The U.S. Supreme Court decision of 1954 on de jure segregation of schools, and recent decisions denying de facto segregation and barring ''freedom-of-choice'' pupil assignments, have forced school districts to devise plans for integrating public schools. Finding a feasible method of achieving racially balanced schools is, at best, difficult. A great number of factors must be taken into consideration. For a given school district the following is known. There are G grades and J schools. Each school has a capacity Cjg for grade g. In each of I neighborhoods in the district, there is student population Sig for neighborhood i and grade g. Let the distance from neighborhood i to school j be represented by di j.
a) Formulate a model to assign all students to schools, minimizing total distance.
b) Now let Sikg = the number of students in neighborhood i of race k and grade g; ak = the maximum percent of racial group k assigned to a school; and bk = the minimum percent of racial group k. Reformulate the model while also satisfying the racial-balance constraints.
c) Minimizing total distance might lead to some students having to travel great distances by bus, while others would be within walking distance. Reformulate the problems in (a) and (b) to minimize the maximum distance traveled by any student.