# A small city in the northeast must racially balance its 10 elementary schools or sacrifice all…

A small city in the northeast must racially balance its 10 elementary schools or sacrifice all federal aid being issued to the school system. Since the recent census indicated that approximately 28% of the city’s population is composed of minorities, it has been determined that each school in the city must have a minority student population of 25% to 30% to satisfy the federal definition of ‘‘racial balance.’’ The decision has been made to bus children in order to meet this goal. The parents of the children in the 10 schools are very concerned about the additional travel time for the children who will be transferred to new schools. The School Committee has promised these parents that the busing plan will minimize the total time that the children of the city have to travel. Each school district is divided into 2 zones, one which is close to the local school and one which is far from the school, as shown in Fig. E12.1.

The School Committee has also promised the parents of children who live in a ‘‘close zone’’ that they will attempt to discourage the busing of this group of children (minority and nonminority) away from their present neighborhood school. The School Committee members are intent on keeping their promises to this group of parents.

An additional problem plaguing the Committee is that any school whose enrollment drops below 200 students must be closed; this situation would be unacceptable to the Mayor and to the taxpayers who would still be supporting a ‘‘closed school’’ serving no one.

a) Formulate the problem as a linear program. [Hint. To discourage the busing of students who live close to their neighborhood school, you may add a penalty, p, to the travel time to any student who lives in the close zone of school district i and is assigned to school district j (i 6= j). Assume that a student who lives in the close zone of school i and is assigned to school i does not have to be bused.]

b) There is only a small-capacity minicomputer in the city, which cannot solve the linear program in its entirety. Hence, the decomposition procedure could be applied to solve the problem. If you were a mathematical programming specialist hired by the School Committee, how would you decompose the program formulated in part (a)? Identify the subproblem, the weighting program, and the proposal-generating program. Do not attempt to solve the problem.

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