effective for the problems of with (). For the problem of with () NSGAII is more effective. For other cases, the three techniques are almost identical. For the five types of problems studied with the makespan criterion, the aggregative approaches and Pareto are identical.
The results for the total tardiness criterion described in Figure 6 show that the Pareto approach is more effective for the problems of with () , with() and with (). For the problem of with () NSGAII method is the most effective. In other cases, both techniques (NSGAII and Pareto) are almost identical. The aggregative selection method is less effective for problems of , and with ().
The numerical results of the total completion time criterion show that the Pareto technique is more effective in many cases (see Table 7). By looking at these numerical values, we conclude that the three approaches are almost identical and the averages values found are very close. Figure 7 summarizes the numerical results found in average for the three approaches we investigated for the total completion time criterion. The results found by three approaches are close to the lower bounds for the makespan criterion. In many cases, the results for the total completion time are close to the lower bounds for NSGAII and Pareto approach. For total tardiness, the results are quite far from lower bounds with the three approaches studied.
5 Conclusions and perspectives
We studied a multiobjective scheduling problem on two dedicated processors to optimize three criteria; the makespan, the total tardiness and the total completion time. In this study, we exploited the lower bound constructed for each criterion to assess the quality of the solutions found by NSGAII algorithm, Pareto and aggregative methods proposed for solving the multiobjective problem. To generate the weight for the aggregative approach, we used the method of Uniform Design (proposed by Leung and Wang[28]) to choose a variety of solutions uniformly dispersed.
The results of the studied problems are encouraging and promising. Therefore, it is interesting to study other extensions of these problems in a future work, like the study of scheduling problem on parallel processors. For example minimizing the makespan for the problem where is the number of processors required by the task . This problem was proved NPhard in the strong sense in Blazewicz et al.[7]. Therefore, it is interesting to test the proposed methods for scheduling these problems on parallel processors (Venkata et al.[38].
Compliance with Ethical Standards
Conflict of interest The authors declare that they have no conflict of interest.
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