This paper focuses on the multi-objective optimization of the reentrant hybrid flowshop scheduling problem (RHFSP) with machines turning on and off control strategy. RHFSP exhibits significance in many industrial applications, but scheduling with both energy consumption consideration and reentrant concept is relatively unexplored at present. In this study, an improved Multi-Objective Multi-Verse Optimizer (IMOMVO) algorithm is proposed to optimize the RHFSP with objectives of makespan, maximum tardiness, and idle energy consumption. To solve the proposed model more effectively, a series of improved operations are carried out, including population initialization based on Latin hypercube sampling (LHS), individual position updating based on Lévy flight, and chaotic local search based on logical self-mapping. In addition, a right-shift procedure is used to adjust the start time of operations aiming to minimize the idle energy consumption without changing the makespan. Then, Taguchi method is utilized to study the influence of different parameter settings on the scheduling results of the IMOMVO algorithm. Finally, the performance of the proposed IMOMVO algorithm is evaluated by comparing it with MOMVO, MOPSO, MOALO, and NSGA-II on the same benchmark set. The results show that IMOMVO algorithm can solve the RHFSP with machines turning on and off control strategy effectively, and in terms of convergence and diversity of non-dominated solutions, IMOMVO is obviously superior to other algorithms. However, the distribution level of the five algorithms has little difference. Meanwhile, by turning on and off the machine properly, the useless energy consumption in the production process can be reduced effectively.
Hybrid flowshop scheduling problem usually involves several stages, each of which contains a certain number of parallel machines and each job passes through all the stages only once in sequence. However, in some special industries, a job needs to access some stages more than once, such as semiconductor wafer manufacturing and TFT-LCD (thin film transistor liquid crystal display) panel manufacturing. RHFSP has been proved to be NP-hard [
Since Graves and Meal et al. [
In recent years, some scholars have studied the multi-objective RHFSP. Cho and Bae et al. [
At present, energy saving plays an increasingly significant role in manufacturing industries, especially energy-intensive industries. Optimizing production schedule helps to reduce unnecessary energy consumption. Luo and Du et al. [
In the current context of rising energy prices and increasingly stringent environmental concerns, it is particularly important to reduce energy cost and achieve green and sustainable development for the manufacturing industry. According to the latest data released by the State Energy Administration and China Telecom Federation, the energy consumption of manufacturing industry accounts for about one-third of the total energy consumption. However, both purchasing more energy-efficient equipment and building more energy-efficient production lines require huge financial investments. The benefits could not be easily enjoyed by most of manufacturing companies, especially those small and medium sized enterprises. In practice, it was observed that in an eight-hour shift, the bottleneck machines still stay idle 16% of the time on average. If the machines are turned off during the idle periods, 13% energy will be saved [
The remainder of this paper is organized as follows: the RHFSP is described with the makespan, max tardiness, and idle energy consumption objectives in Section
The RHFSP addressed in this study can be described as follows [
Additionally, the following assumptions are made: all the jobs and machines are ready at zero time; at any time, each machine can process at most one job, and each job can only be processed by at most one machine; all the jobs do not affect each other; the number of re-entrance of each job, the processing time of each operation, and the production route of each job through the shop are known in advance; the buffer capacity between any two consecutive processing stages is infinite; preemption is not allowed, and once the job is processed, it cannot be interrupted; the machine failure and the machine adjustment time are not considered.
The mathematical symbols involved in the model and their meanings are as follows:
It is generally assumed in RHFSP that the machine stops until all the jobs have been processed. Inevitably, machines will be idle during the waiting time. Kordonowy [
Power graph for turning.
From Figure
Machine state transition diagram.
First of all, if turns off the machine after an operation, the conversion time should be less than the interval between the start processing time of the next job and the completion time of it, that is to say, the constraint (
Secondly, the energy consumption of starting machine is greater than that of idling state. When choosing to shut down the machine, the interval between adjacent jobs should be greater than a critical time length, that is, the constraint (
Supposing
Based on the existing literature [
In
Plugging
Subject to
Equations (
In a three-objective optimization problem, according to the definition of dominance [
Although many intelligent optimization algorithms have been used to solve the RHFSP, “No Free Lunch Theorems” [
The Multi-Verse Optimizer (MVO) [
MVO algorithm is mainly used to solve optimization problems of continuous functions, so it cannot be directly used to deal with discrete optimization problems. In this paper, we adopt random key coding. Each gene is represented by random numbers within
Take the RHFSP with 4 jobs and 3 stages as an example, in which the number of identical parallel machines at each stage is 3, 2, and 2, respectively. It is assumed that all the following parameters of the machines are the same, the working energy consumption per unit time
The processing times of the four-job and three-stage example.
Processing |
Operations (first) | Reentrant 1 | Reentrant 2 | Due Date | ||||||
---|---|---|---|---|---|---|---|---|---|---|
Stage 1 | Stage 2 | Stage 3 | Stage 1 | Stage 2 | Stage 3 | Stage 1 | Stage 2 | Stage 3 | ||
Job 1 | 2 | 2 | 1 | 0 | 2 | 1 | 0 | 1 | 2 | 13.2 |
Job 2 | 1 | 3 | 0 | 0 | 2 | 1 | 0 | 2 | 1 | 14.5 |
Job 3 | 3 | 1 | 2 | 2 | 1 | 0 | 0 | 0 | 0 | 8.6 |
Job 4 | 2 | 1 | 1 | 2 | 0 | 0 | 3 | 1 | 0 | 11.7 |
If the processing time of one operation
Random key coding.
|
1 | 2 | 3 | 4 |
individual | 0. 6555 | 0. 3922 | 0. 7431 | 0. 1712 |
ascending order | 0.1712 | 0. 3922 | 0. 6555 | 0. 7431 |
job sequence | 4 | 2 | 1 | 3 |
Gantt charts of the four-job and three-stage example. (a) Scheduling Job 4. (b) Scheduling Job 2. (c) Scheduling Job 1. (d) Scheduling Job 3.
In this study, we propose a right-shift procedure to further improve the quality of the solutions. On the premise of not changing the job sequence, the procedure can reduce the
Schematic diagram of right-shift procedure.
When the optimal solution space cannot be predicted, the solution space feature of the initial population can maximize the information of all the individuals within a limited number. Therefore, the distribution of the initial population seriously affects the convergence performance of the algorithm. In this paper, the Latin hypercube sampling technique is used to initialize the population. Assuming that m samples are extracted in an n-dimensional vector space, the steps for Latin hypercube sampling are as follows.
In case of 2-dimensional vector and 100 samples, the distribution of the initial population constructed by completely random and Latin hypercube sampling is shown in Figure
Distribution of complete random and Latin hypercube sampling.
The Lévy distribution was a probability distribution model proposed by the famous French scientist Paul Pierre Lévy in the 1930s. Lévy flight is a random search method following the Lévy distribution. It usually moves in short distance and occasionally in long distance, so as to avoid the repeated movement in one place. The diagram is shown in Figure
Lévy flight diagram.
In this paper, the Mantegna algorithm was used to simulate the Lévy distribution. The specific principles are as follows [
where
Generating random step size according to Lévy distribution,
Introducing logical self-mapping into MOMVO algorithms, 20% of individuals in the external archives are randomly selected as elites. Then, chaotic optimization algorithm is used to search within the neighborhood of the elites, and the search space is gradually narrowed as the iteration progresses. If a better solution is detected, it will replace the solution in the external archives. If the search does not produce a better solution, it jumps to the next elite until the traversal is completed. In this paper, the logical self-mapping function is adopted to generate the chaotic sequences. The mathematical expressions are as follows:
MOMVO and MVO have similar search mechanism using white holes, black holes, and wormholes to improve the solutions. A leader selection mechanism is employed to select solutions from the archives and create tunnels between solutions. Specifically, the crowding distance between each solution in the archives is first selected, and the number of solutions in the neighborhood is counted as a measure of coverage or diversity in the approach. Then, to improve the distribution of solutions in the archives across all objectives, a roulette wheel from the less populated regions of the archives is applied to select solutions. The pseudocode of the IMOMVO algorithm is as shown in Algorithm
In order to validate the effectiveness of the IMOMVO algorithm, four multi-objective optimization algorithms, including MOMVO, MOPSO, MOALO, and NSGA-II, are selected for comparative study. The simulation environment is windows 7, Intel Core i7-4770 cpu@3.40GHz, 8G memory. The algorithm is programmed by MATLAB R2017a.
The experiments were conducted on the benchmark set randomly generated by Cho et al. [
This paper selected three performance indicators: SP, GD, and IGD, in which SP and GD are described in literature [
SP is used to measure the distribution uniformity of the non-dominated solutions on the Pareto front. As we all know, the smaller the SP, the better the result. When SP=0, the non-dominated solutions on the Pareto front are evenly distributed. The SP is calculated as follows:
GD is used to evaluate the approximation degree between the front obtained by the algorithm and the real Pareto front of the problem which is calculated as follows:
The performances of metaheuristics often depend on the parameter setting. The IMOMVO algorithm mainly involves three key parameters, population size
Levels of parameters.
Parameters | Factor level | |||
---|---|---|---|---|
1 | 2 | 3 | 4 | |
|
40 | 60 | 80 | 100 |
|
1.2 | 1.5 | 1.7 | 1.9 |
|
5 | 6 | 7 | 8 |
Orthogonal arrays and RV values.
Number | Factor | RV | ||
---|---|---|---|---|
|
|
|
||
1 | 40 | 1.2 | 5 | 1.1892 |
2 | 40 | 1.5 | 6 | 0.7710 |
3 | 40 | 1.7 | 7 | 1.0505 |
4 | 40 | 1.9 | 8 | 1.3805 |
5 | 60 | 1.2 | 6 | 1.1824 |
6 | 60 | 1.5 | 5 | 1.0333 |
7 | 60 | 1.7 | 8 | 0.8261 |
8 | 60 | 1.9 | 7 | 0.8349 |
9 | 80 | 1.2 | 7 | 0.6824 |
10 | 80 | 1.5 | 8 | 0.7110 |
11 | 80 | 1.7 | 5 | 0.8578 |
12 | 80 | 1.9 | 6 | 0.8531 |
13 | 100 | 1.2 | 8 | 0.9243 |
14 | 100 | 1.5 | 7 | 1.0596 |
15 | 100 | 1.7 | 6 | 1.0430 |
16 | 100 | 1.9 | 5 | 0.9107 |
ARV values and significance ranks.
Level |
|
|
|
---|---|---|---|
1 | 1.0978 | 0.9946 | 0.9978 |
2 | 0.9692 | 0.8937 | 0.9624 |
3 | 0.7761 | 0.9444 | 0.9069 |
4 | 0.9844 | 0.9948 | 0.9605 |
Delta | 0.3217 | 0.1011 | 0.0909 |
Rank | 1 | 2 | 3 |
Main effect plot for RV.
From Table
In this paper, six small-sized and six large-sized problems are randomly selected for the benchmark set. Each problem runs 10 times independently using the five algorithms, and each run gets a set of [
Comparisons of IMOMVO and the four algorithms for small-sized problems.
Test problems | Performance measures | IMOMVO | MOMVO | MOPSO | MOALO | NSGA-II | |
---|---|---|---|---|---|---|---|
Sproblem-01-20 | SP | Min | 4.2538 | 3.3425 |
|
4.8011 | 3.4495 |
Avg | 5.5441 | 6.1904 | 6.7477 | 9.2005 |
|
||
Std |
|
2.8282 | 2.9487 | 3.1855 | 1.5064 | ||
GD | Min |
|
1.0905 | 1.7217 | 1.4066 | 1.3952 | |
|
|
2.0821 | 2.2098 | 5.1492 | 2.0520 | ||
|
|
1.1803 | 0.4345 | 2.3362 | 0.3902 | ||
IGD |
|
|
1.9842 | 2.2384 | 2.7163 | 2.1045 | |
|
|
0.7035 | 0.4396 | 0.6491 | 0.5931 | ||
Std | 1.0887 |
|
1.6941 | 1.6562 | 1.3959 | ||
|
|||||||
Sproblem-02-20 | SP | Min | 6.4828 | 6.0011 | 7.8466 |
|
3.9211 |
Avg | 9.7764 | 15.8743 | 14.8757 |
|
14.3121 | ||
Std | 4.0321 | 9.8391 | 5.0085 |
|
5.4210 | ||
GD | Min |
|
1.5297 | 2.1647 | 3.3802 | 2.5898 | |
Avg |
|
4.0680 | 3.6181 | 5.3739 | 3.8907 | ||
Std |
|
1.8535 | 1.2529 | 1.3984 | 1.3274 | ||
IGD | Min |
|
2.6203 | 2.6529 | 3.3949 | 3.2842 | |
Avg |
|
5.1764 | 4.6116 | 6.1921 | 4.7595 | ||
Std |
|
2.8636 | 1.5003 | 2.7036 | 2.1415 | ||
|
|||||||
Sproblem-03-20 | SP | Min | 2.567 |
|
2.7398 | 4.4499 | 2.3531 |
Avg | 4.9246 | 4.9596 |
|
6.1176 | 6.0466 | ||
Std | 2.0407 | 2.2553 |
|
1.9027 | 3.0756 | ||
GD | Min |
|
0.5079 | 1.5609 | 1.8643 | 0.9265 | |
Avg |
|
2.4554 | 2.1164 | 2.9399 | 1.8288 | ||
Std | 0.8034 | 1.4898 |
|
0.7599 | 0.5991 | ||
IGD | Min |
|
2.0295 | 1.3708 | 2.1544 | 1.4214 | |
Avg |
|
3.1044 | 2.7329 | 2.7603 | 2.5920 | ||
Std |
|
1.0239 | 1.3596 | 0.5844 | 1.1414 | ||
|
|||||||
Sproblem-04-20 | SP | Min | 4.4390 |
|
4.1757 | 6.0383 | 5.036 |
Avg |
|
6.7734 | 7.4176 | 9.9239 | 7.5878 | ||
Std |
|
1.8944 | 2.2194 | 2.6719 | 2.4746 | ||
GD | Min |
|
2.0327 | 1.9053 | 4.3381 | 1.8788 | |
Avg |
|
2.8152 | 3.5303 | 7.1940 | 3.0729 | ||
Std |
|
0.6537 | 2.9476 | 2.2363 | 0.8076 | ||
IGD | Min |
|
1.2991 | 2.1892 | 3.399 | 2.1246 | |
Avg |
|
2.4576 | 3.0508 | 5.4666 | 2.8488 | ||
Std |
|
0.7041 | 0.7275 | 1.1314 | 0.7762 | ||
|
|||||||
Sproblem-05-20 | SP | Min |
|
3.7677 | 2.5162 | 3.8762 | 3.4323 |
Avg |
|
8.5614 | 7.9275 | 7.2344 | 6.4903 | ||
Std |
|
3.7584 | 9.4040 | 3.0191 | 3.3006 | ||
GD | Min | 0.4654 | 2.7581 |
|
4.2029 | 3.5653 | |
Avg |
|
4.3472 | 3.8166 | 8.6313 | 4.5525 | ||
Std |
|
2.0990 | 1.9706 | 3.5671 | 0.9877 | ||
IGD | Min |
|
2.0496 | 3.0676 | 5.155 | 2.7927 | |
Avg |
|
3.5092 | 3.8419 | 7.9333 | 4.0812 | ||
Std |
|
1.1922 | 0.8280 | 2.9383 | 1.0016 | ||
|
|||||||
Sproblem-06-20 | SP | Min | 5.6331 | 7.7566 |
|
7.3206 | 5.7043 |
Avg | 10.1854 | 13.3589 | 9.5592 | 12.1647 |
|
||
Std | 3.58811 | 5.2599 | 4.1399 | 3.5184 |
|
||
GD | Min |
|
2.8996 | 3.3732 | 5.2997 | 3.1649 | |
Avg |
|
5.1802 | 5.5500 | 14.7124 | 5.1470 | ||
Std |
|
1.9553 | 2.3292 | 7.9477 | 1.5420 | ||
IGD | Min |
|
3.067 | 3.0608 | 4.3261 | 3.0502 | |
Avg |
|
4.6015 | 5.2186 | 12.3180 | 4.5948 | ||
Std |
|
1.1447 | 1.3723 | 5.1917 | 0.8459 |
The average and standard deviation can only represent the problem-solving performance from the macroscopic perspective. It can be seen from the student’s t-tests whether there are significant differences between two algorithms. To show the statistical difference between the IMOMVO and other algorithms, the results are listed in Tables
T-test of IMOMVO vs. MOMVO, MOPSO, MOALO, and NSGA-II for small-sized problems.
Test problems | Performance |
p value of IMOMVO vs. | |||
---|---|---|---|---|---|
MOMVO | MOPSO | MOALO | NSGA-II | ||
Sproblem01-20 | SP | 0.518 | 0.258 |
|
0.901 |
GD |
|
|
|
|
|
IGD |
|
|
|
|
|
|
|||||
Sproblem02-20 | SP | 0.095 |
|
0.625 |
|
GD |
|
|
|
|
|
IGD |
|
|
|
|
|
|
|||||
Sproblem03-20 | SP | 0.977 | 0.464 | 0.193 | 0.349 |
GD |
|
|
|
|
|
IGD |
|
|
|
|
|
|
|||||
Sproblem04-20 | SP | 0.675 | 0.251 |
|
0.215 |
GD |
|
|
|
|
|
IGD |
|
|
|
|
|
|
|||||
Sproblem05-20 | SP |
|
0.405 | 0.118 | 0.360 |
GD |
|
|
|
|
|
IGD |
|
|
|
|
|
|
|||||
Sproblem06-20 | SP | 0.132 | 0.722 | 0.229 | 0.351 |
GD |
|
|
|
|
|
IGD |
|
|
|
|
Comparisons of IMOMVO and the four algorithms for large-sized problems.
Test problems | Performance measures | IMOMVO | MOMVO | MOPSO | MOALO | NSGA-II | |
---|---|---|---|---|---|---|---|
Lproblem-01-20 | SP | Min |
|
37.9386 | 39.2627 | 45.2234 | 32.5645 |
Avg |
|
63.0421 | 48.9852 | 83.7999 | 56.2081 | ||
Std | 14.6542 | 24.1826 |
|
24.5639 | 19.7613 | ||
GD | Min |
|
20.0731 | 40.3205 | 58.4695 | 14.7962 | |
Avg |
|
33.6796 | 49.1722 | 87.9470 | 28.7340 | ||
Std |
|
11.0479 | 8.2321 | 19.4371 | 15.4064 | ||
IGD | Min |
|
14.2001 | 32.0123 | 36.2466 | 21.3965 | |
Avg |
|
31.7573 | 49.3988 | 63.4794 | 27.3624 | ||
Std |
|
11.5299 | 11.0756 | 28.5335 | 4.5460 | ||
|
|||||||
Lproblem-02-20 | SP | Min | 39.2138 | 43.3812 | 41.9372 | 43.5818 |
|
Avg |
|
80.0144 | 58.5973 | 78.0675 | 48.6288 | ||
Std |
|
42.4928 | 11.7668 | 22.4166 | 16.4953 | ||
GD | Min |
|
13.7379 | 36.8747 | 31.5268 | 25.0612 | |
Avg |
|
53.0043 | 52.3202 | 52.7521 | 35.1891 | ||
Std |
|
26.9211 | 12.8599 | 11.8538 | 16.3095 | ||
IGD | Min |
|
12.3588 | 17.6584 | 31.939 | 22.0302 | |
Avg |
|
50.5360 | 52.2226 | 64.0970 | 33.5677 | ||
Std |
|
32.6129 | 33.5188 | 27.0935 | 19.0259 | ||
|
|||||||
Lproblem-03-20 | SP | Min | 12.9284 | 14.5336 |
|
17.2225 | 14.2715 |
Avg |
|
20.6562 | 17.6746 | 28.972 | 17.6740 | ||
Std | 2.6819 | 5.0773 | 5.2745 | 16.9026 |
|
||
GD | Min |
|
6.5092 |
|
20.6104 | 0.6099 | |
Avg |
|
16.8459 | 15.5075 | 32.7396 | 9.3958 | ||
Std |
|
7.7080 | 7.1737 | 10.1721 | 5.5497 | ||
IGD | Min |
|
10.1753 | 11.3452 | 26.5771 | 1.3472 | |
Avg |
|
17.9577 | 24.5695 | 39.4279 | 16.1063 | ||
Std |
|
7.8994 | 20.5780 | 909037 | 3.9406 | ||
|
|||||||
Lproblem-04-20 | SP | Min | 19.7206 |
|
19.5014 | 23.1202 | 22.6913 |
Avg |
|
31.3330 | 30.8081 | 35.5944 | 27.2626 | ||
Std |
|
12.3887 | 9.3875 | 5.5558 | 5.7137 | ||
GD | Min |
|
2.0338 | 6.719 | 23.9144 | 8.2262 | |
Avg |
|
13.6868 | 15.6351 | 34.7515 | 15.0124 | ||
Std |
|
10.2058 | 5.1480 | 5.7331 | 9.2964 | ||
IGD | Min |
|
5.8252 | 9.7547 | 20.8213 | 7.7169 | |
Avg |
|
12.9112 | 15.1598 | 33.2070 | 12.0886 | ||
Std |
|
5.3789 | 4.7880 | 12.1977 | 4.7048 | ||
|
|||||||
Lproblem-05-20 | SP | Min |
|
|
|
|
|
Avg | 19.7206 | 26.1458 | 24.2800 |
|
21.7567 | ||
Std | 31.3350 | 33.8072 |
|
31.3129 | 27.0370 | ||
GD | Min |
|
|
5.1640 |
|
|
|
Avg |
|
14.9427 | 19.6865 | 26.1215 | 16.9535 | ||
Std |
|
8.2743 | 20.7626 | 15.1955 | 8.0954 | ||
IGD | Min | 6 |
|
7 | 21.7401 | 11.3529 | |
Avg |
|
22.7388 | 25.2810 | 29.1370 | 27.4365 | ||
Std |
|
16.6252 | 14.5438 | 3.1175 | 17.4456 | ||
|
|||||||
Lproblem-06-20 | SP | Min |
|
27.5350 | 29.1025 | 42.2188 | 33.6225 |
Avg | 56.4040 | 57.7383 |
|
63.3400 | 52.4290 | ||
Std | 17.8382 | 19.7179 | 14.8274 | 19.3504 |
|
||
GD | Min |
|
10.4265 | 4.7540 | 18.6695 | 8.2554 | |
Avg |
|
21.3337 | 11.9000 | 27.2603 | 14.1198 | ||
Std |
|
10.5578 | 8.0866 | 8.5111 | 5.9599 | ||
IGD | Min |
|
17.6675 | 17.8134 | 18.5859 | 16.1167 | |
Avg |
|
27.938 | 21.5491 | 38.2586 | 23.4192 | ||
Std |
|
8.9999 | 4.1238 | 17.7398 | 7.8422 |
T-test of IMOMVO vs. MOMVO, MOPSO, MOALO, and NSGA-II for large-sized problems.
Test problems | Performance measures | p value of IMOMVO vs. | |||
---|---|---|---|---|---|
MOMVO | MOPSO | MOALO | NSGA-II | ||
Lproblem01-20 | SP | 0.091 |
|
0.256 | 0.734 |
GD |
|
|
|
|
|
IGD |
|
|
|
|
|
|
|||||
Lproblem02-20 | SP | 0.127 | 0.129 |
|
0.943 |
GD |
|
|
|
|
|
IGD |
|
|
|
|
|
|
|||||
Lproblem03-20 | SP | 0.072 | 0.799 |
|
0.690 |
GD |
|
|
|
|
|
IGD |
|
|
|
|
|
|
|||||
Lproblem04-20 | SP | 0.261 | 0.205 |
|
0.711 |
GD |
|
|
|
|
|
IGD |
|
|
|
|
|
|
|||||
Lproblem05-20 | SP | 0.665 | 0.725 | 0.928 | 0.878 |
GD |
|
|
|
|
|
IGD |
|
|
|
|
|
|
|||||
Lproblem06-20 | SP | 0.928 | 0.125 | 0.671 | 0.392 |
GD |
|
|
|
|
|
IGD |
|
|
|
|
Taking Sproblem-04-02 as an example to analyze the scheduling results after adding the turning on and off control strategy. There are 16 jobs, 8 machines, 1 re-entrance, and 6 stages in the problem. Meanwhile, the number of identical parallel machines in each stage is 1, 2, 1, 2, 1, and 1. The Pareto front obtained by each of the five algorithms is shown in Figure
Pareto fronts of Sproblem-04-02.
Gantt charts of a non-dominated solution with machines turning on and off control strategy.
Gantt charts of a non-dominated solution without machines turning on and off control strategy.
Taking six small-sized problems as examples, each problem runs 10 times independently, and the relative variation of the average makespan and idle energy consumption of each problem is shown in Figure
Relative change of makespan and idle energy consumption.
To sum up, it can be seen from the experimental results that the RHFSP considering machines turning on and off control strategy reduced the total energy consumption of the machines effectively without sacrificing the production efficiency. The useless energy consumption in the production process can be reduced by reasonably setting the turning on and off of the machines. The proposed IMOMVO is more effective than other algorithms for solving the multi-objective RHFSP.
In this paper, the RHFSP with the objectives of makespan, maximum tardiness and idle energy consumption was solved by the MOMVO algorithm. In order to establish a high-performance approach for this problem, the MOMVO algorithm was improved, including population initialization based on Latin hypercube sampling (LHS), individual position update based on Lévy flight, and chaotic local search operation based on logical self-mapping. The effectiveness of the improved operations was shown by numerical tests. Experimental results demonstrated the superiority of the proposed IMOMVO to the other algorithms. Specifically, the comparative results showed that IMOMVO was significantly better than other algorithms in terms of the convergence and diversity of the non-dominated solutions. Regarding the distribution of the non-dominated solutions, there was no significant difference in the five algorithms. At present, the research on the RHFSP is not profound enough. The RHFSP model considering the machine turning on and off control strategy in this paper can reduce the energy consumption of the machine effectively. It is mainly suitable for the scenarios in which the turning on and off is convenient and the turning on and off energy consumption is relatively low. In the future, the work could focus on the RHFSP considering time-of-use tariffs and the joint optimization considering maintenance.
The data used to support the findings of this study were supplied by Professor Cho HM. These datasets are cited at relevant places within the text as references [
The authors declare that they have no conflicts of interest.
The authors would like to thank Professor Cho HM for providing us the data sets. The research of this paper is made possible by the generous support from National Natural Science Foundation of China (71840003); Science and Technology Development Program of University of Shanghai for Science and Technology (2018KJFZ043); Ministry of Education “Cloud Number Integration Science and Education Innovation” Fund Project (2017A01109); and Henan Province Science and Technology Research Project (182102210113).