EMIS 3360

Mathematical Programming Examples Continued

Example 5: Reallocating Students to Achieve Racial Balance.

To comply with recent judicial guidelines on racial composition, a school district must reallocate its white and nonwhite high school students to its five schools. This reallocation will be achieved by busing students. At present, all students attend the school in their neighborhood, giving rise to the racial imbalance. The school district's Board of Education (BOE) has compiled the following data on the enrollment of the five schools and the district as a whole:
 
 

School Total Enrollment Nonwhite Students White Students % White
1 1000 200 800 80
2 800 200 600 75
3 500 50 450 90
4 1200 900 300 25
5 500 450 50 10
District 4000 1800 2200 55

 
  1. The proportion of white students in any school must fall within 10% of the proportion of white high school students in the entire cite (i.e. 55%).
  2. The enrollment in any school cannot increase by more than 5% nor decrease by more than 15% of the present enrollment.
  3. White students can be assigned only to their present school or to a school that is currently predominantly nonwhite and nonwhite students can be assigned only to their present school or to a school that is predominately white.
  4. The objective is to minimize the total distance traveled by the bused students (as measured by the distance from a student's former school to his or her new one). The distances between the schools (in miles) are given in the following table:

 
 
  To School 1 To School 2 To School 3 To School 4 To School 5
To School 1
0
3
5
2
2
To School 2
3
0
1
4
2
To School 3
5
1
0
5
3
To School 4
2
4
5
0
1
To School 5
2
2
3
1
0

 

Formulate a mathematical program that the BOE can use to determine an acceptable busing plan that minimizes the total distance traveled by the students.
 
 

Solution:

Decision variables:

Let Wij and Nij be the number of white and nonwhite students reallocated from school i to school j, respectively.

Let Ei be the enrollment at school i after the reallocation. Let EWi and ENi be the white and nonwhite enrollment at school i after the reallocation, respectively.

Objective Function:

Let dij be the distance from school i to school j as given in the table above. Since dii = 0, the total distance traveled by the bused students is given by the objective function

d11 (W11 + N11) + d12 (W12 + N12) + d13 (W13 + N13) + d14 (W14 + N14) + d15(W15 + N15) + ... +
d51 (W51 + N51) + d52 (W52 + N52) + d53 (W53 + N53) + d54 (W54 + N54) + d55 (W55 + N55).

We wish to minimize this function subject to the following sets of constraints:

  1. BOE Guideline 1
The number of white students at school j will be
EWj = W1j + W2j + W3j + W4j + W5j

The number of nonwhite students will be
ENj = N1j + N2j + N3j + N4j + N5j.

The total enrollment at school j will be
Ej = EWj + ENj.

The proportion of white students will be  EWj/Ej.

Thus, for each school j have the following constraints:

EWj = W1j + W2j + W3j + W4j + W5j

ENj = N1j + N2j + N3j + N4j + N5j

Ej = EWj + ENj

0.45 <= EWj/Ej <= 0.65.

These last constraints are not linear as written, so we rewrite them as
0.45 Ej <= EWj
EWj     <= 0.65 Ej

2) BOE Guideline 2

If E1 is greater than 1000, it means that the enrollment at school 1 will increase.
Since the enrollment at any school may not increase by more than 5%,
we must add the constraint  E1 <= 1050.

If E1 is less than 1000, it means that the enrollment at school 1 will decrease.
Since the enrollment at any school may not decrease by more than 15%,
we must add the constraint E1 >= 850.

We will have four more sets of similar constraints for the other schools.

3) BOE Guideline 3

Schools 1, 2 and 3 are predominantly white. Thus,

W12, W13, W21, W23, W31 and W32 = 0.

Schools 4 and 5 predominantly nonwhite, Thus

N14, N15, N24, N25, N34, N35, N45 and N54 = 0.

Finally, we require all variables to be >=0.