Currently, energy conservation draws wide attention in industrial manufacturing systems. In recent years, many studies have aimed at saving energy consumption in the process of manufacturing and scheduling is regarded as an effective approach. This paper puts forwards a multi-objective stochastic parallel machine scheduling problem with the consideration of deteriorating and learning effects. In it, the real processing time of jobs is calculated by using their processing speed and normal processing time. To describe this problem in a mathematical way, a multi-objective stochastic programming model aiming at realizing makespan and energy consumption minimization is formulated. Furthermore, we develop a multi-objective multi-verse optimization combined with a stochastic simulation method to deal with it. In this approach, the multi-verse optimization is adopted to find favorable solutions from the huge solution domain, while the stochastic simulation method is employed to assess them. By conducting comparison experiments on test problems, it can be verified that the developed approach has better performance in coping with the considered problem, compared to two classic multi-objective evolutionary algorithms.

Currently, the energy crisis has become one of the most prominent and urgent environmental issues [

Previous work on parallel machine scheduling tends to set the processing time as a fixed constant, which obviously deviates from the actual situation. In real-life, the processing time of jobs always changes with their start time. For example, the steel industry wants to process the job under high temperatures. However, if the processing time of the unprocessed job is very late, the temperature will drop accordingly. Therefore, due to the later start time, the processing time of this job will be prolonged, i.e., the later the job starts, the longer the corresponding processing time is. In general, it is called the deteriorating effect. Kang et al. [

In practical scheduling problems, not only single objective needs to be considered, but also multi-objective needs to be solved optimally [

Through the analysis of existing studies, we find that an energy-aware PMSP with the consideration of deteriorating and learning effects and uncertainty is not be adequately considered by the existing studies. In an actual process, energy consumption optimization needs to be considered, and thus a multi-objective model should be addressed to find a balance between energy-saving and time-related criteria. Additionally, because the details of machines and jobs usually cannot be foreseen, jobs’ processing time is uncertain [

We propose a stochastic energy-aware parallel machine scheduling problem with deteriorating and learning effects. Its goal is to minimize the makespan and energy consumption. Furthermore, a stochastic multi-objective model is formulated to describe it mathematically.

We design a multi-objective multi-verse optimization incorporating a stochastic simulation approach to solve the proposed problem. In it, a multi-objective multi-verse optimization is applied to search for favorable solutions from the entire solution domain, and the stochastic simulation approach is employed to evaluate the performance of obtained solutions. Via conducting experiments on a set of test problems and comparing the designed approach with its peers, i.e., the nondominated sorting genetic algorithm II and multi-objective evolutionary algorithm based on decomposition, we demonstrate its effectiveness in solving the considered problem.

In this paper, the investigated problem is demonstrated as: There are

A viable scheduling scheme needs to obey the following restricted conditions: 1) Only one machine is chosen for each job to perform processing operations; 2) For each machine, only one job rather than multiple jobs can be processed on it at a time; 3) No machines can be interrupted. Note that the actual processing time of jobs can be calculated as:

According to the above notations and definitions, the considered problem can be established as follows:

Recently, a global optimization algorithm named Multi-verse optimizer (MVO) is initially put forward by Hatamlou et al. [

Due to its advantages, such as easy implementation, powerful exploration and exploitation abilities, MVO has been successfully employed to solve many optimization problems [

For solving the considered PMSP, it is necessary to make three decisions as: job assignment among machines, job sequence on machines and job processing speed. In this work, we use a double-chain structure to indicate a solution [

The decode rules are used to transform a solution into a feasible scheme as: Each job is allocated to a machine having earliest time and its actual processing time is calculated as

In MOMVO, an initial population consists of 100 randomly viable solutions. Besides, to restore the obtained non-dominated solutions in the search process, an external archive is developed and a Pareto rule is employed to update it [

Notice that the objective function is expressed as an expected value, and we cannot directly evaluate it. Monte Carlo simulation is one of the most effective stochastic simulation methods. It has been effectively employed to deal with various stochastic problems [

In the proposed MOMVO, we randomly select an individual in the current population as a black hole. And a binary selection approach is regarded as wormhole, and it is employed to choose an individual as a white hole.

Select two parent individuals by using a binary selection method, i.e.,

Perform an order-based crossover approach to acquire child individuals

Generate two binary strings

The following mutation operations with a probability are performed on each newly generated offspring individual. Different ways of mutation operations are adopted for parts

For the solution algorithm designed to tackle multi-objective problems, finding a solution set with outstanding approximation and distribution is desired. Thus, the rank and crowding distance methods [

In addition, MOMVO uses the newly obtained population to update the external archive with maximum capacity constraints by employing a Pareto dominance rule [

To test MOMVO’s effectiveness in coping with the problem studied, we choose two classical and popular multi-objective optimization methods, i.e., Nondominated sorting genetic algorithm II (NSGA-II) [

To validate the performance of MOMVO in solving the problem under consideration, we randomly produce 30 test problems which are the combinations of

MOMVO, NSGA-II and MOEA/D have two identical parameters as: population size and mutation probability. For three algorithms, the population size is equal to 100. The values of mutation probability for them are equal to 0.3, 0.3 and 1.0, respectively. The maximum capacity of external archive is set as 100 and the number of fitness evaluation 100

Approximation and distribution are two key indicators for evaluating the multi-objective optimization algorithms’ performance. Therefore, this work employs C-metric by Deb et al. [

C-metric is employed to measure a dominated percentage of two sets acquired by two optimizers.

IGD-metric is adopted to assess both convergence and diversity of a solution set by measuring the average distance between an optimal solution set

In this part, MOMVO’s performance in tackling the considered problem is validated. MOMVO and its peers run 10 times for each problem, and the mean and variance values regarding C-metric and IGD-metric are calculated.

C(X, Y) | C(Y, X) | C(X, Z) | C(Z, X) | ||||||
---|---|---|---|---|---|---|---|---|---|

Mean | Variance | Mean | Variance | Mean | Variance | Mean | Variance | ||

2 | 20 | 0.6214 | 0.2100 | 0.1906 | 0.1208 | 0.0616 | 0.0093 | 0.8008 | 0.0444 |

40 | 0.7212 | 0.0900 | 0.1569 | 0.0479 | 0.3000 | 0.2233 | 0.6473 | 0.2000 | |

60 | 0.4277 | 0.1832 | 0.3737 | 0.0768 | 0.7235 | 0.1441 | 0.2436 | 0.1269 | |

80 | 0.6875 | 0.1740 | 0.1117 | 0.0405 | 0.9000 | 0.0989 | 0.1000 | 0.0989 | |

100 | 0.8143 | 0.1506 | 0.0583 | 0.0336 | 1.0000 | 0.0000 | 0.0000 | 0.0000 | |

4 | 20 | 0.8139 | 0.1049 | 0.1273 | 0.0743 | 0.2034 | 0.1718 | 0.5919 | 0.1314 |

40 | 0.7425 | 0.1613 | 0.2130 | 0.1248 | 0.8000 | 0.1067 | 0.1500 | 0.0336 | |

60 | 0.7167 | 0.0863 | 0.0581 | 0.0059 | 0.5958 | 0.2201 | 0.2401 | 0.1200 | |

80 | 0.4000 | 0.2489 | 0.4616 | 0.1648 | 0.8086 | 0.1145 | 0.0917 | 0.0586 | |

100 | 0.7000 | 0.2233 | 0.3000 | 0.2233 | 1.0000 | 0.0000 | 0.0000 | 0.0000 | |

6 | 20 | 0.4545 | 0.1405 | 0.3459 | 0.1832 | 0.1813 | 0.1443 | 0.6732 | 0.1414 |

40 | 0.6517 | 0.1773 | 0.1303 | 0.0139 | 1.0000 | 0.0000 | 0.0000 | 0.0000 | |

60 | 0.7000 | 0.1789 | 0.1142 | 0.0106 | 0.7000 | 0.2233 | 0.2886 | 0.2071 | |

80 | 0.7665 | 0.1578 | 0.2040 | 0.1586 | 0.8000 | 0.1067 | 0.0880 | 0.0396 | |

100 | 0.6000 | 0.2489 | 0.2376 | 0.1295 | 0.9000 | 0.0989 | 0.0938 | 0.0869 | |

8 | 20 | 0.6157 | 0.1306 | 0.2277 | 0.1238 | 0.6071 | 0.2405 | 0.3173 | 0.1724 |

40 | 0.6759 | 0.1435 | 0.0597 | 0.0217 | 1.0000 | 0.0000 | 0.0000 | 0.0000 | |

60 | 0.5435 | 0.2148 | 0.2791 | 0.1217 | 0.8000 | 0.1733 | 0.0290 | 0.0083 | |

80 | 0.4563 | 0.1598 | 0.2876 | 0.1473 | 0.6879 | 0.1741 | 0.1711 | 0.1143 | |

100 | 0.5066 | 0.2493 | 0.1158 | 0.0761 | 0.6763 | 0.2115 | 0.1348 | 0.0988 | |

10 | 20 | 0.9607 | 0.0015 | 0.0000 | 0.0000 | 0.8035 | 0.1192 | 0.1731 | 0.1265 |

40 | 0.5072 | 0.2022 | 0.3083 | 0.1670 | 0.9000 | 0.0989 | 0.0511 | 0.0258 | |

60 | 0.6085 | 0.1295 | 0.1000 | 0.0989 | 1.0000 | 0.0000 | 0.0000 | 0.0000 | |

80 | 0.6000 | 0.2489 | 0.3281 | 0.1914 | 0.8000 | 0.1733 | 0.0928 | 0.0702 | |

100 | 0.5592 | 0.2255 | 0.1859 | 0.1046 | 0.6000 | 0.2489 | 0.2275 | 0.1279 | |

Average | 0.6345 | 0.1696 | 0.1990 | 0.0984 | 0.7139 | 0.1241 | 0.2082 | 0.0813 |

MOMVO | NSGA-II | MOEA/D | |||||
---|---|---|---|---|---|---|---|

Mean | Variance | Mean | Variance | Mean | Variance | ||

2 | 20 | 0.1741 | 0.0032 | 0.2678 | 0.0140 | 0.1132 | 0.0014 |

40 | 0.3277 | 0.0125 | 0.4417 | 0.0335 | 0.4677 | 0.1499 | |

60 | 0.2492 | 0.0295 | 0.2840 | 0.0158 | 0.7633 | 0.2149 | |

80 | 0.2324 | 0.0344 | 0.2569 | 0.0349 | 0.9803 | 0.1094 | |

100 | 0.0865 | 0.0047 | 0.1627 | 0.0021 | 1.1800 | 0.0309 | |

4 | 20 | 0.2253 | 0.0035 | 0.3866 | 0.0092 | 0.1740 | 0.0079 |

40 | 0.2485 | 0.0205 | 0.4047 | 0.0390 | 0.7080 | 0.1492 | |

60 | 0.2998 | 0.0238 | 0.4787 | 0.0325 | 0.4954 | 0.0689 | |

80 | 0.4265 | 0.0983 | 0.4907 | 0.0559 | 0.9583 | 0.0556 | |

100 | 0.2892 | 0.0081 | 0.3759 | 0.0734 | 1.0325 | 0.0442 | |

6 | 20 | 0.3008 | 0.0117 | 0.3435 | 0.0223 | 0.2212 | 0.0280 |

40 | 0.2533 | 0.0298 | 0.4727 | 0.0447 | 0.8333 | 0.0444 | |

60 | 0.3773 | 0.0396 | 0.5236 | 0.0338 | 0.7398 | 0.0727 | |

80 | 0.3043 | 0.0461 | 0.5035 | 0.0996 | 0.7997 | 0.0544 | |

100 | 0.4409 | 0.0326 | 0.5640 | 0.0852 | 0.9696 | 0.0676 | |

8 | 20 | 0.3704 | 0.0551 | 0.5118 | 0.0112 | 0.4987 | 0.0612 |

40 | 0.3325 | 0.0452 | 0.6160 | 0.0783 | 0.8188 | 0.0489 | |

60 | 0.3851 | 0.0440 | 0.4236 | 0.0596 | 0.8806 | 0.0654 | |

80 | 0.3437 | 0.0449 | 0.4731 | 0.0462 | 0.7981 | 0.1081 | |

100 | 0.4782 | 0.0473 | 0.5670 | 0.0479 | 0.6998 | 0.0571 | |

10 | 20 | 0.1756 | 0.0045 | 0.4148 | 0.0157 | 0.3749 | 0.0898 |

40 | 0.2702 | 0.0471 | 0.3927 | 0.0647 | 0.8161 | 0.0843 | |

60 | 0.3886 | 0.0683 | 0.6103 | 0.0970 | 0.9527 | 0.0509 | |

80 | 0.9907 | 0.0401 | 0.8963 | 0.1609 | 1.2023 | 0.0211 | |

100 | 0.9636 | 0.1632 | 1.1385 | 0.0294 | 1.1636 | 0.0213 | |

Average | 0.3573 | 0.0383 | 0.4800 | 0.0482 | 0.7456 | 0.0683 |

To visually analyze the experiment results, we draw the boxplot graphs of some test problems of 10 runs in

This work addresses a stochastic multi-objective parallel machine scheduling problem with deteriorating and learning effects to reach makespan and energy consumption minimization. By fully considering the characteristics of the proposed problem, we design a multi-objective multi-verse optimization incorporating a stochastic simulation approach to deal with it. The experimental results show that it is an excellent optimization algorithm for solving our considered problem. The results can assist decision-makers in making effective decisions under stochastic environments when handling a parallel machine scheduling problem with multiple optimization objectives, deteriorating and learning effects. The studied problem can help the manufacturing industry improve production efficiency and decrease energy consumption, thereby protecting resources and reducing environmental pollution.

In future work, we plan: 1) To propose a rescheduling approach based on the formulated model; 2) To study an effective approach to solve parallel machine scheduling problems according to the formulated model.