1} 2) 3} Formulate the school assignment problem as an optimization problem, incorporating capacity constraints at each school as well as constraints that all students must be assigned to a school. To balance enrollments across schools, the school district also wants to ensure that no school has enrollment less than 75% of its maximum capacity. a} Find a plan that minimizes the total distance that students must travel to school. What is the total travel distance required by this plan? How many students change schools with this plan? b} Find a plan that minimizes the number of students who will be required to change schools. What is the total travel distance required by this plan? How many students change schools with this plan? c) To balance the objectives in parts (a) and (b), find a plan that minimizes (the number of students changing schools} plus (the total travel distance). Such an objective assumes that a student changing schools has the same "cos " as commuting an extra mile to school. You should construct a linear model and allow Solver to split segments by assigning fractions of segments to different schools; don't worry about making sure an integer number of children are assigned to a school. In addition to minimizing travel distances and disruptions caused by children changing schools, the school district wants to "maintain diversity in the level of student achievement that is reasonably reflective of the school system" and "maintain the socio-economic diversity of families that is reasonably reective of the school system as a whole." (The quotes are from the charge to the redistricting committee.) Consider the following two diversity constraints: 0 17.4% of the students in the district qualify for the free-lunch program. Suppose we want each school to have between 15% and 20% of their students qualifying for the program. 0 5.5% of the students in the district have below gradelevel test scores. Suppose we want each school to have between 2.5% and 7.5% oftheir students in the "low test'I category. How do these constraints affect the total travel distance and/or the number of students changing schools? Use the combined objective function from 1c. As in the BrightLink exercise, formulate these constraints as linear constraints. For example, rather than having nonlinear constraints ofthe form: (Number of students in free lunch program at school E} (Total number of students at school i} S 20% ' instead consider linear constraints of the form: (Number of students in free lunch program at school i} - 20% (Total number of students at school i} s 0. You should assume that freelunch and lowtest students must be assigned in the same way as other students in the same segment (i.e., all students in the segment must be assigned to the same school or in the same proportions if the segment is split}. Again, don't worry about making sure an integer number of children are assigned to a school. How would adding additional capacity at each of the schools improve the plans? Please use a sensitivity report that uses the combined objective and incorporates the diversity constraints as in problem 2
