To solve the distributed hybrid flow shop scheduling problem (DHFS) in raw glass manufacturing systems, we investigated an improved hyperplane assisted evolutionary algorithm (IhpaEA). Two objectives are simultaneously considered, namely, the maximum completion time and the total energy consumptions. Firstly, each solution is encoded by a three-dimensional vector, i.e., factory assignment, scheduling, and machine assignment. Subsequently, an efficient initialization strategy embeds two heuristics are developed, which can increase the diversity of the population. Then, to improve the global search abilities, a Pareto-based crossover operator is designed to take more advantage of non-dominated solutions. Furthermore, a local search heuristic based on three parts encoding is embedded to enhance the searching performance. To enhance the local search abilities, the cooperation of the search operator is designed to obtain better non-dominated solutions. Finally, the experimental results demonstrate that the proposed algorithm is more efficient than the other three state-of-the-art algorithms. The results show that the Pareto optimal solution set obtained by the improved algorithm is superior to that of the traditional multi-objective algorithm in terms of diversity and convergence of the solution.

The hybrid flow shop scheduling problem (HFS) has been investigated and employed in lots of realistic industrial applications [

With the development of industries, more and more researches have focused on distributed scheduling problem, including the distributed flow shop scheduling problem (DFSSP) [

In realistic industry system, including the glass manufacturing system, the improvement of glass raw materials processing has been studied by many researches [

Recently, multiobjective optimization algorithms have been applied and considered in many domains [

Therefore, to solve DHFS in glass manufacturing systems, we propose an improved hyperplane assisted evolutionary algorithm (IhpaEA). The main contributions of this study are as follows: (1) each solution is represented with a three-dimensional vector, including the factory assignment, machine assignment, and operation scheduling; (2) an efficient initialization strategy is developed to increase the diversity of the population (3) an improved crossover operator is designed to enhance the global search abilities of the proposed algorithm; and (4) a cooperative search method is designed to enhance the local search abilities of the proposed algorithm deeply.

The structure of the rest paper is as follows. The problem descriptions are given in

The DHFS addressed in this study can be described as follows. There are

Each job should be released at time zero and be operated from the first stage to the next stage;

All machines are available at time zero and remain continuously available over the entire production horizon;

A job can be processed on exactly one machine at a time, and a machine can process exactly one job at a time;

At each stage, one job can select one suitable machine from the parallel machine;

There is unlimited buffer between stages;

All machines belonging to the same stage have similar processing abilities.

Index of the machines.

Index of the jobs.

Index of the factories.

Index of the stages

Number of jobs.

Number of machines.

Number of stages.

Number of factories.

Number of machines in

Job

Machine

i

Number of jobs that are processed on

r

Completion time of

Makespan, i.e., the maximum completion time.

Machine power of

Machine working time of

Total energy consumption.

A binary decision variable, which equals to 1 when job

A binary decision variable, which equals to 1 when job

The makespan (

A detailed illustration of the considered realistic DHFS is presented in a glass manufacturing casting system in

A common production flow is shared by different glass manufacturing systems, i.e., raw glass should experience preprocessing, melting, and forming processes in sequence, as shown in

Raw material preprocessing: Crush large raw materials (soda, quartz sand, feldspar, limestone, etc.) to dry raw materials which are wet, and then remove iron from raw materials to ensure glass quality.

Compound preparation.

Glass melting: In order to make the glass raw materials meet the forming requirements of uniform, bubble free and molten liquid glass, the glass raw materials need to be placed in the pool kiln or crucible kiln and heated at high temperature (1500–1600 degrees).

Glass forming: Liquid glass is processed into the required shape of the specific products.

Heat treatment: Through annealing, quenching and other processes to change the structural state of glass.

In this section, the proposed IhpaEA algorithm is presented to solve the considered DHFS problem. The first part describes the main framework of the proposed IhpaEA. Then the encoding, decoding, initialization, crossover, and other problem-specific heuristics are presented, respectively.

The main framework of the proposed IhpaEA algorithm is an enhanced inverted GA (genetic algorithm) indicator based hpaEA [

Algorithm 1 represents the framework of IhpaEA, where the first step is to initialize four parameters (1) an initial population P (line 1); (2) vectors V (line 2); (3) the number of prominent solutions (line 3) and (4) the evaluation functions (line 4); and the loop of IhpaEA (lines 5–12). Each generation performs three steps in the algorithm: (1) mating selection; (2) offspring population generation, and (3) environmental selection. The mating selection tries to assign more evolutionary results to the prominent solutions, and select better solutions. The set

Each solution is represented by a three-dimensional vector as follows.

The first dimensional vector is called scheduling vector, and the length of it equals to the total number of operations

The name of the second dimensional vector is called the machine assignment vector

The third dimensional vector is named as the factory assignment vector, and the length of factory assignment vector equals to the total number of jobs

Step 1: The assigned jobs are scheduled based on the sequence in the scheduling vector which is the first stage of each factory.

Step 2: After determining the factory, each job should select a suitable machine following the earliest available time rule.

Step 3: For the other stages, each job is scheduled as soon as possible after completing its previous operations. The first available suitable machine is also selected.

To solve the considered problem, a solution is encoded with two dispatching rules. The longest processing time at the first stage (LPTF) rule, and the shortest processing time at the first stage (SPTF) rule. Based on the non-increasing total processing times, LPTF generates a permutation. Meanwhile, based on the non-decreasing total processing times, SPTF produces a permutation by sorting the jobs.

To produce an effect initial population, the following technique is used. Suppose the population size is

The first

One individual is generated by LPTF. First, all the jobs’ processing time are calculated in each stage. Then, every job in every stage has a processing time and the summation of these time is called total processing time. Finally, the individual is generated by permuting the total processing time in non-increasing order.

SPTF generates the last individual. The first two steps are the same with LPTF. However, the third step is to permute the processing time in a non-decreasing order.

Based on the encoding representation, we proposed a novel crossover heuristic including two parts.

PTL crossover

The first type of crossover is PTL, which can be described as follows:

Randomly select two different elements from the first parent.

Copy the block of jobs which are cut by the two points from the first parent. And then move the block to the rightmost or leftmost part of the offspring.

Place the empty elements of jobs which are remaining from the second parent.

The process of PTL for generating offspring is depicted in

Two-cut PTL crossover
Two-cut PTL crossover
Two-cut PTL crossover
P1
5
2
3
P1
2
3
P1
4
2
3
P2
3
5
2
P2
5
2
3
P2
3
4
2
O1
3
5
2
O1
5
2
3
O1
3
4
2
O2
3
5
2
O2
5
2
3
O2
1
4
2

ISJOXI crossover

The second type of crossover operator is Improve Similar Job Order Crossover I or ISJOXI, with which the building blocks of jobs are directly copied to the offspring. In

The main steps of ISJOXI crossover are described in Algorithm 2.

Suppose

(1) Mutation for the factory assignment

Select two jobs

Select two jobs

Randomly insert a job which is removed from

Randomly insert a job which is removed from

(2) Mutation for the scheduling vector

Randomly choose two different jobs from

Randomly choose two different jobs from

Insert a job which is randomly selected into a random location in

Insert a job which is randomly selected into a random location in

(3) Mutation for the machine vector

The procedure of the mutation operator is as follows. Firstly, a position

The following multi-objective cooperation local search operator is embedded to achieve good diversity and convergence.

First, in each generation, the maximum completion time of solution

Second, the value

An example is provided in

Solutions | Objective | f = 1 | f = 2 | f = 3 | |||
---|---|---|---|---|---|---|---|

Makespan | 66 | 98 | 117 | 117 | 8936 | 76.3760 | |

TEC | 1265 | 1648 | 6023 | ||||

Makespan | 122 | 102 | 94 | 122 | 22144 | 181.5081 | |

TEC | 9744 | 11984 | 416 | ||||

Makespan | 290 | 64 | 71 | 290 | 26339 | 90.8241 | |

TEC | 24784 | 532 | 1023 | ||||

Makespan | 98 | 152 | 154 | 154 | 18502 | 120.1428 | |

TEC | 3728 | 5112 | 9662 |

The computational experiments to test the performance of IhpaEA algorithm is discussed in this section. The improved algorithm was implemented in the PlatEMO v3.0 on an Intel Core i7 3.4-GHz PC with 16 GB of memory. To test the performance of IhpaEA algorithm, 20 different scales of instances are generated according to the realistic flow shop.

All the compared algorithms are used to solving the considered problem, including the encoding, and decoding method, and the initialization procedure. The parameters are set according to their literatures. For each instance, the stop condition is set to 3000 iterations.

30 independent runs are used to test the performance of the proposed algorithm, the results of non-dominated solutions found by all the compared algorithms were collected for performance comparisons. The relative percentage increase (RPI) is used for the ANOVA comparison, which is formulated as follows:

20 large-scale test instances of DHFS problem are randomly generated to solve the DHFS problem and test the validity of the hpaEA algorithm based on the actual production data. For example, instance 1 can be denoted with 20 jobs, 2 stages, as well as 3 parallel machines in the first stages as well as 5 parallel machines in the second stages wherein the index of jobs are {20, 30, 50, 80, 100}, the parameter of machines are {2, 3, 4, 5}, the parameter of stages are {2, 3, 5, 10}, and the parameter of factories are {2, 3, 4, 5, 6}, respectively. The four algorithms ran 30 times.

Two types of IhpaEA algorithms are coded to test the initialization heuristic discussed in

Instances | HV | IGD | ||
---|---|---|---|---|

IhpaEA –NI | IhpaEA | IhpaEA –NI | IhpaEA | |

Instance 1 | 0.4931 | 14.2508 | ||

Instance 2 | 0.5198 | 3.3700 | ||

Instance 3 | 0.5002 | 90.9069 | ||

Instance 4 | 0.5113 | 38.6131 | ||

Instance 5 | 0.4809 | 8.5562 | ||

Instance 6 | 0.4642 | 8.8448 | ||

Instance 7 | 0.4964 | 2.8100 | ||

Instance 8 | 0.5209 | 844.9814 | ||

Instance 9 | 0.5029 | 11.5812 | ||

Instance 10 | 0.5250 | 176.0458 | ||

Instance 11 | 0.5018 | 34.5704 | ||

Instance 12 | 0.4981 | 398.4254 | ||

Instance 13 | 0.4963 | 13.5969 | ||

Instance 14 | 0.5081 | 166.6750 | ||

Instance 15 | 0.5037 | 6.7975 | ||

Instance 16 | 0.5034 | 263.8726 | ||

Instance 17 | 0.5463 | 125.8685 | ||

Instance 18 | 0.5029 | 205.5073 | ||

Instance 19 | 0.4954 | 112.0908 | ||

Instance 19 | 0.4926 | 169.1495 | ||

Instance 20 | 0.5229 | 3.9088 | ||

Mean | 0.502852 | 128.5916 |

It can be concluded from the comparison results that: (1) IhpaEA algorithm obtains 16 better results by considering the HV values of the IhpaEA-NI algorithm, and the slightly worse results for the other two instances; (2) for the IGD values, IhpaEA obtains 20 better results out of the given 20 different scale instances; and (3) from the average performance in HV and IGD given in the last line and the ANOVA results from

In order to test the performance of the crossover operators discussed in

From the comparison results given in

Instances | HV | IGD | ||
---|---|---|---|---|

IhpaEA -NC | IhpaEA | IhpaEA -NC | IhpaEA | |

Instance 1 | 0.5563 | 8.2956 | ||

Instance 2 | 0.4924 | 7.5307 | ||

Instance 3 | 0.5455 | 10.1593 | ||

Instance 4 | 0.4209 | 25.0598 | ||

Instance 5 | 0.4673 | 32.3624 | ||

Instance 6 | 0.5931 | 19.3699 | ||

Instance 7 | 0.6289 | 14.6482 | ||

Instance 8 | 0.5209 | 10.6421 | ||

Instance 9 | 0.6259 | 60.2584 | ||

Instance 10 | 0.5350 | 30.1638 | ||

Instance 11 | 0.5316 | 10.1558 | ||

Instance 12 | 0.4381 | 38.4254 | ||

Instance 13 | 0.4963 | 7.2531 | ||

Instance 14 | 0.5081 | 37.927 | ||

Instance 15 | 0.5437 | 5.1988 | ||

Instance 16 | 0.5345 | 11.2876 | ||

Instance 17 | 0.4715 | 14.2800 | ||

Instance 18 | 0.5295 | 12.3323 | ||

Instance 19 | 0.4485 | 7.0617 | ||

Instance 19 | 0.5265 | 2.1003 | ||

Instance 20 | 0.6565 | 15.3439 | ||

Mean | 0.5335 | 16.8632 |

Two different types of IhpaEA algorithms are coded to test the performance of the mutation operator discussed in

From the comparison results given in

Instances | HV | IGD | ||
---|---|---|---|---|

IhpaEA -NS | IhpaEA | IhpaEA -NS | IhpaEA | |

Instance 1 | 0.5258 | 12.0756 | ||

Instance 2 | 0.5801 | 54.039 | ||

Instance 3 | 0.4906 | 10.6876 | ||

Instance 4 | 0.6444 | 23.2886 | ||

Instance 5 | 0.4281 | 16.7376 | ||

Instance 6 | 0.5355 | 7.0794 | ||

Instance 7 | 0.5933 | 13.0640 | 28.934 | |

Instance 8 | 0.4768 | 4.0355 | ||

Instance 9 | 0.5997 | 15.6516 | ||

Instance 10 | 0.5249 | 20.4162 | ||

Instance 11 | 0.5704 | 4.5155 | ||

Instance 12 | 0.6273 | 18.0674 | ||

Instance 13 | 0.5708 | 11.6355 | ||

Instance 14 | 0.5927 | 10.4286 | ||

Instance 15 | 0.5975 | 9.0578 | ||

Instance 16 | 0.6717 | 8.0385 | ||

Instance 17 | 0.5728 | 7.1544 | ||

Instance 18 | 0.4824 | 2.9527 | ||

Instance 19 | 0.6619 | 0.5908 | 5.0999 | |

Instance 19 | 0.5709 | 2.6596 | ||

Instance 20 | 0.5088 | 6.3669 | ||

Mean | 0.5677 | 12.3486 |

To evaluate the performance of the local search heuristic discussed in

From the comparison results given in

Instances | HV | IGD | ||
---|---|---|---|---|

IhpaEA -NL | IhpaEA | IhpaEA -NL | IhpaEA | |

Instance 1 | 0.6162 | 14.2508 | ||

Instance 2 | 0.5895 | 3.3700 | ||

Instance 3 | 0.5879 | 90.9069 | ||

Instance 4 | 0.6068 | 38.6131 | ||

Instance 5 | 0.5673 | 8.5562 | ||

Instance 6 | 0.5932 | 8.8448 | ||

Instance 7 | 0.5489 | 2.8100 | ||

Instance 8 | 0.5526 | 844.9814 | ||

Instance 9 | 0.5766 | 11.5812 | ||

Instance 10 | 0.6107 | 176.0458 | ||

Instance 11 | 0.6015 | 34.5704 | ||

Instance 12 | 0.6049 | 398.4254 | ||

Instance 13 | 0.6523 | 13.5969 | ||

Instance 14 | 0.5873 | 166.6750 | ||

Instance 15 | 0.5671 | 6.7975 | ||

Instance 16 | 0.6040 | 263.8726 | ||

Instance 17 | 0.5722 | 125.8685 | ||

Instance 18 | 0.6153 | 205.5073 | ||

Instance 19 | 0.5585 | 112.0908 | ||

Instance 19 | 0.5798 | 169.1495 | ||

Instance 20 | 0.6544 | 3.9088 | ||

Mean | 0.0606 | 128.5916 |

Three algorithms are selected, namely, NSGAII [

Instance | HV | |||
---|---|---|---|---|

NSGAII | BiGE | GFMMOEA | IhpaEA | |

Instance 1 | 0.0653 | 0.0000 | 0.0653 | |

Instance 2 | 0.0432 | 0.0118 | 0.0440 | |

Instance 3 | 0.0345 | 0.0000 | 0.0387 | |

Instance 4 | 0.1209 | 11.0856 | 0.1288 | |

Instance 5 | 0.1673 | 0.0000 | 0.1795 | |

Instance 6 | 0.0593 | 0.0000 | 0.0594 | |

Instance 7 | 0.0289 | 0.0000 | 0.0340 | |

Instance 8 | 0.0275 | 7.6034 | 0.0255 | 0.0000 |

Instance 9 | 0.1565 | 0.0000 | 0.1648 | |

Instance 10 | 0.035 | 0.0306 | 0.0000 | |

Instance 11 | 0.0316 | 0.0000 | 0.0341 | |

Instance 12 | 0.0381 | 0.3611 | 0.0374 | |

Instance 13 | 0.1163 | 0.0000 | 0.1168 | |

Instance 14 | 0.0381 | 0.0358 | 0.0000 | |

Instance 15 | 0.0437 | 0.0118 | 0.0000 | |

Instance 16 | 0.0345 | 0.0000 | 0.0345 | |

Instance 17 | 0.0715 | 0.0681 | 0.9585 | |

Instance 18 | 0.0295 | 0.0303 | 0.0000 | |

Instance 19 | 0.0485 | 0.0000 | 0.0489 | |

Instance 20 | 0.0000 | 0.0258 | 0.0000 | |

Mean | 0.063269 | 2.4609 | 0.064 |

Instance | IGD | |||
---|---|---|---|---|

NSGAII | BiGE | GFMMOEA | IhpaEA | |

Instance 1 | 116.2298 | 158.8989 | 36.7109 | |

Instance 2 | 8.1941 | 34.8277 | 32.5033 | |

Instance 3 | 83.8725 | 22.5379 | 59.8702 | |

Instance 4 | 19.8748 | 94.4182 | 31.8385 | |

Instance 5 | 47.9635 | 179.0395 | 273.2809 | |

Instance 6 | 12.8869 | 170.6979 | 36.779 | |

Instance 7 | 19.4737 | 18.5112 | 277.2231 | |

Instance 8 | 75.6480 | 67.4810 | 28.3924 | |

Instance 9 | 73.9556 | 70.0808 | 265.1931 | |

Instance 10 | 64.1465 | 41.8955 | 16.3510 | |

Instance 11 | 53.4103 | 24.1171 | 81.9518 | |

Instance 12 | 19.5608 | 36.0141 | 112.583 | |

Instance 13 | 41.2228 | 33.5263 | 96.2000 | |

Instance 14 | 93.9044 | 78.7471 | 137.8545 | |

Instance 15 | 57.8876 | 50.251 | 98.1652 | |

Instance 16 | 13.8127 | 8.3406 | 56.0629 | |

Instance 17 | 80.9432 | 79.2301 | 41.8200 | |

Instance 18 | 83.1895 | 31.1598 | 18.7015 | |

Instance 19 | 23.9740 | 43.5960 | 157.5631 | |

Instance 20 | 37.4045 | 16.7215 | 29.9731 | |

Mean | 51.3776 | 63.0046 | 91.4573 |

A multifactor analysis of variance (ANOVA) is also performed to verify the difference from the above tables, based on three compared algorithms namely, NSGAII, GFMMOEA and BiGE.

From the above discussed comparison results, the efficient performance of the proposed IhpaEA algorithm has been tested. The main advantages of IhpaEA are as follows: (1) the proposed initialization heuristic, which can enhance the population diversity and quality; (2) the Pareto-based crossover operator enhance the global search abilities; (3) the mutation operator enhance the convergence of optimization process; and (4) a cooperation of search operators improve the local search abilities.

This paper studies a DHFS problem with makespan and total energy consumption. To solving the problem, a multiobjective optimization algorithm is proposed. The contributions are as follows: (1) an efficient encoding and decoding mechanism is embedded; (2) in the initialization phase of the algorithm, considering the constraints in the model, two heuristics are developed; (3) a Pareto-based crossover operator is designed; and (4) a cooperation of search operator is developed to further improve the quality of the solution and accelerate the convergence speed of the algorithm. To further illustrate the effectiveness of the proposed algorithm, IhpaEA is compared with three other multi-objective algorithms, including NSGA-II, BiGE, and GFMMOEA, The Pareto frontier is closer to the optimal solution than the other three algorithms.

We test the IhpaEA algorithm with different scales and compare several efficient algorithms with the IhpaEA algorithm. The robustness as well as efficiency is shown by experimental results. There are some works need to be focused as follows: (1) to improve the search capabilities, more local optimization methods or other efficient heuristics need to be introduced; (2) some useful dynamic and rescheduling strategies should be considered in flow shop scheduling problem; (3) more conflict objectives such as maximum workload and parallel batch workload need to be focused on.