The job shop scheduling problem, which has been dealt with by various traditional optimization methods over the decades, has proved to be an NP-hard problem and difficult in solving, especially in the multiobjective field. In this paper, we have proposed a novel quadspace cultural genetic tabu algorithm (QSCGTA) to solve such problem. This algorithm provides a different structure from the original cultural algorithm in containing double brief spaces and population spaces. These spaces deal with different levels of populations globally and locally by applying genetic and tabu searches separately and exchange information regularly to make the process more effective towards promising areas, along with modified multiobjective domination and transform functions. Moreover, we have presented a bidirectional shifting for the decoding process of job shop scheduling. The computational results we presented significantly prove the effectiveness and efficiency of the cultural-based genetic tabu algorithm for the multiobjective job shop scheduling problem.
The scheduling problem is one of the most important and hardest combinatorial optimization problems on account of its complexity and frequency in practical applications. The purpose of scheduling generally is to allocate a set of resources to tasks by the definition of Pinedo. Since the first appearance of the systematic method to scheduling problems was in the mid-1950s, thousands of articles on different scheduling problems have arisen in the literature, which can be categorized in accordance with shop environments, including single machine, parallel machines, flow shop, flexible flow shop, job shop, open shop, and others.
Job shop scheduling problem (JSSP) is one of the most difficult ones among all the scheduling problems. Literature [
Among all these algorithms, the cultural algorithm has been paid attention to gradually and applied to solve scheduling problems. Similar to the development of evolutionary computation, Reynolds developed a model of the evolution of cultural systems and subsequently the cultural algorithms in 1994 [
It is common sense that practical scheduling problems are multiobjective by nature. And a number of solutions should be provided to decision makers to make a conclusion. Smith started the researches on multiobjective scheduling in 1956 by providing a Smith algorithm for some scheduling problems with weighted sum of completion time minimization, subject to all jobs being completed by their due dates [
However, different from the research in the area of the single objective scheduling problem, researches that pay attention to multiple criteria have been scarce. The goal of multiobjective JSSP is to find as many different promising schedules as possible. In this work, we apply makespan and mean flow time as the objectives of our algorithm. After considering the advantage and disadvantage of the cultural algorithm, we modified the structure and strategies of the original one to coordinate with multiobjective JSSP.
In this paper, we present a novel quadspace cultural genetic tabu algorithm to solve the multiobjective scheduling problem, which consists of four spaces, including double brief spaces and double population spaces. The search process is divided into two parts, which have their own cultural structures while exchanging with each other at a predefined frequency, which is one of the novelties of our paper. The other novelty is to design two different brief spaces and their influence functions along with the multiobjective selection strategy. Last but not least, a bidirectional shifting is presented for decoding the presentation, which could help improve the solutions efficiently. The goals of this paper are twofold: first, to analyze the effects when changing the parameters of our algorithm and to verify the proposed cultural algorithm, second, use the algorithm to deal with the real multiobjective JSSP. To justify our approach, we compared our proposed approach with other well established MOEAs- (NSGAII [
The reminder of the paper is organized as follows. In Section
JSSP is one of the most famous and hardest combinatorial optimization problems. During the past decades, a bunch of literature has been published, but no efficient algorithm has been presented yet for solving it to optimality in polynomial time.
Suppose we are given
The indices and variables of the model are enumerated as follows:
Functions (
The conflicting character, where improving one objective may only be achieved when worsening another objective, exists in objective functions generally. Therefore, obtaining an optimal scheduling solution that optimizes all the objectives is nearly impossible. However, there exists a set of equally efficient, nondominated, or noninferior solutions, known as the Pareto-optimal set. We recall the basic notion of efficient solution: a feasible solution
Many metaheuristic techniques have been proposed in the literature to search for near-optimal scheduling solutions. But the researches [
The concept of culture can be defined in plenty of ways. Durham [
Cultural algorithm is an evolution model through observing the cultural process in nature. It is a dual inheritance system that characterizes evolution at both the macroevolutionary level, which occurs at the brief space, and at the microevolutionary level, which takes place within the population space. These two spaces interface with each other through two functions: an acceptance function and an influence function. The cultural algorithm framework is shown in Figure
The cultural algorithm framework.
As described in Figure
It is well known that GA is not good at fine-tuning the solutions that are already close to the optimal solution, which means its local search ability is not as good as the global one. Hence, it is necessary to incorporate local search methods to find more effective optimal solutions, which is tabu search in this paper. We are not going to apply some local search after GA like many other researches do. The way we do it is to employ the two searching processes simultaneously with some interactions. The advantage of our approach is to gain more near-optimal solutions without suffering from premature convergence. The overview of the two processes is shown in Figure
Overview of the exploration and exploitation processes.
The proposed model, which consists of four spaces and dual evolution processes, takes advantage of a parallel and thorough search process compared to the original one. GA and TS are for the purpose of global and local search separately. The better individuals generated by genetic search in junior space are sent to the senior population space for further local search at regular or irregular intervals. The number and frequency of the individuals transmitted are determined by the grade function. Corresponding to the dual evolution process, there are two belief spaces guiding them separately toward the promising area through the influence functions and updated by the acceptance functions. And the best individuals in belief space of TS will be sent to the belief space of GA at predefined intervals by transform function. It is obvious that the advantage of the double search processes, besides the exploration and exploitation simultaneously, is that users can design different specific knowledge and influence functions for different goals. The flowchart of the advanced cultural algorithm is shown in Figure
The framework of QSCGTA.
Generally speaking, heuristic methods own advantages than exact methods in solving combinatorial optimization problems. Because it can provide more near-optimal solutions to decision makers. Therefore, we proposed the cultural genetic tabu algorithm to deal with the multiobjective JSSP.
Chromosome representation is a key point in designing efficient evolutionary algorithms for constrained JSSPs. The reason is that different formulations in solutions correspond to different search spaces and different difficulties for further optimization operators. Although there have been all sorts of representation methods, generally speaking, these representations can be classified into the following two encoding approaches: direct approach and indirect approach [
In our work, direct representation that belongs to operation-based representation with a schedule builder is used. This representation encodes a schedule as a sequence of operations and each gene represents one operation. To avoid the infeasibility raised by the precedence constraints, all operations for a job are named with the same symbol and interpreted by the order of occurrence in the given chromosome. This representation was employed by Bierwirth [
By searching the permutation from left to right, a task
The initial solutions are generated randomly with the length of
After the representation, the chromosome must be transformed into a feasible schedule. And computational experiments performed in [
The schedule builder used in this paper is bidirectional decoding, which performs a kind of local search. The decoding allocates each operation on its assigned machine one by one in the order represented by the coding. When operation
The pseudocode of our proposed bidirectional shifting is as shown in Algorithm
Begin for if else end End % For each end
end
Through the bidirection shifting, the actual processing order of operations
Genetic algorithms are stochastic search methods, containing complex interactions among parameters. The mechanics of the complex parameter interactions play an essential role in the performance of GA. Based on probability calculations and simulation results, Deb and Agrawal [
Crossover is considered as the backbone of GA, which aims to inherit information of two parent solutions to offspring. Provided that the parents keep different aspects of better solutions, such as in multiobjective problems, crossover owns a good opportunity to find better offspring. Considering the repetition structure of the representation, crossover operators containing more genes should be applied instead of the one-point or two-point crossover.
In this paper, we applied two kinds of crossover and each applied with half possibility. Firstly the generalized order crossover [
Illustration of the procedure of binary crossover.
Mutation is another important operator in GA and is usually performed after the crossover with a small probability. Considering the validation of the chromosome, we apply in this paper a swap mutation that needs no repair. Two different operations are picked randomly and then they exchange their positions.
The new individuals will be compared with the corresponding parent solutions by the following rules:
Tabu search is one of the most efficient local search strategies for scheduling problems. It is obtained by transforming one solution to the next according to some neighborhood structures. The main elements of TS are the neighborhood structures, the tabu list length, and stopping rules. In our paper, because TS is a kind of point-search, the size should be much smaller than GA. It worked every ten iterations and shares the same stopping rules with global search. As for the neighborhood structure, we applied two kinds of neighborhood structures. Firstly, we chose the two-job exchange mutation as the neighborhood and an example of (
Secondly, the reversal of any two successive operations
The cultural mechanism in this paper plays a key role in guiding the evolution to promising areas, which consists of dual belief space, acceptance functions, influence functions, grade function, and transform function.
The grade function is proposed to decide how many and how frequent the individuals from GA should be sent to go through TS. Because of the single-point search nature of TS, we send a small population to TS compared to the population in GA. The number of grade population is self-adapted as shown in formula (
Generally speaking, the belief space of the cultural algorithm consists of several knowledge structures and is updated at a certain frequency. In our structure, there are two brief spaces that have different knowledge structures. Because our purpose is to enlarge the search space for GA in order to process global search while reduce the search space to guide the TS for local search, in brief space for GA, situational and topographical knowledge are adopted while situational and normative knowledge are applied for TS, while the history and domain knowledge are not applied for they are usually effective when the fitness landscape is dynamic.
The situational knowledge for GA consists of the best exemplars found along the evolution process. The structure is
The normative knowledge represents the best district in the objective space and consists of two members
The topographical knowledge is used to record the distribution of solutions to later help adapt the global acceleration. The space represented by normative knowledge is divided into grids of
During each iteration, this knowledge will be updated to rebuild a new cell following the normative knowledge.
The knowledge of current belief space is updated by the individuals selected by acceptance function. The nondominated sets of populations are chosen to update the belief spaces.
The situational and normative knowledge are updated as follows:
Influence function is one of the key issues in cultural algorithms in guiding the evolutionary search. In this paper, we propose two totally different influence function structures for GA and TS.
The influence function applied on the genetic search consists of three parts. The first one is sending 10% individuals randomly to GA. This method will improve the quality of the population and then improve the evolution process.
The second one is applied to the mutation procedure. We apply the formula (
And the last one is towards crossover. This is inspired by the phenomenon called Atavism, which is a theory in heredity holding that the reappearance of a characteristic in an organism after several generations of absence is usually caused by the chance recombination of genes. Therefore, we make the best individuals in situational knowledge crossover with the worst individuals in GA expecting that the offspring in several generations will inherit the good gene from the best individuals. We use the topographical knowledge stored in the belief space to select the best individual. Firstly, we use the roulette wheel selection to choose the least populated cell, and then randomly select a nondominated individual from that cell to be the best individual. Each cell is assigned a fitness of
The influence function for TS consists of a combined effect by both situational knowledge and normative knowledge. Its strategy is, as formula (
We design a limited archive to store the best solutions. And each new nondominated solution from the two processes will be compared with members in the archive. If
To sum up, we proposed a novel hypergenetic algorithm incorporated with tabu search under the frame of cultural algorithm. The flowchart of our QSCGTA is shown in Figure
The flowchart of multiobjective QSCGTA.
Computational experiments are carried out to investigate the performance of our proposed cultural genetic tabu search. In order to evaluate the performance of our algorithm, we run the algorithm on a series of benchmark problems from the OR-Library (
All the programs in the experiments were written in Matlab and all the experiments were running on platform using Intel Core 4 Quad 2.4 GHZ CPU with 2 GB RAM. First we applied orthogonal experiments to decide our parameter settings in the QSCGTA. And then we compared our algorithm with other well-known multiobjective evolutionary algorithms, which are NSGAII, SPEA2, and MPSO.
A set of solutions that are superior to the rest of the solutions exists in multiobjective optimization. Therefore, new approaches, which differ from the single objective, are required to compare the performance of the algorithms. The performance measures that are used are as follows. Number of Pareto solution (NPS): this performance measurement is calculated by counting the number of nondominated solutions obtained. A larger number corresponds to better performance. Hyperarea ratio (HR) [ Spread of nondominance solutions (SNS): this criterion, which is known as an indicator of diversity, is calculated through the following formula. Larger values of this criterion correspond to higher quality solutions:
where
The parameters, which can be uncertainty, had a great influence on the performance of the algorithm [
If the authors apply the factorial experiment to design the parameters, it requires
The orthogonal experiment used here is to design experiments to investigate how different parameters affect the mean and variance of a process performance characteristic. The experiment design involves using orthogonal arrays to organize the parameters, which affect the process, and levels at which the parameters should be varies.
Generally first we considered the hyperarea ratio of problem MT10, LA25, LA28, LA36, and ABZ7 for ten times at 500 generations as the objective. Then we chose the four levels for population size of genetic search (PS), the crossover possibility (CP), the mutation possibility (MP), tabu list length (TL). The levels of all the parameters are listed in Table
Levels of the parameters.
Levels | PS (A) | CP (B) | MP (C) | TL (D) |
---|---|---|---|---|
1 | 50 | 0.6 | 0.1 | 5 |
2 | 100 | 0.7 | 0.2 | 10 |
3 | 150 | 0.8 | 0.3 | 15 |
4 | 200 | 0.9 | 0.4 | 20 |
Here we chose the L16 orthogonal array and Table
Range analysis of the orthogonal experiment for MT10.
A | B | C | D | Empty | Results (HR) | |
---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | ||
1 | 1 | 1 | 1 | 1 | 1 | 0.0027 |
2 | 1 | 2 | 2 | 2 | 2 | 0.0635 |
3 | 1 | 3 | 3 | 3 | 3 | 0.5675 |
4 | 1 | 4 | 4 | 4 | 4 | 0.2822 |
5 | 2 | 1 | 2 | 3 | 4 | 0.4233 |
6 | 2 | 2 | 1 | 4 | 3 | 0.2439 |
7 | 2 | 3 | 4 | 1 | 2 | 0.5076 |
8 | 2 | 4 | 3 | 2 | 1 | 0.0566 |
9 | 3 | 1 | 3 | 4 | 2 | 0.5441 |
10 | 3 | 2 | 4 | 3 | 1 | 0.8328 |
11 | 3 | 3 | 1 | 2 | 4 | 0.6879 |
12 | 3 | 4 | 2 | 1 | 3 | 0.4633 |
13 | 4 | 1 | 4 | 2 | 3 | 0.9712 |
14 | 4 | 2 | 3 | 1 | 4 | 0.5825 |
15 | 4 | 3 | 2 | 4 | 1 | 0.4239 |
16 | 4 | 4 | 1 | 3 | 2 | 0.4315 |
|
0.2290 | 0.4853 | 0.3415 | 0.3890 | 0.3290 | |
|
0.3079 | 0.4307 | 0.3435 | 0.4448 | 0.3867 | |
|
|
|
0.4377 |
|
0.5615 | |
|
0.6023 | 0.3084 |
|
0.3735 | 0.4940 | |
Range |
|
0.2381 | 0.3070 | 0.1903 | 0.2325 | |
Order of test factors | A > C > B > D | |||||
Optimal level | A3 | B3 | C4 | D3 |
The parameter settings for the other four instances are applied the same way as MT10 and the results are shown in Tables
Results of parameter settings for LA25.
LA25 | A | B | C | D |
---|---|---|---|---|
|
1 | 2 | 3 | 4 |
|
0.4387 | 0.4048 | 0.5426 | 0.3867 |
|
|
0.5185 | 0.3827 | 0.4956 |
|
0.2409 |
|
0.3646 | 0.5208 |
|
0.4611 | 0.4747 |
|
|
Range |
|
0.1324 | 0.2807 | 0.1453 |
Results of parameter settings for LA28.
LA28 | A | B | C | D |
---|---|---|---|---|
|
1 | 2 | 3 | 4 |
|
0.4724 | 0.3571 | 0.3343 | 0.3309 |
|
0.2277 |
|
0.4581 |
|
|
0.4347 | 0.4316 |
|
0.3810 |
|
|
0.3556 | 0.4961 | 0.3924 |
Range |
|
0.2543 | 0.2315 | 0.1592 |
Results of parameter settings for LA36.
LA36 | A | B | C | D |
---|---|---|---|---|
|
1 | 2 | 3 | 4 |
|
0.4552 | 0.3858 | 0.5321 | 0.4192 |
|
0.3872 | 0.3948 | 0.4202 | 0.3370 |
|
|
|
0.4628 |
|
|
0.4889 | 0.5127 |
|
0.4614 |
Range | 0.2473 |
|
0.1305 | 0.1112 |
Results of parameter settings for ABZ7.
ABZ7 | A | B | C | D |
---|---|---|---|---|
|
1 | 2 | 3 | 4 |
|
0.2854 | 0.3627 | 0.2734 | 0.3707 |
|
0.2108 | 0.2395 | 0.3141 | 0.3541 |
|
|
0.1939 |
|
|
|
0.4304 |
|
0.3798 | 0.2416 |
Range | 0.2196 |
|
0.2095 | 0.1422 |
Multiobjective optimization has two goals: one is to find a set of solutions as close as possible to the Pareto front, while the other is to find a set of solutions as diverse as possible.
The parameter settings of our algorithm are the same as above and the one for MPSO is the same as literature [ The initial population is randomly generated and the number is set to 150. Tournament selection is chosen. The GOX and swap are used as crossover and mutation operators. The ratios of GOX and swap are set to 0.8 and 0.4, respectively. The size of archive is set to 40 and number of iteration is set to 500.
Each benchmark problem was tested for twenty times with different seeds. Then all the final generations were combined and the nondominated sorting was performed to constitute the final nondominated solutions.
The results in Table
Results of algorithms on multiobjective JSSP.
|
NSGAII | SPEA2 | MPSO | QSCGTA | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
NPS | HR | SNS | NPS | HR | SNS | NPS | HR | SNS | NPS | HR | SNS | ||
LA03 | 10 * 5 | 2 | 0.44 | 5.17 | 5 | 0.48 | 2.71 | 4 | 0.48 | 3.12 |
|
|
|
LA04 | 10 * 5 | 1 | 0.42 | — | 2 | 0.94 | 1.96 | 2 | 0.94 | 1.96 |
|
|
|
LA05 | 10 * 5 | 1 | — | — | 1 | — | — | 1 | — | — |
|
— | — |
Mt10 | 10 * 10 | 3 | 0.53 | 1.11 | 1 | 0.69 | — | 3 | 0.93 | 2.13 |
|
|
|
LA16 | 10 * 10 | 2 | 0.45 | 5.19 | 5 | 0.62 | 4.85 | 3 | 0.60 |
|
|
|
10.41 |
LA17 | 10 * 10 | 1 | 0.39 | — | 7 | 0.60 |
|
6 | 0.75 | 12.87 |
|
|
|
LA18 | 10 * 10 | 3 | 0.42 | 7.70 | 3 | 0.52 | 1.75 | 5 |
|
2.81 |
|
0.78 |
|
LA22 | 15 * 10 | 4 | 0.32 | 9.31 |
|
0.69 | 18.23 | 3 | 0.73 | 3.15 | 2 |
|
|
LA23 | 15 * 10 | 3 | 0.51 | 5.25 | 1 | 0.43 | — | 1 | 0.69 | — |
|
|
|
LA25 | 15 * 10 | 2 | 0.54 |
|
7 | 0.86 | 1.68 |
|
0.77 | 9.36 | 5 |
|
6.07 |
LA27 | 20 * 10 | 2 | 0.44 | 4.10 | 3 | 0.59 | 5.86 | 2 | 0.70 |
|
|
|
4.62 |
LA28 | 20 * 10 | 3 | 0.52 | 6.38 | 3 | 0.54 | 3.42 | 3 | 0.71 | 8.88 |
|
|
|
LA29 | 20 * 10 | 1 | 0.29 | — | 2 |
|
5.28 |
|
0.64 |
|
2 | 0.67 | 7.22 |
LA36 | 15 * 15 | 3 | 0.38 | 9.15 | 6 | 0.52 | 3.89 | 3 | 0.63 | 3.92 |
|
|
|
LA38 | 15 * 15 | 1 | 0.30 | — | 2 | 0.75 |
|
2 | 0.79 | 12.23 |
|
|
7.35 |
LA40 | 15 * 15 | 4 | 0.37 | 7.69 | 1 | 0.61 | — | 3 |
|
1.19 |
|
0.81 |
|
ABZ7 | 20 * 15 | 3 | 0.43 | 2.51 |
|
0.52 | 4.03 | 3 | 0.63 | 0.79 | 4 |
|
|
ABZ8 | 20 * 15 | 1 | 0.37 | — | 2 | 0.78 | 0.21 | 1 | 0.52 | — |
|
|
|
ABZ9 | 20 * 15 | 3 | 0.52 | 9.95 |
|
0.70 | 1.79 | 3 |
|
|
2 | 0.73 | 13.32 |
To further test the stability of algorithms, we run the four algorithms twenty times independently, under the aforementioned environment, on randomly selected instances with different sizes, which are LA03, MT10, LA17, LA27, LA40, and ABZ7. The box plots of hyperarea ratio on these instances are shown in Figure
The box plots of hyperarea ratio for different problems. Column numbers refer to (1) NSGAII, (2) SPEA2, (3) MPSO, and (4) QSCGTA.
Computational time of four algorithms for different problems.
Moreover, to further verify the performance of our algorithm, the performance for HR is computed for each algorithm on randomly generated instances. Several problem scenarios were generated by varying one or more of the number of machines and number of jobs. For each scenario, 20 problems were randomly generated by setting operation processing times from a uniform distribution in the interval [5, 100], and the average values across these instances were recorded. All jobs had randomly assigned routing through the system. The results on instances generated randomly, shown in Table
Performance of algorithms on instances generated randomly.
|
NSGAII | SPEA2 | MPSO | QSCGTA | ||||
---|---|---|---|---|---|---|---|---|
HR | SNS | HR | SNS | HR | SNS | HR | SNS | |
10 * 10 | 0.39 | 9.06 | 0.37 |
|
|
5.49 | 0.46 | 6.49 |
10 * 15 | 0.34 | 8.85 | 0.50 | 6.10 | 0.75 | 6.36 |
|
|
10 * 20 | 0.51 | 5.19 | 0.72 | 4.79 |
|
6.89 |
|
|
15 * 15 | 0.22 | 5.32 | 0.48 | 4.66 | 0.70 | 8.15 |
|
|
15 * 20 | 0.43 | 4.58 | 0.61 | 6.34 | 0.78 | 6.67 |
|
|
20 * 20 | 0.37 | 3.14 | 0.50 | 5.17 | 0.68 |
|
|
7.24 |
Multiobjective scheduling has become the main research field in scheduling problems because of the multiobjective character, by nature, of many real-world scheduling problems. Due to the complexity of the job shop scheduling problem, many researches have been focused on multiobjective single machine problems or flow shop problems. The researches on multiobjective job shop problems are very rare. Therefore, in this paper, we have proposed QSCGTA for solving the multiobjective JSSP. The GA and TS have been incorporated in the frame of a novel cultural algorithm to search for the Pareto-optimal schedules.
The experiments indicated that our approach is suitable for applied benchmark problems and obviously yielded better performance in terms of solutions, stability, and computation time compared with the other three algorithms. The main strength of our approach is in combination with global and local search under a novel cultural algorithm frame in order to produce diverse solutions while maintaining the convergence of the nondominated solutions. In each generation, only a predefined number of the best solutions of GA is selected for applying TS. It provides more diversity toward Pareto-optimal solutions. All in all, our proposed QSCGTA-based evolutionary scheduling approach accomplished the goals of multiobjective job shop scheduling problems both in convergence and diversity.
More comprehensive studies can be applied to extend the QSCGTA. Other possible criteria in multiobjective optimization will be considered. Furthermore, more local search methods will be analyzed to integrate to the QSCGTA algorithm.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors appreciate the support of National Natural Science Foundation of China (Grant nos. 61174040 and 61104178), Shanghai Commission of Science and Technology (Grant no. 12JC1403400), and Fundamental Research Funds for the Central Universities.