Keywords: Q-value method fair seats
Question restatement: There are ***200 students in three departments, (Department A 100, Department B 60 and Department C 40) representing ***20 seats, and the three departments are allocated 10, 6 and 4 seats respectively. The old situation has become the following. How to distribute it is the fairest. The number of people who have transferred three books now is 103.63.34.
(1) How to allocate 20 seats?
(2) How to allocate the extra 2 1 seats.
Problem analysis:
First of all, the fairness of distribution results is usually measured by the equal or close number of people represented by each representative seat. At present, the customary distribution method is proportional distribution method, namely:
The number of seats allocated by a unit = the proportion of the total number of people in a unit' total seats'
If the number of seats allocated by some participating units according to the above formula is decimal, the seats will be allocated first according to the integer number of allocated seats, and the remaining seats will be allocated in turn according to the decimal size of all participating units. So the initial number of students and student representative seats are as follows
Total number of department names a, b and c
Number of students 100 60 40 200
The proportion of students is 100/200 60/200 40/200.
Seat allocation 10 6 4 20
When students change departments, the number of students in each department and the number of student representative seats become
Total number of department names a, b and c
Number of students 103 63 34 200
The proportion of students is 103/200 63/200 34/200.
Proportional allocation of seats 10.3 6.3 3.4 20
According to the usual seat allocation 10 6 4 20
(1) Such 20 seats should be A series 10 seats, B series 6 seats and C series 4 seats.
Second, the college decided to add another representative seat, and the total representative seat became 2 1. Reassign seats as usual, yes
Total number of department names a, b and c
Number of students 103 63 34 200
The proportion of students is 103/200 63/200 34/200.
Proportional allocation of seats10.8156.6153.57 438+0
According to the usual seat allocation, 1 1 7 3 2 1.
The result of this allocation is that there is one less seat in Part C than before, which makes people feel that the allocation of seats is obviously unfair. How can we be fair? At this time, it is necessary to solve it by mathematical modeling.
Establishment of the model:
Assuming that the seats are fairly distributed by two units, set up
Number of seats per unit Number of representatives per seat
Unit A p 1 n 1
Bpp2n2 unit
In all fairness, there should be =, but this is generally not true. Note that this equation does not exist.
If >, it means that unit A suffers (that is, it is unfair to unit A)
if
Therefore, we can consider using a formula to measure the degree of unfair distribution, but this formula has some shortcomings (the characteristics of absolute numbers), such as:
The number of people and seats in a certain two units is n 1 =n2 = 10, p 1 = 120, p2= 100, so p=2.
The number of people and seats in the other two units are n 1 =n2 = 10, p 1 = 1020, p2= 1000, so p=2.
Although both cases have p=2, it is obvious that the second case is fairer than the first case.
The following uses the relative standard to improve the formula, and defines the relative unfair standard formula for seat allocation:
If it is called a relatively unfair value to A, it is recorded as
If it is called a relatively unfair value to B, it is recorded as
According to the definition, the smaller the unfair value to one party, the more favorable it is for one party to allocate seats, so we can reduce the unfairness in allocation by making the unfair value as small as possible.
Determine the allocation scheme:
Using the unfair value to determine the distribution scheme, we might as well set >; In other words, this is unfair to unit A. When another seat is allocated, the relationship may be
1.& gt shows that this seat is unfair to A after it is given to A;
2.& lt, it means that this seat is unfair to B after being given to A, and the unfair value is
3.& gt, it means that this seat is unfair to A after being given to B, and the unfair value is
4.& lt, impossible.
The above allocation method can determine the allocation of new seats in case 1 and case 3, but it is not easy to determine the allocation of new seats in case 2. The unfair value formula is used to determine the allocation of seats. For the new seat allocation, if any,
Then the extra seat should be given to A, and vice versa. On the inequality Rb (n 1+ 1, N2)
Introduction formula
Therefore, we know that the increased seat allocation can be determined by the maximum value of Qk, and can be extended to the general situation of multiple groups. The method of determining seat allocation with the maximum value of Qk is called Q-value method.
The Q-value method of multi-group (M-group) seat allocation can be described as:
1. First calculate the q value of each group:
Qk，k= 1，2，…，m
2. Find the maximum Q value Qi (if there are multiple maximum values, choose one of them).
3. Assign the seat to the I group corresponding to the maximum Q value Qi.
Solution of the model:
Allocate according to the integer part that should be allocated first, and allocate the rest according to the Q value. The integer quota for this issue * * * allocates 19 seats, specifically:
a 10.8 15n 1 = 10。
B 6.6 15 n2 =6
C 3.570 n3 =3
For the allocation of the 20th seat, calculate the Q value.
q 1 = 1032/( 10’ 1 1)= 96.45； Q2 = 632/(6′7)= 94.5； Q3 = 342/(3′4)= 96.33
Because Q 1 is the largest, the 20th seat should be given to the first department; For the allocation of 2 1 seat, calculate the q value.
q 1 = 1032/( 1 1 ' 12)= 80.37； Q2 = 632/(6′7)= 94.5； Q3 = 342/(3′4)= 96.33
Because Q3 is the largest, we should give the seat of 2 1 to the C system.
(2) The final allocation of seats is: A 1 1 seat B 6, seat C 4.
Conclusion: 20 seats should be allocated as follows: A 10 seats, B 6 seats and C 4 seats.
If there are 2 1 seats, then there should be 1 1 seats in Part A, 6 seats in Part B and 4 seats in Part C.