Class Assignments

*Just need formulation, not solution*

The Salanter Akiba Riverdale (SAR) Academy is a coeducational, private Modern Orthodox Jewish day school located in New York City. Every summer, the SAR Academy must create class assignments for their elementary school students. Each grade of 80-100 students must be divided into different classes. Requests for assignments are made by parents, teachers, and school therapists. These requests include pairs of students that should be placed together, pairs of students that should not be placed together, and requests for students to be placed in classes that better suit their unique or special academic needs. These requests often conflict with each other, and it falls on the administration to prioritize which requests should be fulfilled over others. In this problem, we consider a simplified version of the problem faced by the SAR Academy with 40 students and two classes. The parents or guardians of each of the 40 students are asked to submit preferences for class 1 or class 2. These preferences often depend on the teaching style of the teachers, the teachers older siblings have had in the past, and characteristics of the class (one class is called an inclusion class, which is better for students with academic needs). The parents give a ranking of 1 to the class they prefer (their first choice), and a ranking of 2 to their second choice. This data, as well as the gender of each of the students, is given in the spreadsheet ClassAssignments.xlsx. For the following problems, you will formulate integer optimization problems but you dont have to solve them. For more concise description, you can use the following notation: let ai {1, 2} be the parent preference for Class 1 (i.e., second column in the dataset), bi {1, 2} be the preference for Class 2 (third column), for i = 1, , 40.

(a) The problem faced by the SAR Academy is to decide which students should be assigned to which classes, to satisfy as many of the parent preferences as possible (note that since smaller numbers for the preferences are better, this is a minimization problem). Each student must be assigned to exactly one class, and there should be exactly 20 students in each class. Formulate this problem as an integer optimization problem, and describe the decision variables, objective function, and constraints.