1. EMIS 8373: Integer Programming Mathematical Programming Example 5: Reallocating Students to Achieve Racial Balance updated 18 January 2005

2. slide 1 Decision Variables Let x ij be the number white students reassigned from school i to school j. Let y ij be the number nonwhite students reassigned from school i to school j. Let w i be the number white students in school i after reassignment. Let n i be the number nonwhite students in school i after reassignment. Let t i be the total number students in school i after reassignment.

3. slide 2 Constraints The following constraints ensure that all the students are reassigned to a school:

4. slide 3 Constraints Continued The following constraints the total number of white and nonwhite students in each school after reassignment:

5. slide 4 t 1 = w 1 + n 1 ( 21 ) t 2 = w 2 + n 2 ( 22 ) t 3 = w 3 + n 3 ( 23 ) t 4 = w 4 + n 4 ( 24 ) t 5 = w 5 + n 5 ( 25 ) 850 · t 1 · 1050 ( 26 ) 680 · t 2 · 840 ( 27 ) 425 · t 3 · 525 ( 28 ) 1020 · t 4 · 1260 ( 29 ) 425 · t 5 · 525 ( 30 ) Constraints Continued The following constraints determine the total number of students in each school after reassignment: The following constraints enforce requirement (2):

6. slide 5 w 1 ¸ ( 0 : 45 ) t 1 ( 31 ) w 2 ¸ ( 0 : 45 ) t 2 ( 32 ) w 3 ¸ ( 0 : 45 ) t 3 ( 33 ) w 4 ¸ ( 0 : 45 ) t 4 ( 34 ) w 5 ¸ ( 0 : 45 ) t 5 ( 35 ) w 1 · ( 0 : 65 ) t 1 ( 36 ) w 2 · ( 0 : 65 ) t 2 ( 37 ) w 3 · ( 0 : 65 ) t 3 ( 38 ) w 4 · ( 0 : 65 ) t 4 ( 39 ) w 5 · ( 0 : 65 ) t 5 ( 40 ) Requirement (1) The proportion of white students must fall in the range of 45% to 65%.

7. slide 6 Variable Domains

8. slide 7 Objective Function: Total Distance Traveled by Bussed Students Minimize (46) subject to (1)-(45).