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2005 年全国部分高校研究生数学建模竞赛 A 题
A: Highway Traveling time Estimate and Optimal Routing
Ⅰ
Highway traveling time estimate is crucial to travelers. Hence, detectors are mounted on
some of the US highways. For instance, detectors are mounted on every two-way six-lane
highways of San Antonio city. However, since vehicles tend to change lanes from time to
time, we may ignore vehicle lane change and consider just one lane traffic as shown
below, in which the square boxes stand for detectors.
636 m 417 m 522 m 475 m
travel direction
Detector 1 Detector 2 Detector 3 Detector 4 Detector 5
1. Detectors are able to detect and measure the speed of individual vehicles 24 hours a
day. Average vehicle speed measured within 20 seconds by detectors is reported and
refreshed. The following table provides the real time data (due to the huge volume of
traffic data, only traffic data of the last 20 seconds in every two minutes is in the last 20
seconds. Unit: mile/hour). Please analyze traveling characteristics on highways (for
instance, congestion and its dispersion. Typically, it is not considered as congestion if
traffic speed is higher than 50 mile/hr). If a vehicle passes the detector at time t, how long
will it take for this vehicle to travel to the fifth sensor? Please design an algorithm for
estimating such travel times. Make sure that you demonstrate the rationality and accuracy
of your algorithm. If traffic data is provided every 20 seconds rather than every 2
minutes, how this information is going to affect your estimate?
All the conditions stay the same as in the previous problem. If detectors can measure
not only vehicle speed, but also traffic volume per unit time (see table below. The unit of
Flow is the number of cars/20 seconds), does this additional information help to improve
the rationality and accuracy of your algorithm? If your answer is yes, please re-design
your algorithm.
Time
Sensor 1
Sensor 2 Sensor 3 Sensor 4 Sensor 5
Speed Flow Speed Flow Speed Flow Speed Flow Speed Flow
3:40:07 PM
57
10
54
9.7
62
8.9
20 10.7 58 5.9
3:42:07 PM
62
9.5
68
11.4
63
13.6
21 10 59 12.2
3:44:07 PM
56
11.1
62
10.3
61
12.9
19 14.2 60 8.1
3:46:07 PM
58
12.5
61
9.1
59
9.3
23 14.8 61 10
3:48:07 PM
53
11.8
64
6.7
62
12.3
45 9 59 10.1
3:50:07 PM
58
9.3
66
6.4
63
4.5
63 10 56 13.8
3:52:07 PM
55
11.9
63
13
60
9.2
55 11.4 59 18
3:54:07 PM
59
8
62
5.3
63
9.7
65 11 55 11
3:56:07 PM
52
9.8
73
9.7
56
8.3
65 8 62 12
3:58:07 PM
60
14.1
63
13.7
59
12
66 5.1 61 5
4:00:07 PM
55
4.8
64
12.3
61
12.8
65 5.1 59 11
4:02:07 PM
60
13.7
61
7.9
60
12.6
65 11.1 66 0
4:04:07 PM
59
8.7
62
9.1
60
4.1
64 10.6 57 14
4:06:07 PM
58
8
64
8.3
64
14.5
45 7 59 14
4:08:07 PM
55
5.4
63
15.3
63
16
1 9.3 57 7
4:10:07 PM
62
13.7
61
9.1
57
5.4
15 11.5 64 8
4:12:07 PM
59
9.9
63
6.7
62
5
64 6.1 52 14
4:14:07 PM
58
8.5
63
12
62
11.9
40 8.7 60 10
4:16:07 PM
56
8.3
58
9
60
13.7
61 11.5 58 13
4:18:07 PM
60
9.3
64
10.3
65
10
52 10 51 9
4:20:07 PM
59
10.4
66
9.9
68
5
45 14.7 55 15.9
4:22:07 PM
53
14
56
8.6
63
8
55 13.7 40 12.7
4:24:07 PM
57
13
62
13.5
62
13
29 14 54 11.9
4:26:07 PM
58
13.4
64
11.5
63
14
62 13 30 12.9
4:28:07 PM
55
8.3
62
11.3
58
12
8 10.9 33 11.9
4:30:07 PM
59
11
78
8
61
13
39 5.1 51 10.1
4:32:07 PM
59
17.1
67
11.8
61
11
57 11.9 59 16.9
4:34:07 PM
54
7.3
64
10.6
59
9
62 10 60 11.1
4:36:07 PM
59
12
66
7.8
68
5.1
62 10 27 8.3
4:38:07 PM
56
11.7
68
10.4
60
8
20 8.2 49 13.9
4:40:07 PM
57
8.4
60
11.2
65
14.9
37 11 51 11.1
4:42:07 PM
55
10.4
63
12.3
66
6
40 9.9 45 14.2
4:44:07 PM
59
12.1
65
9.2
62
6.9
49 8.8 51 14.6
4:46:07 PM
49
13.4
69
11.5
66
8.1
54 7.8 20 9.2
4:48:07 PM
51
11.8
66
14
61
10.2
21 14.4 30 12.3
4:50:07 PM
53
11.6
60
14.7
62
10.1
2 5.5 34 11.9
4:52:07 PM
53
13.8
62
12.2
50
9.2
21 12 36 7.5
4:54:07 PM
49
14.3
59
12.7
59
10.1
5 3.8 38 14.8
4:56:07 PM
53
13.5
62
13.6
56
11
38 13.2 26 10.1
4:58:07 PM
55
12.8
62
12.8
57
11
23 8.2 37 13.8
5:00:07 PM
56
11.9
62
13
57
10.1
4 14.8 34 11
5:02:07 PM
56
7.6
63
9.7
59
8.9
2 11.9 28 11.6
5:04:07 PM
60
6.6
65
14.7
64
14
47 9.1 29 10.8
5:06:07 PM
55
11.9
63
11
66
7.6
36 12.9 28 9.8
5:08:07 PM
59
10.2
65
8
60
9
40 12.1 37 14.3
5:10:07 PM
60
8.2
64
11
66
9.2
26 14.8 29 12.2
5:12:07 PM
60
8.7
64
5.6
64
7.5
13 9.4 38 13.2
5:14:07 PM
55
13.1
62
8.4
66
5.9
63 8.2 34 12
5:16:07 PM
56
15.5
57
12.4
61
10
61 10.8 27 13
5:18:07 PM
56
13.3
63
5.5
64
6.1
23 7.9 17 9.4
5:20:07 PM
57
13.1
57
12
65
11.9
4 8.8 14 10
5:22:07 PM
51
14
60
10.9
59
7.5
4 9.2 19 6.4
5:24:07 PM
48
12.5
59
8.9
27
9
17 8 21 9
5:26:07 PM
46
11.6
57
7.4
2
8.9
5 7.8 28 9.7
5:28:07 PM
52
12.4
53
12
26
8.9
3 8.5 26 7.9
5:30:07 PM
57
12.6
42
10.6
12
7.8
3 8.5 18 8.2
5:32:07 PM
51
14.4
38
11.6
10
8.5
2 7.9 19 7.8
5:34:07 PM
51
11.4
19
4.7
21
9.2
22 8.3 33 5.6
5:36:07 PM
53
12.4
14
3.7
12
5
2 6.8 42 9.7
5:38:07 PM
40
10.2
13
8.3
2
7.5
3 7.4 40 10.3
5:40:07 PM
39
7.9
14
6.4
2
7.5
2 9.3 38 9.6
5:42:07 PM
42
10.4
8
7
2
8.3
2 9.7 44 9.4
5:44:07 PM
46
5.8
4
6.4
2
5.6
3 7.7 38 11.3
