Assignment 1

1. Develop a linear programming model to produce a busing plan for New Haven junior high schools and implement it in GAMS. Run the linear program with and without the goal of integrating the school system. How did you determine the level of integration which was to be accomplished by your programs?
2. Interpret the output of your linear program. How could the Board make use of the results in planning the current year's busing schedule. How could the results help them plan for future years? (Briefly interpret all of the relevant output elements - objective function and variable values, reduced costs, slack, dual prices, and sensitivity analysis)
3. Given the multiple objectives of the Board suggested throughout the case, how did you decide which would be objectives and which constraints in the formulation of your linear program. In the Supreme Court decision in the case of Bakke vs. The University of California it was concluded that precise racial quotas were unconstitutional while trying to achieve racial balance as a goal was advisable. Is your formulation constitutional? In anticipation of the inevitable court cases which will follow its implementation, prepare a short defense of your formulation. Interpret the Supreme Court's decision in relation to linear programming formulation.
4. After analyzing your busing proposal, a citizens protest group suggests that the school enrollments should not be permitted to vary over the full twenty percent range.

   In the past, the only reason the schools varied from the optimal levels was that the schools drew from a fixed set of neighborhoods. Now that you vary the neighborhoods which feed into each school there is no reason to let the schools fluctuate over such a large range. We recommend, therefore, that you rerun your linear program permitting only five or ten percent maximum variance about the optimum values.

   What affect do you think the reduction in enrollment tolerance would have on the results of the linear program, given that the Board must limit the total busing distance in order to keep costs under control? The rulings of the Supreme Court are rather vague in defining what constitute reasonable efforts at desegregation. Would limiting the enrollment tolerance be considered unreasonable? Would the influence that limiting the school enrollment tolerance has upon integration even be noticed?
5. Propose a goal programming formulation of the busing problem. In your formulation, which goals would be used as objectives in the program and which as constraints? What justification is there for keeping any of the goals as constraints rather than weighted objectives? Now that the weights for each objective can be separately determined, how would you go about this in such a way that it can be justified to the public, the government, and to the district administrators and teachers?
6. Having implemented your linear program and integrated the schools, the white students leave the public schools in the phenomenon dubbed ``white flight." The end result is that the levels of segregation have actually increased, with each school now being comprised of 85-90% minority students. Have you in fact implemented an integration solution? Consider the problems associated with working on a problem where all of your solutions influence the parameters of the problem. How could you effectively forecast the actual, eventual integration achieved by your integration proposals? Would it be possible (functionally or politically) to recommend a solution which would maximize eventual integration, although it achieved only very limited integration over the short term?