Page 1Page 2Page 3Page 4Page 5Page 6Page 7Page 8Page 9Page 10Page 11Page 12Page 13Page 14Page 15Page 161982-06 Spring 1982 SOUTHERN METHODIST UNIVI Integrating the Public School System in New Haven Maria Fatima Cupas de Mareno INTEGRATING THE PUBLIC SCHOOL SYSTEMIN NEW HAVEN SENIOR DESIGN DEPARTMENT OF OPERATIONS RESEARCH AND ENGINEERING MANAGEMENTSCHOOL OF ENGINEERING AND APPLIED SCIENCE DALLAS, TEXAS 75275INTEGRATING THE PUBLIC SCHOOL SYSTEM IN NEW HAVEN SENIOR DESIGNMaria Fatima Cupas de Moreno Prof: Richard S. BarrCONTENTS I INTRODUCTION A.- Purpose of the Model II MODEL DESCRIPTION A.- Definition of the Variables and Equations 1.-First Model 2.-Second Model 3.-Third Model III SOLUTIONS A.-Solution for the First Model B.-Solution for the Second Model C.-Solution for the Third Model IV ANALYSIS V CONCLUSION VI BIBLIOGRAPHYIINTRODUCTION I IThis case arose when the Supreme Court, in 1954, declared that "segregation of children in public schools solely on the basis of Irace, even though the physical facilities and other tangible factors may be equal, deprives children of the minority group of equal educa-tional opportunities, in contravention of the Equal protection Clause of the Fourteenth Amendment. The Court further stated that, "the Idoctrine of 'separate but equal' had no place in the field of public Ieducation, since separate educational facilities are inherently uneq-ual. (Supreme Court -- 1954) UAfter this, the New Havenhool Board began to devise plans for Iimproving the racial balance in the city's public schools.-2-A.- Purpose of the Model The purpose of this model is to integrate the New Haven Junior High Schools (4), which are serving nine neighborhoods. At the same time, the Board wants to reduce busing cost and minimize the total distance traveled by children on school buses. To achieve the construction of the model, we used the data given on tables A, B, and C, which are the distance from each neighborhood to each of the four shools, the composition of each neighborhood in terms of number of minority and non-minority students, and the schools capa-cities, respectively. II. MODEL DESCRIPTION In our model, we established our objective fuction in several forms. First, we defined it with the purpose of minimizing just the total distance traveled by children, without taking into account the race factor. Second, we developed a model in which our objective function was to minimize the total distance traveled by the children but this time we introduced the race factor within the constraints. It was develope4in this way to avoid the use of a fi integration fraction. So, modeling the problem in this way, theintegration would depend on the distance traveled by the children. Finally, we developed the model with goal programming minimizing-3-the total distance traveled by children, as well as the sum of the to-tal deviation from the ideal integration fraction (0.41). A.- Definition of the Variables and Equations 1) First Model: Xij= Number of students from the ith neighborhood assigned to the jth school Si = Number of students living in the ith neighborhood Cj = Total student capacity of the jth school Dij= Distance from neighborhood i to school j Objective Function: MINDij Xij Subject to:Xij = Si XijCj Lower limit (1) XijCj Upper limit (1) Xij0 (1) The School Board has established an optimal enrollment for each school and has tried to keep each school within twenty percent of this enrollment; for that reason we have a lower and upper limit for each school.-4-2) Second Model: Xijk = Number of students from the ith neighborhood assigned to the jth school of the kth ethnic group; k=1 for minority k=2 for non-minority Sik = Number of students of the kth ethnic group living in the ith neighborhood Dij = Distance between 11th neighborhood and jth school Cj = Total student capacity of the jth school Objective Function: Minimize >5Dij Xijk I Subject to: a)School Capacity and Optimal Enrollment Xijk ' CjLower limit Xijk4 CjUpper limit b)Availability of students in each neighborhood Xijk = Sik c)NonnegativityXijk 3b 0 3) Third Model: Xijk = Number of students from the ith neighborhood assigned to the jth school of the kth ethnic group. Sik = Number of students of the kth ethnic group living in the ith neighborhood Dij = Distance between ith neighborhood and jth school-5-Cj = Total student capacity of the jth school DIjM= Negative deviation from ideal fraction of integration in each school DIjP= Positive deviation from ideal fraction of integration in each school Objective Function: NinimizeI Dij Xijk + DIjM + DIjp =\ Subôect to: a)School Capacity and Optimal Enrollment Xijk Cj. Lower limit Xijk Cj 4 Upper limit b)Availability of students in each neighborhood Xijk = Sik c) IntegrationXijl = 0.41*Xijk + DIjP - DIjM j = 1,2,3,4 (*) 0.41 + Ideal Fraction of integration for each school in terms of minority students Ideal FractionTotal of Minority Students Total of StudentsVARIABLE X52 X64 X73 X82 X91 X93VALUE 557 427 595 246 289 52 -6-III. SOLUTION A.- Solution for the First Model: The next table will show the solution this model gave us: VARIABLE VALUE Xli264 X12277 X23337 X34368 X41227 X44169 TOTAL DISTANCE = 4781.30 Xiii73 Xi1263 X141405 X2315 X232332 X341166 X342202 X411357 X41239 X52173B.- Solution for the Second Model: VARIABLE VALUEVARIABLE X522 X641 X642 X731 X732 X821 X822 X921 X922VALUE 484 222 205 194 401 1 245 84 257 TOTAL DISTANCE = 2940.74I I I I I I I I I I I I I I I I I I IMM The next table will show the integration achieved by this solu-tion:SCHOOLTOTAL% MINORITY% NON-MINORITY 15320.810.19 211440.140.86 3'9320.210.79 412000.340.66 C.- For the third model, we tried several solutions in which we changed the coefficients of the variables in the objective function, to see the results for the different weights. 1) Both coefficients equal one. Solution: VARIABLESVALUEVARIABLESVALUE X131171X52173 X141307X522254 X14263X641222 X2315X642205 X232131X731194 X242201X732401 X321166X8211 X342202X822245 X411357X92184 X41239X922257 X512230 TOTALDISTANCE = 2940.7 TOTALDEVIATION=256.32-8-2) For this version of the solution we altered the coefficient of the total distance, increasing it by 7. The coefficient of total deviation remained constant. This gave us the following results: VARIABLEVALUEVARIABLEVALUE Xlii136X52173 X141342X522484 X14263X641222 X2315X642205 X232332X731194 X341166X732401 X342202X822245 X411357X92184