Effective Heuristic Algorithms Solving the Jobshop Scheduling Problem with Release Dates
Abstract
:1. Introduction
- A JSS model with the TQCT criterion is established, where each job is available at its own release date. This scheduling model simulates the production environment in which jobs arrive to the system over time.
- An HDDE algorithm is presented to achieve high-quality schedules in a given time, where tri-point insertion in crossover operator and local search scheme with exchange neighbourhood enhance the quality of final solutions.
- An innovative heuristic algorithm combining SPT and dense scheduling rule is proposed, which provides dominant feasible solutions on large-scale instances.
- A preemptive single-machine-based lower bound is proposed to estimate the optimal schedule, which can server as an estimation of the optimal solution to evaluate the performance of approximate algorithms.
2. Literature Review
3. MIP Model
4. SPT-DS Heuristic and Lower Bound
4.1. SPT-DS Heuristic
4.2. Well-Designed Lower Bound
5. Effective HDDE Algorithm
5.1. Encoding and Decoding
5.2. Initialization
Procedure 1: Initialization | |
1 | Input: n,Λ,m //n is the number of jobs, m is the number of machines, Λ is the size of the population |
2 | Output: pop[Λ] [m × n]←Array to store initial population; |
3 | begin |
4 | j ← 1; |
5 | for (i from 0 to m × n − 1) do |
6 | for (x from i to i + m − 1) do |
7 | pop[0][x] ← j; |
8 | end for |
9 | i ← i + m; j ← j + 1; |
10 | end for |
11 | for (i from 1 to Λ) do |
12 | for (j from 0 to m × n − 1) do |
13 | P[j] ← pop[i − 1][j]; |
14 | end for |
15 | t ← rand(1,m × n); |
16 | f ← rand(1,m × n); |
17 | if (t ! = f) |
18 | temp ← P[t − 1]; |
19 | P[t − 1] ← P[f − 1]; |
20 | P[f − 1] ← temp; |
21 | else |
22 | i ← i − 1; |
23 | Continue; |
24 | end if |
25 | for (j from 0 to m × n − 1) do |
26 | pop[i][j]←P[j]; |
27 | end for |
28 | end for |
29 | return pop[Λ] [m × n]; |
30 | end |
5.3. Mutation and Crossover
Procedure 2: Mutation operator | |
1 | Input: pop[Λ] [m × n];//initial population |
2 | Output: |
3 | Begin |
4 | do |
5 | { |
6 | αx, βx ← random(1,Λ);//x = 1,2 |
7 | } while(αx, βx (x = 1,2) is pairwise different); |
8 | /Operation of operator ⊗ */ |
9 | ←An individual randomly selected from the population; |
10 | ←Another non-repeating individual randomly selected from the population; |
11 | for (j from 1 to m × n) do //n is the number of jobs and m is the number of machines |
12 | γ←random(0,1); |
13 | if (γ < Z) |
14 | = − ; |
15 | else |
16 | = 0; |
17 | end if |
18 | end for |
19 | /* Operation of operator ⊕ */ |
20 | for (j from 1 to m × n) do |
21 | ; |
22 | end for |
23 | return ; |
24 | end |
Procedure 3: Crossover operator | |
1 | Input: |
2 | Output: |
3 | Begin |
4 | for (j from 0 to m × n − 1) do //n is the number of jobs and m is the number of machines |
5 | r ← random (0,1); |
6 | if (r >= Y) do |
7 | ←Remove ∈ ; |
8 | end if |
9 | end for |
10 | ←; |
←Randomly split into three parts; | |
11 | Generate three random insertion points of ; |
12 | The divided three parts of is inserted into the three random insertion points of in sequence; |
13 | ←Remove duplicate operations; |
14 | return |
15 | End |
5.4. Hill-Climbing-Based Improvement Strategy
Procedure 4: Hill-climbing-based improvement strategy | |
1 | Input: θ, |
2 | Output: |
3 | Begin |
4 | q ← random(0,1); |
5 | If (q > θ) |
6 | for (i from 0 to m × n − 1) |
7 | better[i]←; //store better individual; |
18 | end for |
9 | for (z from 1 to 10) |
10 | init ← the value of the objective function corresponding to better individual. |
11 | for (y from 1 to 20) |
12 | Neighboor[y]←Randomly exchange the two elements of better to produce neighborhood individual. |
13 | obj[y]←the value of objective function of Neighboor[y]; |
14 | end for |
15 | best_nei←the best one from 20 neighborhood individuals; |
14 | best_obj←the objective value of best_nei; |
15 | if(best_obj < init) |
16 | for (i from 0 to m × n − 1) |
16 | better[i]←best_nei[i]; |
17 | end for |
18 | end if |
19 | end for |
20 | for (i from 0 to m × n − 1) |
21 | ←better[i] |
22 | end for |
23 | end if |
24 | return ; |
25 | End |
5.5. Selection
5.6. Framework of the HDDE Algorithm
Algorithm 1: the HDDE algorithm: | |
1 | Input: Parameter τmax, Z, Y, Λ, θ |
2 | Output: |
3 | Begin |
4 | /* Initialization phase */ |
5 | pop[Λ] [m × n]←Randomly generate the initial population; |
6 | Evaluate the initial population; |
7 | k←1; |
8 | ←Individual with the optimal objective value in the current population; |
9 | while (k ≤ τmax) do |
10 | for (h from 1 to Λ) do |
11 | /* Mutation phase */ |
12 | ←the mutant individual after the mutation operation; |
13 | /* Crossover phase */ |
14 | ←the variant individual after the crossover operation; |
15 | /* Improvement strategy phase */ |
18 | Enter the improvement strategy operator to update ; |
20 | /* Selection Phase */ |
21 | if (F() < F() // F(X) is the objective value corresponding to the individual X; |
22 | = ; |
23 | else |
24 | = ; |
25 | end if |
26 | end for |
27 | Update ; |
28 | k = k + 1; |
29 | end while |
30 | return ; |
31 | End |
6. Numerical Simulation Experiment
6.1. Performance of SPT-DS Heuristic
6.2. Improvement of the HDDE Algorithm
6.3. Comparison between HDDE and Other Optimization Algorithms
6.3.1. Comparison between HDDE and ACO
6.3.2. Comparison between HDDE and PSO
6.3.3. Comparison between HDDE and GA
6.4. Comparison under JSS Problem Benchmarks
7. Conclusions
Author Contributions
Funding
Conflicts of Interest
Appendix A
Level | Z | Y | Λ | τmax | θ |
---|---|---|---|---|---|
1 | 0.1 | 0.1 | 200 | 100 | 0.75 |
2 | 0.2 | 0.2 | 300 | 150 | 0.8 |
3 | 0.3 | 0.3 | 400 | 200 | 0.85 |
4 | 0.4 | 0.4 | 500 | 250 | 0.9 |
5 | 0.5 | 0.5 | 600 | 300 | 0.95 |
No. | Z | Y | Λ | τmax | θ | MIP |
---|---|---|---|---|---|---|
1 | 1 | 1 | 1 | 1 | 1 | 63.047 |
2 | 1 | 2 | 2 | 2 | 2 | 56.857 |
3 | 1 | 3 | 3 | 3 | 3 | 62.44 |
4 | 1 | 4 | 4 | 4 | 4 | 61.173 |
5 | 1 | 5 | 5 | 5 | 5 | 56.198 |
6 | 2 | 1 | 2 | 3 | 4 | 68.973 |
7 | 2 | 2 | 3 | 4 | 5 | 63.566 |
8 | 2 | 3 | 4 | 5 | 1 | 62.135 |
9 | 2 | 4 | 5 | 1 | 2 | 51.484 |
10 | 2 | 5 | 1 | 2 | 3 | 63.784 |
11 | 3 | 1 | 3 | 5 | 2 | 76.275 |
12 | 3 | 2 | 4 | 1 | 3 | 48.861 |
13 | 3 | 3 | 5 | 2 | 4 | 47.98 |
14 | 3 | 4 | 1 | 3 | 5 | 57.46 |
15 | 3 | 5 | 2 | 4 | 1 | 56.328 |
16 | 4 | 1 | 4 | 2 | 5 | 65.379 |
17 | 4 | 2 | 5 | 3 | 1 | 60.104 |
18 | 4 | 3 | 1 | 4 | 2 | 57.64 |
19 | 4 | 4 | 2 | 5 | 3 | 55.235 |
20 | 4 | 5 | 3 | 1 | 4 | 44.366 |
21 | 5 | 1 | 5 | 4 | 3 | 58.799 |
22 | 5 | 2 | 1 | 5 | 4 | 64.755 |
23 | 5 | 3 | 2 | 1 | 5 | 40.386 |
24 | 5 | 4 | 3 | 2 | 1 | 42.218 |
25 | 5 | 5 | 4 | 3 | 2 | 48.76 |
Appendix B
Level | α | β | ρ | Q | Λ | tmax |
---|---|---|---|---|---|---|
1 | 0.6 | 0.2 | 0.9 | 0.9 | 200 | 100 |
2 | 0.7 | 0.3 | 1 | 0.8 | 300 | 150 |
3 | 0.8 | 0.4 | 0.8 | 0.7 | 400 | 200 |
4 | 0.9 | 0.5 | 0.6 | 0.6 | 500 | 250 |
5 | 1 | 0.6 | 0.7 | 0.5 | 600 | 300 |
Level | α | β | ρ | Q | Λ | tmax | MIP |
---|---|---|---|---|---|---|---|
1 | 1 | 1 | 1 | 1 | 1 | 1 | 19.133 |
2 | 1 | 2 | 2 | 2 | 2 | 2 | 17.13 |
3 | 1 | 3 | 3 | 3 | 3 | 3 | 21.664 |
4 | 1 | 4 | 4 | 4 | 4 | 4 | 25.079 |
5 | 1 | 5 | 5 | 5 | 5 | 5 | 24.833 |
6 | 2 | 1 | 2 | 3 | 4 | 5 | 12.356 |
7 | 2 | 2 | 3 | 4 | 5 | 1 | 17.048 |
8 | 2 | 3 | 4 | 5 | 1 | 2 | 22.346 |
9 | 2 | 4 | 5 | 1 | 2 | 3 | 22.964 |
10 | 2 | 5 | 1 | 2 | 3 | 4 | 19.987 |
11 | 3 | 1 | 3 | 5 | 2 | 4 | 15.748 |
12 | 3 | 2 | 4 | 1 | 3 | 5 | 17.635 |
13 | 3 | 3 | 5 | 2 | 4 | 1 | 21.003 |
14 | 3 | 4 | 1 | 3 | 5 | 2 | 20.2 |
15 | 3 | 5 | 2 | 4 | 1 | 3 | 24.685 |
16 | 4 | 1 | 4 | 2 | 5 | 3 | 14.161 |
17 | 4 | 2 | 5 | 3 | 1 | 4 | 22.272 |
18 | 4 | 3 | 1 | 4 | 2 | 5 | 29.234 |
19 | 4 | 4 | 2 | 5 | 3 | 1 | 23.501 |
20 | 4 | 5 | 3 | 1 | 4 | 2 | 22.543 |
21 | 5 | 1 | 5 | 4 | 3 | 2 | 11.02 |
22 | 5 | 2 | 1 | 5 | 4 | 3 | 23.61 |
23 | 5 | 3 | 2 | 1 | 5 | 4 | 26.278 |
24 | 5 | 4 | 3 | 2 | 1 | 5 | 30.529 |
25 | 5 | 5 | 4 | 3 | 2 | 1 | 25.445 |
Appendix C
Level | ω | vmax | smax | c1 | c2 |
---|---|---|---|---|---|
1 | 0.9 | 5 | 6 | 3 | 1 |
2 | 1.0 | 4 | 5 | 2.5 | 2 |
3 | 1.1 | 3 | 4 | 2 | 3 |
4 | 1.2 | 2 | 3 | 1.5 | 2.5 |
5 | 1.3 | 1 | 2 | 1 | 1.5 |
NO. | ω | vmax | smax | c1 | c2 | MIP |
---|---|---|---|---|---|---|
1 | 1 | 1 | 1 | 1 | 1 | 75.467 |
2 | 1 | 2 | 2 | 2 | 2 | 80.659 |
3 | 1 | 3 | 3 | 3 | 3 | 87.806 |
4 | 1 | 4 | 4 | 4 | 4 | 44.144 |
5 | 1 | 5 | 5 | 5 | 5 | 43.162 |
6 | 2 | 1 | 2 | 3 | 4 | 68.973 |
7 | 2 | 2 | 3 | 4 | 5 | 64.049 |
8 | 2 | 3 | 4 | 5 | 1 | 46.286 |
9 | 2 | 4 | 5 | 1 | 2 | 68.278 |
10 | 2 | 5 | 1 | 2 | 3 | 74.803 |
11 | 3 | 1 | 3 | 5 | 2 | 60.749 |
12 | 3 | 2 | 4 | 1 | 3 | 88.783 |
13 | 3 | 3 | 5 | 2 | 4 | 58.687 |
14 | 3 | 4 | 1 | 3 | 5 | 44.56 |
15 | 3 | 5 | 2 | 4 | 1 | 65.843 |
16 | 4 | 1 | 4 | 2 | 5 | 40.564 |
17 | 4 | 2 | 5 | 3 | 1 | 60.727 |
18 | 4 | 3 | 1 | 4 | 2 | 58.335 |
19 | 4 | 4 | 2 | 5 | 3 | 48.848 |
20 | 4 | 5 | 3 | 1 | 4 | 48.898 |
21 | 5 | 1 | 5 | 4 | 3 | 58.814 |
22 | 5 | 2 | 1 | 5 | 4 | 44.572 |
23 | 5 | 3 | 2 | 1 | 5 | 36.048 |
24 | 5 | 4 | 3 | 2 | 1 | 57.198 |
25 | 5 | 5 | 4 | 3 | 2 | 49.239 |
Appendix D
Level | Z1 | Y1 | Λ | τmax |
---|---|---|---|---|
1 | 0.1 | 0.1 | 200 | 100 |
2 | 0.2 | 0.2 | 300 | 150 |
3 | 0.3 | 0.3 | 400 | 200 |
4 | 0.4 | 0.4 | 500 | 250 |
5 | 0.5 | 0.5 | 600 | 300 |
NO. | Z1 | Y1 | Λ | τmax | MIP |
---|---|---|---|---|---|
1 | 1 | 1 | 1 | 1 | 15.438 |
2 | 1 | 2 | 2 | 2 | 19.518 |
3 | 1 | 3 | 3 | 3 | 28.87 |
4 | 1 | 4 | 4 | 4 | 24.818 |
5 | 1 | 5 | 5 | 5 | 29.768 |
6 | 2 | 1 | 2 | 3 | 22.523 |
7 | 2 | 2 | 3 | 4 | 28.46 |
8 | 2 | 3 | 4 | 5 | 28.97 |
9 | 2 | 4 | 5 | 1 | 20.75 |
10 | 2 | 5 | 1 | 2 | 26.568 |
11 | 3 | 1 | 3 | 5 | 32.46 |
12 | 3 | 2 | 4 | 1 | 24.465 |
13 | 3 | 3 | 5 | 2 | 22.413 |
14 | 3 | 4 | 1 | 3 | 29.891 |
15 | 3 | 5 | 2 | 4 | 35.587 |
16 | 4 | 1 | 4 | 2 | 22.361 |
17 | 4 | 2 | 5 | 3 | 27.872 |
18 | 4 | 3 | 1 | 4 | 36.13 |
19 | 4 | 4 | 2 | 5 | 30.854 |
20 | 4 | 5 | 3 | 1 | 21.569 |
21 | 5 | 1 | 5 | 4 | 24.833 |
22 | 5 | 2 | 1 | 5 | 38.551 |
23 | 5 | 3 | 2 | 1 | 27.793 |
24 | 5 | 4 | 3 | 2 | 33.15 |
25 | 5 | 5 | 4 | 3 | 31.272 |
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Variables | Meanings | |
---|---|---|
n | total number of jobs; | |
m | total number of machines; | |
rj | release date of job j; | |
pi,j | processing time of job j on machine i; | |
Ci,j | completion time of job j on machine i; | |
Cj | maximum completion time of job j; | |
aj,i,l | = | 1, if job j is processed on machine i before machine l; 0, otherwise. |
zi,j,k | = | 1, if job k precedes job j immediately on machine i; 0, otherwise. |
yi,j | = | 1, if the first operation of job j is processed on machine i; 0, otherwise. |
M | a large positive number. |
Jobs | Processing Routes | Processing Times | Release Dates |
---|---|---|---|
J1 | M2, M3, M1 | 4, 3, 2 | 1 |
J2 | M3, M2, M1 | 6, 2, 2 | 0 |
J3 | M1, M3, M2 | 2, 5, 1 | 2 |
Individual | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 |
---|---|---|---|---|---|---|---|---|---|
Xα1 | 2 | 1 | 3 | 1 | 2 | 1 | 3 | 2 | 3 |
Xβ1 | 2 | 3 | 1 | 3 | 2 | 3 | 1 | 1 | 2 |
Xα1 − Xβ1 | 0 | −2 | 2 | −2 | 0 | −2 | 2 | 1 | 1 |
rand | 0.20 | 0.37 | 0.06 | 0.18 | 0.26 | 0.68 | 0.86 | 0.50 | 0.15 |
0 | 0 | 2 | −2 | 0 | 0 | 0 | 0 | 1 |
Individual | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 |
---|---|---|---|---|---|---|---|---|---|
Xα2 | 3 | 2 | 3 | 1 | 2 | 1 | 3 | 1 | 2 |
Xβ2 | 2 | 1 | 1 | 3 | 2 | 3 | 1 | 3 | 2 |
Xα2 − Xβ2 | 1 | 1 | 2 | −2 | 0 | −2 | 2 | −2 | 0 |
rand | 0.56 | 0.02 | 0.38 | 0.26 | 0.66 | 0.18 | 0.59 | 0.89 | 0.15 |
0 | 1 | 0 | −2 | 0 | −2 | 0 | 0 | 0 |
Individual | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 |
---|---|---|---|---|---|---|---|---|---|
2 | 3 | 1 | 2 | 1 | 3 | 3 | 1 | 2 | |
0 | 0 | 2 | −2 | 0 | 0 | 0 | 0 | 1 | |
0 | 1 | 0 | −2 | 0 | −2 | 0 | 0 | 0 | |
2 | 4 | 3 | −2 | 1 | 1 | 3 | 1 | 2 | |
2 | 1 | 3 | 1 | 1 | 1 | 3 | 1 | 2 | |
rand | 0.56 | 0.28 | 0.32 | 0.48 | 0.66 | 0.18 | 0.59 | 0.89 | 0.15 |
1 | 3 | 1 | 1 | 2 |
n/m | m = 3 | m = 5 | m = 8 |
---|---|---|---|
n = 100 | 0.7808 | 1.0644 | 1.4948 |
n = 300 | 0.7349 | 1.2143 | 1.4221 |
n = 500 | 0.7217 | 1.1150 | 1.4003 |
m × n | SDDE | HDDE | ||||||
---|---|---|---|---|---|---|---|---|
ARDP | MaxRDP | MinRDP | SD | ARDP | MaxRDP | MinRDP | SD | |
3 × 50 | 26.11 | 55.36 | 16.42 | 14.98 | 4.81 | 48.13 | 0 | 14.43 |
3 × 100 | 14.64 | 20.39 | 9.35 | 3.91 | 0 | 0 | 0 | 0 |
3 × 150 | 7.38 | 11.75 | 0 | 3.73 | 0.40 | 4.04 | 0 | 1.21 |
5 × 50 | 13.82 | 23.55 | 6.59 | 4.34 | 0 | 0 | 0 | 0 |
5 × 100 | 7.81 | 15.51 | 2.36 | 4.03 | 0 | 0 | 0 | 0 |
5 × 150 | 5.82 | 12.29 | 0 | 3.44 | 0.06 | 0.55 | 0 | 0.17 |
8 × 50 | 4.78 | 10.8 | 0 | 3.66 | 0.05 | 0.48 | 0 | 0.14 |
8 × 100 | 7.32 | 13.5 | 3.53 | 2.91 | 0 | 0 | 0 | 0 |
8 × 150 | 3.95 | 11.38 | 0 | 3.34 | 0.09 | 0.91 | 0 | 0.27 |
m × n | ACO | HDDE | ||||||
---|---|---|---|---|---|---|---|---|
ARDP | MaxRDP | MinRDP | SD | ARDP | MaxRDP | MinRDP | SD | |
3 × 50 | 27.54 | 50.31 | 14.36 | 12.88 | 0 | 0 | 0 | 0 |
3 × 100 | 9.07 | 22.34 | 0 | 7.48 | 0.81 | 5.70 | 0 | 1.78 |
3 × 150 | 2.67 | 10.76 | 0 | 3.75 | 1.89 | 12.43 | 0 | 3.72 |
5 × 50 | 13.62 | 26.73 | 0 | 9.69 | 0.43 | 4.32 | 0 | 1.29 |
5 × 100 | 41.12 | 55.66 | 33.50 | 6.00 | 0 | 0 | 0 | 0 |
5 × 150 | 37.16 | 43.71 | 32.24 | 3.35 | 0 | 0 | 0 | 0 |
8 × 50 | 5.76 | 16.99 | 0 | 6.53 | 0.28 | 2.84 | 0 | 0.85 |
8 × 100 | 34.16 | 39.27 | 28.12 | 3.51 | 0 | 0 | 0 | 0 |
8 × 150 | 30.88 | 35.04 | 25.83 | 3.12 | 0 | 0 | 0 | 0 |
m × n | PSO | HDDE | ||||||
---|---|---|---|---|---|---|---|---|
ARDP | MaxRDP | MinRDP | SD | ARDP | MaxRDP | MinRDP | SD | |
3 × 50 | 45.96 | 79.70 | 19.87 | 19.89 | 0 | 0 | 0 | 0 |
3 × 100 | 44.23 | 65.99 | 23.06 | 14.33 | 0 | 0 | 0 | 0 |
3 × 150 | 30.60 | 48.25 | 13.48 | 9.878 | 0 | 0 | 0 | 0 |
5 × 50 | 62.99 | 83.20 | 44.53 | 12.414 | 0 | 0 | 0 | 0 |
5 × 100 | 48.66 | 55.63 | 40.24 | 5.29 | 0 | 0 | 0 | 0 |
5 × 150 | 42.89 | 48.38 | 37.98 | 3.353 | 0 | 0 | 0 | 0 |
8 × 50 | 51.02 | 63.66 | 41.41 | 7.13 | 0 | 0 | 0 | 0 |
8 × 100 | 41.63 | 53.69 | 33.94 | 5.52 | 0 | 0 | 0 | 0 |
8 × 150 | 37.70 | 43.38 | 32.99 | 3.34 | 0 | 0 | 0 | 0 |
m × n | GA | HDDE | ||||||
---|---|---|---|---|---|---|---|---|
ARDP | MaxRDP | MinRDP | SD | ARDP | MaxRDP | MinRDP | SD | |
3 × 50 | 29.09 | 62.36 | 9.19 | 16.67 | 0 | 0 | 0 | 0 |
3 × 100 | 20.07 | 46.35 | 0.58 | 12.69 | 0 | 0 | 0 | 0 |
3 × 150 | 10.34 | 30.00 | 0 | 9.04 | 1.42 | 8.18 | 0 | 3.04 |
5 × 50 | 36.05 | 59.27 | 18.34 | 12.91 | 0 | 0 | 0 | 0 |
5 × 100 | 18.02 | 33.99 | 0.99 | 10.70 | 0 | 0 | 0 | 0 |
5 × 150 | 14.92 | 25.37 | 4.26 | 6.73 | 0 | 0 | 0 | 0 |
8 × 50 | 23.23 | 32.33 | 16.81 | 5.70 | 0 | 0 | 0 | 0 |
8 × 100 | 30.18 | 51.12 | 13.14 | 11.54 | 0 | 0 | 0 | 0 |
8 × 150 | 32.02 | 46 | 16.94 | 9.24 | 0 | 0 | 0 | 0 |
m × n | HDDE | ACO | PSO | GA | gap1 | gap2 | gap3 |
---|---|---|---|---|---|---|---|
15 × 50 | 5.34 × 108 | 7.88 × 108 | 9.18 × 108 | 1.28 × 109 | 47.68 | 71.92 | 139.70 |
5.44 × 108 | 7.87 × 108 | 9.35 × 108 | 1.27 × 109 | 44.52 | 71.84 | 132.34 | |
5.12 × 108 | 7.54 × 108 | 8.34 × 108 | 1.45 × 109 | 47.17 | 62.95 | 182.17 | |
5.15 × 108 | 7.99 × 108 | 8.90 × 108 | 1.15 × 109 | 55.23 | 73.03 | 123.65 | |
5.15 × 108 | 7.92 × 108 | 8.84 × 108 | 1.39 × 109 | 53.94 | 71.84 | 170.79 | |
5.54 × 108 | 8.36 × 108 | 9.35 × 108 | 1.17 × 109 | 50.98 | 68.90 | 111.56 | |
5.32 × 108 | 8.22 × 108 | 8.91 × 108 | 1.51 × 109 | 54.50 | 67.38 | 183.35 | |
5.71 × 108 | 8.40 × 108 | 9.94 × 108 | 1.38 × 109 | 47.06 | 74.12 | 142.28 | |
5.16 × 108 | 7.43 × 108 | 8.86 × 108 | 1.28 × 109 | 44.02 | 71.71 | 147.24 | |
5.50 × 108 | 8.13 × 108 | 8.67 × 108 | 1.48 × 109 | 47.87 | 57.99 | 168.68 | |
20 × 50 | 7.56 × 108 | 1.05 × 109 | 1.18 × 109 | 2.01 × 109 | 39.26 | 55.67 | 166.45 |
7.81 × 108 | 1.16 × 109 | 1.24 × 109 | 2.31 × 109 | 47.97 | 59.22 | 195.26 | |
7.08 × 108 | 1.02 × 109 | 1.13 × 109 | 1.79 × 109 | 43.55 | 59.82 | 152.24 | |
6.49 × 108 | 9.59 × 109 | 1.10 × 109 | 1.77 × 109 | 47.66 | 69.11 | 172.34 | |
7.01 × 108 | 1.03 × 109 | 1.25 × 109 | 2.04 × 109 | 46.63 | 78.74 | 190.4 | |
7.16 × 108 | 1.08 × 109 | 1.23 × 109 | 2.07 × 109 | 50.28 | 71.38 | 189.73 | |
6.92 × 108 | 1.04 × 109 | 1.20 × 109 | 2.20 × 109 | 50.31 | 72.82 | 217.56 | |
6.91 × 108 | 9.99 × 108 | 1.17 × 109 | 1.82 × 109 | 44.70 | 69.10 | 162.85 | |
7.21 × 108 | 1.11 × 109 | 1.19 × 109 | 2.29 × 109 | 53.57 | 65.19 | 218.07 | |
7.57 × 108 | 1.08 × 109 | 1.20 × 109 | 2.08 × 109 | 42.33 | 58.00 | 174.95 | |
20 × 100 | 4.68 × 109 | 6.86 × 109 | 7.54 × 109 | 1.20 × 1010 | 46.60 | 61.20 | 156.75 |
4.31 × 109 | 6.14 × 109 | 6.89 × 109 | 1.55 × 1010 | 42.60 | 59.89 | 260.32 | |
4.67 × 109 | 6.62 × 109 | 7.35 × 109 | 1.32 × 1010 | 41.79 | 57.41 | 182.91 | |
4.49 × 109 | 6.44 × 109 | 6.99 × 109 | 1.57 × 1010 | 43.37 | 55.8 | 250.04 | |
4.58 × 109 | 6.56 × 109 | 7.19 × 109 | 1.53 × 1010 | 43.01 | 56.74 | 233.26 | |
4.51 × 109 | 6.41 × 109 | 6.82 × 109 | 1.60 × 1010 | 42.13 | 51.11 | 254.59 | |
4.71 × 109 | 6.20 × 109 | 6.94 × 109 | 1.66 × 1010 | 31.57 | 47.45 | 252.63 | |
4.44 × 109 | 6.40 × 109 | 6.86 × 109 | 1.64 × 1010 | 43.91 | 54.35 | 268.87 | |
4.62 × 109 | 6.16 × 109 | 6.88 × 109 | 1.55 × 1010 | 33.14 | 48.78 | 235.07 | |
4.42 × 109 | 6.23 × 109 | 6.97 × 109 | 1.40 × 1010 | 41.19 | 57.91 | 217.75 |
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Ren, T.; Zhang, Y.; Cheng, S.-R.; Wu, C.-C.; Zhang, M.; Chang, B.-y.; Wang, X.-y.; Zhao, P. Effective Heuristic Algorithms Solving the Jobshop Scheduling Problem with Release Dates. Mathematics 2020, 8, 1221. https://doi.org/10.3390/math8081221
Ren T, Zhang Y, Cheng S-R, Wu C-C, Zhang M, Chang B-y, Wang X-y, Zhao P. Effective Heuristic Algorithms Solving the Jobshop Scheduling Problem with Release Dates. Mathematics. 2020; 8(8):1221. https://doi.org/10.3390/math8081221
Chicago/Turabian StyleRen, Tao, Yan Zhang, Shuenn-Ren Cheng, Chin-Chia Wu, Meng Zhang, Bo-yu Chang, Xin-yue Wang, and Peng Zhao. 2020. "Effective Heuristic Algorithms Solving the Jobshop Scheduling Problem with Release Dates" Mathematics 8, no. 8: 1221. https://doi.org/10.3390/math8081221