Introduction
Related work
Distributed permutation flow shop scheduling
Distributed heterogeneous flow shop scheduling
Research gap and discussion
Problem statement and modeling
Problem statement
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All the factories start processing operations at time zero. Moreover, all jobs and machines are available at time zero in each factory.
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Each machine can only process one operation at the same time. Meanwhile, interruption is not considered for each operation.
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The processing times and energy consumption are certain values.
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Every job can only select one factory and each job can only be processed on one machine at a moment.
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The processing time of each operation on different machine is different in each factory. Meanwhile the machine speed \(v_{f,i,k}\in [1,5]\).
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Transportation time, setup time, and energy consumption are not considered. Moreover, machine breakdown and dynamic events would not happen.
\(p_{f,i,k}^{o}\) | \(v_{f,i,k}\) | |||||||
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\(F_1\) | \(F_2\) | \(F_1\) | \(F_2\) | |||||
\(M_{1,1}\) | \(M_{1,2}\) | \(M_{2,1}\) | \(M_{2,2}\) | \(M_{1,1}\) | \(M_{1,2}\) | \(M_{2,1}\) | \(M_{2,2}\) | |
\(J_1\) | 10 | 5 | 8 | 4 | 5 | 5 | ||
\(J_2\) | 6 | 4 | 7 | 5 | 3 | 2 | ||
\(J_3\) | 2 | 3 | 4 | 2 | 2 | 1 | ||
\(J_4\) | 4 | 5 | 3 | 6 | 3 | 2 | ||
\(J_5\) | 2 | 1 | 4 | 2 | 1 | 1 | ||
\(J_6\) | 5 | 6 | 3 | 4 | 2 | 2 | ||
\(J_7\) | 7 | 5 | 8 | 6 | 4 | 2 | ||
\(J_8\) | 3 | 3 | 6 | 4 | 2 | 2 |
MILP model for DHPFSP-FMS
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\(i,i'\): indices of all jobs, \(i\in \{1,\ldots ,n\}\);
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\(k,k'\): indices of machines, \(k\in \{1,\ldots ,m\}\);
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\(v,v'\): indices of speed, \(v\in \{1,\ldots ,n_v\}\);
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f: indices of factories, \(f\in \{1,\ldots ,n_f\}\);
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t: job position of machine \(M_k\) in factory f, \(l\in \{1,\ldots ,n_t\}\);
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n: the number of all jobs;
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\(n_f\): the number of factories;
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m: the number of all machines;
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\(n_v\): the max level of machine speed;
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\(n_t\): the number of all positions;
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I: set for jobs and \(I=\{1,2,\ldots ,n\}\);
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F: set of factories and \(F=\{1,2,\ldots ,n_f\}\);
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M: set of machines and \(M=\{1,2,\ldots ,m\}\);
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\(T_{f,k}\): set of job positions on machines \(M_K\) in factory f and \(T_{f,k}=\{1,2,\ldots ,n_t\}\);
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\(T_{f,k}^{'}\): set of top \(n_t-1\) positions on machines \(M_K\) in factory f and \(T_{f,k}=\{1,2,\ldots ,n_t-1\}\);
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V: set of speeds and \(V=\{1,2,\ldots ,n_v\}\);
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\(p_{f,i,k}^o\): The original processing time for operation \(O_{i,k}\) job \(I_i\) processed by machine \(M_k\) in factory f;
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\(W_{I}\) the idle power when machine stands by;
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\(W_{O}\) processing power when machine processing operation;
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L: a large integer for keeping the consistency of the inequality;
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\(O_{i,k}\): The \(k_{th}\) operation of job \(I_i\) on machine \(M_k\);
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\(v_{f,i,k}\): The speed selection of operation \(O_{i,k}\) on machine \(M_k\) in factory f;
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\(p_{f,i,k}^{r}\): The real processing time of operation \(O_{i,k}\) on machine \(M_k\) in factory f;
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\(S_{f,i,k}\): the starting time of operation \(O_{i,j}\) in factory f;
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\(F_{f,i,k}\): the finishing time of operation \(O_{i,j}\) in factory f;
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\(B_{f,k,t}\): the beginning time of machine \(M_k\) at position t in factory f;
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\(C_{f,k,t}\): the completion time of machine \(M_k\) at position t in factory f;
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\(E_P\): total processing energy consumption;
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\(E_I\): total idle energy consumption;
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\(E_{f,k,t}^M\): idle energy consumption of machine \(M_k\) at position t in factory f;
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\({\text {TEC}}\): total energy consumption;
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\(C_{\max }\): the makespan of a schedule;
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\({\textbf{X}}_{f,i,k,v,t}\): The value is equals to 1, if operation \(O_{i,k}\) is processed on machine \(M_k\) with speed \(V_v\) at position t in factory f; otherwise it is set to 0;
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\({\textbf{Y}}_{i,f}\): The value is equals to 1, if job \(I_i\) is assigned to factory f; otherwise is set to 0;
Problem features
Our approach: BRCE
Framework of BRCE
Encoding and decoding
Initialization
Global search
Knowledge-driven local search
Energy-saving strategy
Experimental results
Instances and metrics
MOEAs | HV | GD | Spread | |||
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Rank | p value | Rank | p value | Rank | p value | |
BRCE-C | 4.455 | 1.24E-13 | 4.092 | 6.54E-10 | 3.409 | 2.51E-13 |
BRCE | 4.092 | 3.682 | 3.091 | |||
BRCE + E | 2.955 | 2.091 | 2.727 | |||
BRCE + EV | 2.500 | 1.409 | 4.773 | |||
BRCE + EVH | 1.000 | 3.727 | 1.000 |
Parameters calibration
MOEAs | HV | GD | Spread | |||
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Rank | p value | Rank | p value | Rank | p value | |
NSGA-II | 2.318 | 7.19E−15 | 1.909 | 1.77E−14 | 2.636 | 4.48E−13 |
MOEA/D | 3.773 | 4.546 | 2.909 | |||
PMMA | 4.773 | 4.318 | 4.000 | |||
KCA | 3.136 | 2.864 | 4.455 | |||
BRCE | 1.000 | 1.364 | 1.000 |