1 Introduction
1.1 Contribution
1.2 Paper Outline
2 Problem Formulation
2.1 Objectives and Constraints
2.2 Example
3 Related Literature
3.1 Vehicle Routing Problem with Time Windows
3.2 Parallel Evolutionary Algorithms
4 Parallel Memetic Algorithm
4.1 Algorithm Outline
4.2 Initial Population
4.3 Selection
4.4 Crossover
4.5 Repair, Education, and Mutation
Move | Removed edges | Added edges |
---|---|---|
2-opt* |
\(\langle v_{i},v_{(i+1)}\rangle \), \(\langle v_{j},v_{(j+1)}\rangle \)
|
\(\langle v_{i},v_{(j+1)}\rangle \), \(\langle v_{j},v_{(i+1)}\rangle \)
|
Out-exchange |
\(\langle v_{(i-1)},v_{i}\rangle \), \(\langle v_{i},v_{(i+1)}\rangle \), |
\(\langle v_{(i-1)},v_{j}\rangle \), \(\langle v_{j},v_{(i+1)}\rangle \), |
\(\langle v_{(j-1)},v_{j}\rangle \), \(\langle v_{j},v_{(j+1)}\rangle \)
|
\(\langle v_{(j-1)},v_{i}\rangle \), \(\langle v_{i},v_{(j+1)}\rangle \)
| |
Out-relocate |
\(\langle v_{(i-1)},v_{i}\rangle \), \(\langle v_{i},v_{(i+1)}\rangle \), |
\(\langle v_{(j-1)},v_{i}\rangle \), \(\langle v_{i},v_{j}\rangle \), |
\(\langle v_{(j-1)},v_{j}\rangle \)
|
\(\langle v_{(i-1)},v_{(i+1)}\rangle \)
| |
In-exchange |
\(\langle v_{(i-2)},v_{(i-1)}\rangle \), \(\langle v_{(i-1)},v_{i}\rangle \), |
\(\langle v_{(i-2)},v_{(i+1)}\rangle \), \(\langle v_{(i+1)},v_{i}\rangle \), |
\(\langle v_{i},v_{{(i+1)}}\rangle \), \(\langle v_{(i+1)},v_{{(i+2)}}\rangle \)
|
\(\langle v_{i},v_{(i-1)}\rangle \), \(\langle v_{(i-1)},v_{(i+2)}\rangle \)
| |
In-relocate |
\(\langle v_{(i-1)},v_{i}\rangle \), \(\langle v_{i},v_{(i+1)}\rangle \), |
\(\langle v_{(i-1)},v_{(i+1)}\rangle \), \(\langle v_{(i+1)},v_{i}\rangle \), |
\(\langle v_{(i+1)},v_{{(i+2)}}\rangle \)
|
\(\langle v_{i},v_{(i+2)}\rangle \)
| |
GENIUS-exchange |
\(\langle v_{(i-1)},v_{i}\rangle \), \(\langle v_{i},v_{(i+1)}\rangle \), |
\(\langle v_{(i-1)},v_{(i+1)}\rangle \), \(\langle v_{(j-2)},v_{i}\rangle \), |
\(\langle v_{(j-2)},v_{{(j-1)}}\rangle \), \(\langle v_{j},v_{{(j+1)}}\rangle \), |
\(\langle v_{i},v_{(j+1)}\rangle \), \(\langle v_{(j+1)},v_{(j-1)}\rangle \), | |
\(\langle v_{(j+1)},v_{{(j+2)}}\rangle \)
|
\(\langle v_{j},v_{(j+2)}\rangle \)
|
4.6 Co-operation
Characteristic \(\downarrow \)
| Ring | R-EAX | KS |
---|---|---|---|
Migration topology | Figure 4a | Figure 4b | Figure 4c |
Number of emigrants | 1 (\(p^B\)) | 1 (\(p^B\)) | 1 (\(p^B\)) |
Number of immigrants | 1 | 1 |
\(N_b\), \(N>N_b\ge 1\)
|
Immigration policy | Replace \(p^B\), if \(T(p^B)>T(p^I)\)
| Replace \(p^B\), if \(T(p^B)>T(p^I)\), otherwise \(p_c^I\leftarrow \mathcal {X}(p^B,p^I)\), and \(p_c^I\) replaces \(p^B\), if \(T(p^B)>T(p_c^I)\)
| Replace \(N_b\) worst individuals |
Emigration policy | Keep emigrants in the sending island’s population | ||
Migration interval (\(\delta \)) |
\(\delta =2\)
or
\(\delta =20\) (the number of consecutive generations) | ||
Time complexity \((\mathcal {T}_\mathrm{C})^\mathrm{a}\)
|
\(nM\)
|
\(nM^2\)
|
\(2nM + n\log n\)
|
5 Experimental Results
5.1 Setup
5.2 Dataset
Characteristic \(\downarrow \)
| C1 | C2 | R1 | R2 | RC1 | RC2 |
---|---|---|---|---|---|---|
Structure | Clustered | Clustered | Random | Random | Mixed | Mixed |
Time windows | Tight | Wide | Tight | Wide | Tight | Wide |
Vehicle capacities | Small | Large | Small | Large | Small | Large |
5.3 Analysis and Discussion
Id \(\downarrow \)
| C1 | C2 | R1 | R2 | RC1 | RC2 |
---|---|---|---|---|---|---|
Ring
| ||||||
1 |
42,478.95
|
16,879.24
| 53,687.82 |
42,188.86* | 46,725.88 | – |
2 | 42,387.85 |
17,126.39
|
49,630.74
| 33,634.05 | 44,494.17 | – |
3 | 40,446.33 | – | 45,793.13 |
24,965.63* |
42,772.49
|
20,050.71* |
4 | 39,659.12 |
15,742.92
| 43,320.69 |
17,926.45* |
41,835.34
| 15,869.82 |
5 |
42,469.18
|
16,561.57
|
52,186.14
|
36,243.51* | 45,981.43 | – |
6 | – | – | 48,411.62 | 30,123.53* | 45,892.32 | 26,910.35* |
7 | – | – | 45,050.20 |
23,257.36* |
45,245.68
|
25,116.15* |
8 | – | – |
43,017.17
| 17,585.79* |
44,565.66
| 23,845.66 |
9 | 41,031.46 | 16,421.32* | 51,042.39 | 33,064.26 | 44,841.57 | 23,112.13* |
10 | 42,478.95 |
15,982.98
|
48,986.21
|
30,257.93* |
44,214.03
| 22,048.98 |
Avg. | 41,251.06 |
16,452.40
|
48,112.61
|
28,924.74
| 44,656.86 | 22,421.97 |
R–EAX
| ||||||
1 |
42,478.95
|
16,879.24
| 53,772.62 | 42,261.34 |
46,612.69
| – |
2 |
42,278.45* |
17,126.39
| 49,832.97 | 33,570.65 | 44,781.82 | – |
3 |
40,338.02
| – | 46,237.88 | 25,029.43* | 42,830.96 | 20,132.61 |
4 | 39,589.01 | 15,805.27 | 43,659.12 | 19,180.23 | 42,043.04 | 15,992.44 |
5 |
42,469.18
| 16,563.10 | 52,713.43 | 36,289.00* |
45,850.40
| – |
6 | – | – | 48,664.85 | 30,270.10 |
45,889.93
| 27,097.58 |
7 | – | – | 45,393.80 | 23,730.70 | 45,382.16 | 25,380.63 |
8 | – | – | 43,632.43 | 18,018.54 | 44,780.84 | 24,013.56 |
9 |
40,733.69
| 16,421.14* | 51,364.94 | 33,146.76 | 44,757.77 | 23,292.42 |
10 |
40,167.29
| 16,022.40 | 49,411.66 | 30,428.55* | 44,273.97 | 22,225.09 |
Avg. |
41,150.66
| 16,469.59 | 48,468.37 | 29,192.53 | 44,720.36 | 22,590.62 |
KS
| ||||||
1 |
42,478.95
|
16,879.24
|
53,676.62
| 42,196.47* | 46,694.35 | – |
2 | 42,339.90 |
17,126.39
| 49,877.76 |
33,550.62* |
44,371.97
| – |
3 | 40,343.51 | – |
45749.74
| 24,989.17* | 42,774.24 | 20,068.74 |
4 |
39,548.04
| 15,829.51 |
43,226.05
| 18,262.09 | 41,873.12 |
15,790.47
|
5 |
42,469.18
|
16,561.57
| 52,328.30 | 36,250.95* | 45,962.10 | – |
6 | – | – |
48,386.76
|
30,102.91* | 45,622.36 |
26,883.75* |
7 | – | – |
45,001.07
| 23,270.90* | 45,358.51 | 25,162.18* |
8 | – | – | 43,101.84 |
17,526.72* | 44,684.21 |
23,822.03
|
9 | 40,898.81 |
16,372.32* |
50,882.21
|
33,025.59* |
44,532.34
|
23,089.01* |
10 | 40,357.13 | 15,988.14 | 49,004.42 | 30,270.36* | 44,287.86 |
22,004.78* |
Avg. | 41,205.07 | 16,459.53 | 48,123.48 | 28,944.58 |
44,616.11
|
22,402.99
|
Id \(\downarrow \)
| C1 | C2 | R1 | R2 | RC1 | RC2 |
---|---|---|---|---|---|---|
Ring
| ||||||
1 |
42,478.95
|
16,879.24
|
53,560.85* |
42,188.86* |
46,281.52
| – |
2 | 42,291.63* |
17,126.39
|
49,350.91
| 33,554.71* | 44,441.19 | – |
3 | 40,325.00 | – | 45,822.97 | 24,954.81* |
42,782.41
| 20,084.70 |
4 | 39,638.56 | 15,755.22 |
43,152.26
| 17,997.69* | 41,951.81 | 15,770.18 |
5 |
42,469.18
|
16,561.57
| 52,230.01 | 36,251.43* |
45,801.99
| – |
6 | – | – |
48,187.98
| 30,099.45* |
45,674.32
| 26,864.81* |
7 | – | – | 45,099.17 |
23,344.99* | 45,164.73 |
25,112.77* |
8 | – | – |
43,059.13
|
17,495.51* | 44,679.94 |
23,709.29* |
9 | 40,709.26 |
16,370.44* |
50,653.74
| 33,019.95* |
44,498.21
| 23,063.76* |
10 | 40,296.60 |
15,981.89
|
48,779.67
| 30,242.35* |
44,227.46
|
21,965.94* |
Avg. | 41,172.74 |
16,445.79
|
47,989.67
| 28,914.98 |
44,550.36
|
22,367.35
|
R–EAX
| ||||||
1 |
42,478.95
|
16,879.24
| 53,846.08 | 42,259.70 | 46,833.99 | – |
2 |
42,289.09* |
17,126.39
| 50,190.39 | 33,912.29 | 44,772.67 | – |
3 | 40,340.90 | – | 46,280.64 | 25,106.05 | 42,877.25 | 20,229.46 |
4 |
39,555.09
| 15,908.75 | 44,507.48 | 19,421.25 | 42,400.06 | 17,124.27 |
5 |
42,469.18
| 16,563.10 | 52,659.74 | 36,352.12 | 45,927.56 | – |
6 | – | – | 48,919.69 | 30,309.43 | 45,827.36 | 27,086.98 |
7 | – | – | 45,808.16 | 23,636.36 | 45,418.16 | 25,471.85 |
8 | – | – | 43,528.46 | 17,859.14 | 44,879.47 | 24,063.00 |
9 |
40,694.49
| 16,481.81 | 51,494.07 | 33,116.32* | 44,878.70 | 23,297.83 |
10 |
40,231.23
| 16,116.81 | 49,742.52 | 30,499.52* | 44,247.53 | 22,347.35 |
Avg. |
41,151.28
| 16,512.68 | 48,697.72 | 29,247.22 | 44,806.28 | 22,802.96 |
KS
| ||||||
1 |
42,478.95
|
16,879.24
| 53,615.67* |
42,188.86* | 46,496.78 | – |
2 | 42,338.01 |
17,126.39
| 49,671.22 |
33,512.83* |
44,365.22
| – |
3 |
40,310.85
| – |
45,719.52
|
24,940.32* | 42,790.99 |
20,076.27
|
4 | 39,710.03 |
15,746.46
| 43,291.04 |
17,959.91* |
41,780.75
|
15,754.41
|
5 |
42,469.18
|
16,561.57
|
52,218.87
|
36,232.18* | 45,910.92 | – |
6 | – | – | 48,335.40 |
30,091.93* | 45,722.35 |
26,797.76* |
7 | – | – |
45,080.30
| 23,355.92* |
45,096.25
| 25,128.19* |
8 | – | – | 43,156.24 | 17,524.42* |
44,665.94
| 23,796.20 |
9 | 40,910.39 | 16,372.32 | 50,768.29 |
33,002.36* | 44,643.41 |
23,061.90* |
10 | 40,283.92 | 15,993.24 | 48,946.49 |
30,215.24* | 44,235.87 | 21,999.00* |
Avg. | 41,214.48 | 16,446.54 | 48,080.30 |
28,902.40
| 44,570.85 | 22,373.39 |
Id \(\downarrow \)
| C1 | C2 | R1 | R2 | RC1 | RC2 |
---|---|---|---|---|---|---|
Ring
| ||||||
1 |
42,478.95
|
16,879.24
| 53,735.17 |
42,212.86* | 46,954.02 | – |
2 |
42,387.85
|
17,126.39
|
49,726.08
| 33,670.45 | 44,744.58 | – |
3 | 40,528.58 | – | 45,941.36 | 25,079.85 |
42,928.15
|
20,160.25
|
4 | 39,759.69 | 16,089.40 | 43,367.80 |
18,014.51* |
41,931.37
| 15,930.61 |
5 |
42,469.18
| 16,562.49 |
52,346.05
| 36,272.59* | 46,029.71 | – |
6 | – | – | 48,669.45 | 30,222.22* | 46,076.97 | 26,912.34* |
7 | – | – | 45,204.57 | 23,424.77 |
45,460.11
|
25,116.15* |
8 | – | – |
43,107.99
| 17,628.96* | 44,894.12 |
23,881.79
|
9 | 41,266.71 | 16,450.34 | 51,280.59 | 33,093.84* | 44,897.74 |
23,146.48* |
10 | 40,438.51 | 16,023.14 |
49,125.08
|
30,296.18* | 44,449.37 | 22,120.70 |
Avg. | 41,332.78 | 16,521.83 |
48,250.41
|
28,991.62
| 44,836.61 |
22,466.90
|
R–EAX
| ||||||
1 |
42,478.95
|
16,879.24
| 53,807.59 | 42,291.11 | 46,961.67 | – |
2 | 42,406.28 | 17,142.98 | 50,178.31 | 33,640.11 | 44,902.27 | – |
3 |
40,392.98
| – | 46,363.70 | 25,191.62 | 42,955.13 | 20,224.91 |
4 |
39,654.39
|
15,868.05
| 43,809.30 | 19,433.22 | 42,097.47 | 16,106.00 |
5 |
42,469.18
| 16,566.20 | 52,952.11 | 36,347.26 | 46,062.40 | – |
6 | – | – | 48,883.58 | 30,352.69 | 46,080.32 | 27,146.39 |
7 | – | – | 45,554.95 | 23,803.27 | 45,512.01 | 25,494.36 |
8 | – | – | 43,791.72 | 18,182.23 | 44,870.17 | 24,164.79 |
9 |
40,912.96
| 16,498.40 | 51,876.63 | 33,202.82 | 44,873.52 | 23,438.13 |
10 |
40,325.16
| 16,059.15 | 49,726.29 | 30,563.48* |
44,382.68
| 22,427.01 |
Avg. |
41,234.27
| 16,502.34 | 48,694.42 | 29,300.78 | 44,869.76 | 22,714.51 |
KS
| ||||||
1 |
42,478.95
|
16,879.24
|
53,691.52
| 42,214.16* |
46,795.24
| – |
2 | 42,668.32 |
17,126.39
| 49,960.81 |
33,615.75
|
44,546.19
| – |
3 |
40,402.24
| – |
45,851.78
|
25,038.87* | 42,968.44 | 20,169.93 |
4 | 39,736.93 | 15,873.11 |
43,339.93
| 18,365.29 | 41,985.52 |
15,852.43
|
5 |
42,469.18
|
16,561.57
| 52,602.36 |
36,261.01* |
46,008.60
| – |
6 | – | – |
48,625.45
|
30,171.57* |
45,897.57
|
26,908.14* |
7 | – | – |
45,176.82
|
23,412.49
| 45,505.78 | 25,252.48* |
8 | – | – | 43,199.05 |
17,582.20* |
44,809.70
| 23,883.43 |
9 | 40,970.05 |
16,400.61* |
51,024.38
|
33,060.30* |
44,779.06
| 23,161.29 |
10 | 40,477.13 |
15,999.76
| 49,197.66 | 30,313.90* | 44,413.41 |
22,065.36* |
Avg. | 41,314.69 |
16,473.45
| 48,266.98 | 29,003.55 |
44,770.95
| 22,470.44 |
Id \(\downarrow \)
| C1 | C2 | R1 | R2 | RC1 | RC2 |
---|---|---|---|---|---|---|
Ring
| ||||||
1 |
42,478.95
|
16,879.24
|
53,626.82* | 42,209.68* |
46,562.32
| – |
2 |
42,359.36
|
17,126.39
|
49,729.14
|
33,647.83
| 44,615.58 | – |
3 | 40,431.26 | – | 45,921.21 | 24,999.15* |
42,834.01
|
20,125.28
|
4 | 39,744.58 | 15,832.46 |
43,330.69
|
18,056.59
|
42,014.82
|
15,798.70
|
5 |
42,469.18
| 16,562.49 |
52,359.55
| 36,255.76* |
46,022.34
| – |
6 | – | – |
48,387.30
| 30,178.59* |
45,760.65
| 26,879.59* |
7 | – | – | 45,251.39 |
23,387.73
|
45,292.69
|
25,138.07* |
8 | – | – |
43,146.94
|
17,546.31* |
44,732.74
|
23,803.10
|
9 | 40,984.30 |
16,379.48* |
50,901.63
| 33,047.26* |
44,722.24
| 23,100.65 |
10 |
40,369.25
| 16,010.32 |
48,980.19
| 30,284.52* | 44,406.02 |
21,995.50* |
Avg. | 41,262.41 | 16,465.06 |
48,163.49
|
28,961.34
|
44,696.34
|
22,405.84
|
R–EAX
| ||||||
1 |
42,478.95
|
16,879.24
| 53,935.61 | 42,270.62 | 47,182.99 | – |
2 | 42,367.15 |
17,126.39
| 50,459.87 | 33,951.40 | 44,820.79 | – |
3 |
40,386.08
| – | 46,426.27 | 25,170.14 | 43,049.98 | 20,363.40 |
4 |
39,648.46
| 15,958.67 | 44,732.58 | 19,492.46 | 42,475.18 | 17,180.56 |
5 |
42,469.18
| 16,569.10 | 52,950.27 | 36,391.82 | 46,365.72 | – |
6 | – | – | 49,096.64 | 30,413.00 | 46,144.14 | 27,131.66 |
7 | – | – | 45,931.75 | 23,789.55 | 45,591.49 | 25,471.85 |
8 | – | – | 43,731.51 | 17,908.58 | 45,014.27 | 24,199.93 |
9 |
40,876.66
| 16,524.75 | 51,782.76 | 33,252.21 | 44,995.33 | 23,496.33 |
10 | 40,379.99 | 16,137.12 | 49,969.36 | 30,659.73 | 44,456.08 | 22,441.98 |
Avg. |
41,229.50
| 16,532.55 | 48,901.66 | 29,329.95 | 45,009.60 | 22,897.96 |
KS
| ||||||
1 |
42,478.95
|
16,879.24
| 53,649.09* |
42,201.83* | 46,779.46 | – |
2 | 42,422.66 |
17,126.39
| 50,092.45 | 33,654.05 |
44,611.65
| – |
3 | 40,421.20 | – |
45,856.74
|
24,980.55* | 42,938.35 | 20,128.92 |
4 | 39,819.31 |
15,826.95
| 43,616.72 | 18,292.28 | 42,021.67 | 15,992.54 |
5 |
42,469.18
|
16,561.57
| 52,412.51 |
36,247.61* | 46,085.55 | – |
6 | – | – | 48,477.39 |
30,163.96* | 45,920.41 |
26,839.08* |
7 | – | – |
45,244.39
| 23,427.09 | 45,336.30 | 25,166.34* |
8 | – | – | 43,226.30 | 17,558.80* | 44,781.20 | 23,837.32 |
9 | 40,980.37 | 16,383.74* | 51,044.55 |
33,043.74* | 44,746.04 |
23,076.81* |
10 | 40,436.61 |
15,998.07
| 49,097.84 |
30,245.51* |
44,321.03
| 22,047.55* |
Avg. | 41,289.75 |
16,462.66
| 48,271.80 | 28,981.54 | 44,754.17 | 22,441.22 |
Scheme \(\rightarrow \)
| Ring | R-EAX | KS | |||
---|---|---|---|---|---|---|
\(\delta \rightarrow \)
| 2 | 20 | 2 | 20 | 2 | 20 |
#\(T_B\) (out of 50) |
24
|
26
| 11 | 8 |
24
|
26
|
%\(T_B\) (in %) |
48
|
52
| 22 | 16 |
48
|
52
|
#\(T^{\star }\)
|
14
|
18
| 5 | 3 |
14
| 15 |
#\(T_B^A\) (out of 50) | 19 |
32
| 9 | 7 |
30
| 19 |
%\(T_B^A\) (in %) | 38 |
64
| 18 | 14 |
60
| 38 |
#\(T^{A\star }\)
| 10 | 12 | 1 | 0 |
11
|
13
|
Scheme \(\rightarrow \)
| Ring | R-EAX | KS | WB | |||
---|---|---|---|---|---|---|---|
Test \(\downarrow \)
\(\delta \rightarrow \)
| 2 | 20 | 2 | 20 | 2 | 20 | |
C1_10_2 | 42,387.85 | 42,291.63 |
42,278.45
| 42,289.09 | 42,339.90 | 42,338.01 | 42,300.76 |
C2_10_9 | 16,421.32 |
16,370.44
| 16,421.14 | 16,481.81 | 16,372.32 | 16,372.32 | 16,432.53 |
R1_10_1 | 53,687.82 |
53,560.85
| 53,772.62 | 53,846.08 | 53,676.62 | 53,615.67 | 53,657.99 |
R2_10_1 | 42,188.86 | 42,188.86 | 42,261.34 | 42,259.70 | 42,196.47 |
42,188.86
| 42,219.21 |
R2_10_2 | 33,634.05 | 33,554.71 | 33,570.65 | 33,912.29 | 33,550.62 |
33,512.83
| 33,567.91 |
R2_10_3 | 24,965.63 | 24,954.81 | 25,029.43 | 25,106.05 | 24,989.17 |
24,940.32
| 25,053.80 |
R2_10_4 |
17,926.45
| 17,997.69 | 19,180.23 | 19,421.25 | 18,262.09 | 17,959.91 | 18,039.77 |
R2_10_5 | 36,243.51 | 36,251.43 | 36,289.00 | 36,352.12 | 36,250.95 |
36,232.18
| 36,335.72 |
R2_10_6 | 30,123.53 | 30,099.45 | 30,270.10 | 30,309.43 | 30,102.91 |
30,091.93
| 30,223.14 |
R2_10_7 |
23,257.36
| 23,344.99 | 23,730.70 | 23,636.36 | 23,270.90 | 23,355.92 | 23,381.36 |
R2_10_8 | 17,585.79 |
17,495.51
| 18,018.54 | 17,859.14 | 17,526.72 | 17,524.42 | 17,598.63 |
R2_10_9 | 33,064.26 | 33,019.95 | 33,146.76 | 33,116.32 | 33,025.59 |
33,002.36
| 33,131.99 |
R2_10_10 | 30,257.93 | 30,242.35 | 30,428.55 | 30,499.52 | 30,270.36 |
30,215.24
| 30,598.69 |
RC2_10_3 |
20,050.71
| 20,084.70 | 20,132.61 | 20,229.46 | 20,068.74 | 20,076.27 | 20,053.78 |
RC2_10_6 | 26,910.35 | 26,864.81 | 27,097.58 | 27,086.98 | 26,883.75 |
26,797.76
| 26,965.51 |
RC2_10_7 | 25,116.15 |
25,112.77
| 25,380.63 | 25,471.85 | 25,162.18 | 25,128.19 | 25,295.67 |
RC2_10_8 | 23,845.66 |
23,709.29
| 24,013.56 | 24,063.00 | 23,822.03 | 23,796.20 | 23,787.26 |
RC2_10_9 | 23,112.13 | 23,063.76 | 23,292.42 | 23,297.83 | 23,089.01 |
23,061.90
| 23,116.15 |
RC2_10_10 | 22,048.98 |
21,965.94
| 22,225.09 | 22,347.35 | 22,004.78 | 21,999.00 | 22,076.90 |
6 Conclusions and Future Work
GH subclass \(\downarrow \)
| Minimize \(\tau _P\)? | |||
---|---|---|---|---|
Yes | No | |||
Scheme |
\(\delta \)
| Scheme |
\(\delta \)
| |
C1 | R-EAX |
\(2\)
| R-EAX |
\(20\)
|
C2 | KS |
\(2\)
| KS |
\(20\)
|
R1 | KS |
\(2\)
| Ring |
\(20\)
|
R2 | KS |
\(20\)
| Ring |
\(20\)
|
RC1 | KS |
\(20\)
| Ring |
\(20\)
|
RC2 | KS |
\(2\)
| Ring |
\(20\)
|