What is Question B of the 2007 National Mathematical Modeling Competition for College Students? There is also a reference answer!

: integer;

Conclusion: The shortest waiting time in each stage can make the measurement time of the whole process shortest.

Model 2: According to the analysis of Question 2, the time ratio of each student's height, weight and grip strength to standing long jump, vital capacity and step test is about 1: 2: 2: 2. That is to say, when the number of students is about 2: 1: 1, the waiting time is the shortest, but in order to optimize the overall situation, some students assigned to measure their height, weight and grip strength can be equally assigned to the standing long jump, vital capacity and step test groups. From the analysis of this ratio, we can know that the number of people measuring height, weight and grip strength is even greater than that of other groups, so in the second stage, the number of people will change when the time ratio is unchanged, and the number of people measuring height, weight, grip strength, vital capacity and step test is the same. It takes the least time to measure the height, weight and grip strength, and the most time to stand in the long jump, which can also achieve the best. The third stage and the fourth stage, like the first two stages, can achieve time optimization, thus achieving overall optimization. The maximum number of students that this test center can accommodate is 150, so we can regard 150 as a whole, that is, the number of students is continuous.

Solve the problem with LINGO software and get the result. (Appendix 1) The waiting time for testing all students is at least 1575 seconds, of which the longest time for the first stage is 845 seconds, the longest time for the second stage is 805 seconds, the longest time for the third stage is 805 seconds, and the longest time for the fourth stage is 845 seconds, so it can be known that the time for testing all students is 3300 seconds. The number of students assigned to measure the grip strength of height and weight is 2 1, so the number of students in each group should be 39, 37, 37, 37.

When the waiting time of the test is the least, the input time is reduced, so the overall time can be reduced. The way to minimize the input time is to reduce the number of inputs. In the case of class combination, the fewer students are separated in each class, the fewer items there are.

Under 20 19, 17, 17,

20-29 26,20,20,25,20,28,25,20,24,20,20,

30-39 38,37,30,39,35,38,38,30,36,32,33,33,39,37,38,39,37,39,

40-49 4 1,45,44,44,44,42,45,45,45,44,4 1,44,42,40,42,43,4 1,42,45,42,

More than 50 5 1, 50, 50, 75,

According to the above requirements, according to the number of classes, it is proposed to merge as follows:

Serial number serial number

1 39,37,37,37 8 44,44,42,20

2 75,50,25 9 4 1,43,36,30

3 5 1,45,44,20 10 4 1,42, 17,30,20

4 50,42,38,20 1 1 42,38,32,38

5 45,45,40,20 12 39,33,28,35

6 45,45,4 1, 19 13 39,33,38,39

7 44,44,42,20 14 26,25,24, 17

For each class with a combination number of 150, the number of additional entries is (I = 1, 2, 3 ...), which are 0, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, respectively; After running the appendix I program for many times, the corresponding time period (representing the time taken by all students in the I-th combination to complete all five tests) is 3300, 3350, 3375, 3375 and 3335, respectively. The number of class combinations is less than 150, and the other three combinations are 135 and149,92 respectively. By analyzing the time proportion of each measurement, the integer part divisible by 5 is allocated to each test in proportion, and the rest belongs to height, weight and grip strength.

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Using LINGO software to solve the constraint conditions, the results are obtained. (Appendix II) When 135 students serve as a class, the waiting time is at least 1475 seconds, while the longest test time in the first stage is 845 seconds, the longest test time in the second stage is 685 seconds, the longest test time in the third stage is 685 seconds, and the longest test time in the fourth stage is 845 seconds. The combination of different classes is as follows. The total time of measuring 135 students is 3060 seconds. (Appendix III) When 149 students are in a class, the waiting time is at least 1600 seconds, while the longest test time in the first stage is 845 seconds, the longest test time in the second stage is 705 seconds, the longest test time in the third stage is 705 seconds, and the longest test time in the fourth stage is 845 seconds. The combination of different classes is as follows. The total time of measuring 135 students is 3 100 seconds. (Appendix 4) When 92 students are regarded as a class, the waiting time is at least 1080 seconds, while the longest test time in the first stage is 635 seconds, the longest test time in the second stage is 445 seconds, the longest test time in the third stage is 445 seconds, and the longest test time in the fourth stage is 635 seconds. The combination of different classes is as follows. The total time of measuring 135 students is 2 160 seconds.

The combination of different classes

Time (minutes) of the number of students after the merger of the whole school classes.

8:00-9:00 40\43\ 1 1\38 (39,37,37,37) 54.33333333 3260

9:05 - 10:05 54\45\24 (75,50,25) 55. 16666667 33 10

10: 10- 1 1: 10 33/37/ 14/8 (5 1,45,44,20) 55.58333333 3335

1 1: 15- 12: 15 44/4 1/39/9 (50,42,38,20) 55.58333333 3335

13:30- 14:30 2/ 13/35/42 (45,45,40,20) 55.58333333 3335

14:35- 15:35 15/48/50/52 (45,45,4 1, 19) 55.58333333 3335

15:40- 16:40 3/4/7/9 (44,44,42,20) 55.58333333 3335

8:00-9:00 6/ 16/36/46 (44,44,42,20) 55.58333333 3335

9:05 - 10:05 25/26/3 1/47 (4 1,43,36,30) 55.58333333 3335

10: 10- 1 1: 10 1/ 18/27/49/55 (4 1,42, 17,30,20) 55.58333333 3335