Model 1: Consider the situation of a class, analyze the test time and the number of machines, divide the five test items into four groups for synchronous measurement, and divide the students in the class into four groups according to their student numbers, establish a multi-objective linear programming model, and get the first scheme with the least measurement time.
Model 2: Considering the situation of multiple classes, the five test items are also divided into four groups for synchronous measurement, and the students in the class are divided into four groups according to their student numbers, and a linear programming model is established and solved to get the second scheme with the least measurement time.
Model 3: On the basis of model 1 and model 2, considering the number of instruments, site capacity and grouping, the planning model is established again and solved, and the optimal decision-making scheme is put forward.
[Keywords:] physical fitness test waiting time planning model
I. Restatement of the problem
(a) the basic situation and requirements of the problem:
Physical fitness test includes five items: height and weight, standing long jump, vital capacity, grip strength and step test, all of which are automatically measured, recorded and saved by electronic instruments. At present, there are 3 height and weight measuring instruments, standing long jump and vital capacity measuring instrument 1 set, and 2 grip strength and step test measuring instruments.
Height and weight, standing long jump, vital capacity and grip strength, the average test time of each instrument for each student (including student conversion) is 10 second, 20 second, 20 second, 15 second respectively, and it takes 3 minutes and 30 seconds for each instrument to test 5 students.
Each student must enter personal information before testing each item, which takes an average of 5 seconds. After each student measures, the student number of the instrument will automatically move back one place. If you connect the student numbers before and after the exam, you can save input time and connect the student numbers of the same class.
The school arranges the examination time every day at 8: 00- 12: 10 and 13: 30- 16: 45. The five tests were conducted in a small place with a maximum capacity of 150 people, and the test items were not in a fixed order.
(2) Problems to be solved:
(1) The school requires all students in the same class to complete all the exams in the same time period, and try to save students' waiting time when the number of time periods required for the whole exam is the least.
(2) Use mathematical symbols and language to express the examination schedule of each course, give the algorithm of this math problem, and show the examination schedule plan in the form of chart.
(3) In the school's future physical fitness test, we discussed such aspects as "introducing all kinds of measuring instruments", "increasing the staff capacity of the test center" and "whether students in a class need to be grouped during the test".
Second, the problem analysis
Question 1 analysis:
At this time, only the number of students in a class is considered, and the number must be within the accommodation range. Only consider the two conditions of making the entry and waiting time as small as possible. In the former case, students can be grouped continuously. In the latter case, because the total time of height and weight test and grip strength test is shorter than the other three items, it can be regarded as a whole. That is, the class is divided into four groups with continuous student numbers, and synchronous measurement produces four stages. The completion of all synchronous measurement projects marks the end of each stage. In each stage, one of the four groups of projects must have the most and least test time, and the difference between the two groups (the difference between the two groups is recorded as waiting time, which is a variable; The shorter the waiting time of students in the same group, the closer the four groups are to completing the test at this stage. Solve the planning problem and find the objective function of the planning to minimize the sum of the waiting time of these four stages.
Question 2 analysis:
In a given measuring instrument, there are three measuring instruments for height and weight, and the average test time of each instrument for each student (including student conversion) is 10 second, that is, the test time of each student is seconds (from this, it can be known that the number of people tested is 3 or more and can be evenly divided by 3); There are two grip strength measuring instruments, and the average test time of each instrument for each student is 15 seconds, that is, the test time of each student is seconds (it can be known that the number of people being tested is 2 or more and can be divisible by 2), and the sum of the test time of height, weight and grip strength is less than that of standing long jump, vital capacity and step test, so the height, weight and grip strength can be regarded as a whole; There is one standing long jump and one vital capacity measuring instrument, and the average test time of each instrument for each student is 20 seconds, that is, the test time for each student is 20 seconds; There are two bench test measuring instruments, each instrument tests five students at a time, and the average test time of each student is 2 10 second, so the test time of each student is 2 1 second (from this, it can be known that the number of people being tested is more than 10 and can be divisible by 10), so it can be concluded that the time ratio of these four is 65. According to the idea of the first question, the model is established, and finally each kind of data is fitted by fitting method to find the best grouping scheme.
Analysis of question three:
This question is the deepening and extension of the first and second questions, and discusses the number of instruments, site capacity and grouping.
Third, the model hypothesis.
1) The student numbers of each item tested in the same group are continuous;
2) All the tested machines are working normally;
3) Students take exams one after another, with no time interval;
Fourth, the symbol description
: the number of people tested in the first group;
: The time spent on the project with the longest measurement time in the first stage;
: The time spent on the project with the shortest measurement time in the first stage;
The total waiting time of students in the first stage of the test
The total number of students in a class;
N: the total number of students in several classes;
: the number of people who moved from projects with a large number of people to projects with a small number of people at each stage;
Establishment and solution of verb (verb abbreviation) model
Model 1: We divide each class into four groups. This is the number of people in the first group. At the same time, we divide the testing process into four stages, and each team has to complete all the required testing items. When each group enters the next stage of testing after testing a certain item, each test item can be randomly replaced between groups without repeated testing. According to the topic analysis, the sum of per capita time for measuring height, weight and grip strength is less than the sum of per capita time for measuring standing long jump, vital capacity and steps. Therefore, combining the time of measuring height and weight with the time of grip strength, the students are divided into four groups. The objectives are:
Suppose a class is divided into several groups, and the number of people in each group can be divisible by 2, 3, 10, so that the measurement utilization rate of each project is the largest.
The first stage model:
Constraints:
The second stage model:
Constraints:
The mode of the third stage:
Constraint:;
The fourth stage:
;
;
Constraint:;
Among them,
: Represents an intermediate variable of an integer.
Suppose that 40 students in a class are divided into four groups, and the student numbers of the four groups are1-10 respectively; 1 1—20; 2 1—30; 3 1—40。 After the first group is finished, the second group is connected to the test, thus connecting the student number, which can reduce the input time, and a set of plans is drawn up as shown in the following figure:
The overall objective function is:
The first stage:
;
;
Constraint:;
The second stage:
;
;
Constraint:;
The third stage:
;
;
Constraint:;
The fourth stage:
;
;
Constraint:;
Among them,
: 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; By running the appendix I program for many times, we can get the corresponding time period (representing the time taken by all students in the I-th combination to complete all five tests) after multiple teams synthesize 150 people, which are 3300, 3350, 3375, 3375 and 3335 in turn. 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.
;
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
1 1: 15- 12: 15 10/29/2 1/5 1 (42,38,32,38) 55.58333333 3335
13:30- 14:30 34/32/20/23 (39,33,28,35,) 5 1 3060
14:35- 15:35 19/22/30/53 (39,33,38,39,) 5 1.66 3 100
15:40- 16:40 5/ 12/28/56 (26,25,24, 17) 36 2 160
Question 3:
This paper discusses the school physical fitness test from the following aspects: "introducing various measuring instruments", "increasing the capacity of people in the test center" and "whether students in a class need to be grouped during the test".
It can be seen from the above answers that when the ratio of the total measurement times of height, weight and grip strength to the measurement times of standing long jump, vital capacity and step test is about 1: 2: 2: 2, the demand of the shortest waiting time in the first stage can be met, so the ratio of the total measurement times of height, weight and grip strength to the measurement times of standing long jump, vital capacity and step test is close to 2:/. But in the second stage, the proportion of people is (1: 2: 1: 1), (1: 1), (1: 1). The shorter the waiting time, the ratio of the total measurement time of height, weight and grip strength to the measurement time of standing long jump, vital capacity and step test should be close to1:1:1:1. At this time, the standing long jump, vital capacity and step tester should be increased by 1 and 65438+ respectively. When the ratio of people who meet the minimum waiting time in the first stage is1:1:1:1,the waiting time in any stage is the minimum. At this time, the number of new instruments is: 3 height and weight measuring instruments, 2 standing long jumpers, 2 vital capacity measuring instruments, 2 grip strength measuring instruments and 4 step measuring instruments. If funds permit, it is most reasonable to increase tools according to this ratio.
The larger the staff capacity of the investigation point, the less the overall investigation time arranged by the school. Through the analysis of the class size and the class size range in this school, the following table is given:
Class number, class number, proportion of total number of people
Below 20 3 53 0.0535
20-30 1 1 248 0. 1964
30-40 18 648 0.32 14
40-50 20 86 1 0.357 1
Over 50 4 226 0.07 14
Total 56 2036
By calculating their expectations, we can know the capacity of the physical fitness test site in this school. The expected value represents the number of students, so the expected value of each part should be rounded up. The sum of the products of the total number of students in different classes and their corresponding proportions is the expected value: 3+45+209+294+ 17=568.
If students in a class do not take the group exam, only the admission time is considered; The group test of this class should consider the input time and waiting time of each group at the same time (the student number of each group is continuous), and only consider which time is longer in these two cases. Assuming that the number of students in this class is n, the entry time of the first case is 25n；; Group the second case and establish the following model:
;
Constraint condition
The time of the second case is (5*25+Z). Compared with the input time in the first case, if the time in the second case is shorter, it should be grouped, and vice versa.
Evaluation of intransitive verb model
Advantages:
1), and it is scientific to use LINGO software to solve the model strictly.
2) The established model is universal and easy to popularize;
3) The second model is closely linked with practice, and the existing problems are fully considered, which makes the model more applicable.
Disadvantages:
1) In the establishment of the model 1, the model is idealized, only the measurement time and the overall investment time are considered, and no specific scheme is calculated;
2) In the establishment of model 2, the number of people is fitted by fitting method, and the result is not necessarily optimal;
3) Due to time, we can't get a clear answer by asking whether model 3 is grouped.
Seven, model expansion
The establishment of this model solves the problem of the time arrangement of physical fitness test. The dynamic objective linear programming is used to establish the relevant model, and then the time ratio is used to establish the single objective programming, which reflects a proportion of the number of students. Finally, LINGO software is used to solve the problem. Therefore, the model can also be applied to other similar time arrangements, such as the testing time of parts, the installation time of parts, and the reasonable arrangement of course selection.