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The Elimination of Racially Identifiable Schools∗

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The Elimination of Racially Identifiable Schools∗
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  Proflssiot M Gqfiii/W 32 4), 1il0 · 41~42 © Copyrlglll . by AUodltiOn ol Amlricln eograpllln Recent federal court decisions have emphasized the need to eliminate schools whose racial composition varies from that of the whole district by more than a fixed percent. A linear programming model is presented to assist school administrators in developing desegregation plans that comply with these guidelines. An efficient solutional technique that exp o ts the spec1al structure of this model increases problem-size capabilities. A study of the Columbus City School District examines the tradeoffs involved at different levels of desegregation. THE ELIMINATION O RACIALLY IDENTIFIABLE SCHOOLS* Mark Woodall The Ohio State University Robert G. Cromley University of Kentucky R Keith Semple The Ohio State University Milford B. Green  The Ohio State University SINCE the Supreme Court's decision in 1954 ruled that dual educational systems for blacks and whites were inherently unequal [2], federal courts in conjunction with the Department of Health, Education and Welfare have gradually formulated a set of guidelines for the desegregation of public schools. Although the actual implementation of desegregation plans was initially very slow, various branches of the federal government have initiated a vigorous desegregation program since 1968. The emphasis of the late 1960's on speed in establishing unitary school systems meant that many school districts would be required to bus students. In Swann v Charlotte Mecklenburg the Supreme Court ruled that busing was a egitimate and effective technique for accomplishing desegregation [ 16 ]. The court reasoned that busing school children was nothing new to school districts and that many students had previously been bused to maintain dual school systems. Nothing in the history of the American educational system has created the furor that has been generated by the school busing issue during the past decade. However, judging from the accumulation of precedents favoring school desegregation at district and possibly metropolitan-area scales, it seems reasonable to expect that the trend will continue toward mandatory student transfers in school systems that are ruled segregated. In light of the Court's rejection of voluntary desegregation plans, the busing of school children should' remain an important part of desegregation plans in the foreseeable future. Therefore, it is incumbent upon school administrators to seek methods that assign students to schools in a manner that efficiently and effectively desegregates the schools. To help solve the assignment problem, geographers and other researchers have proposed • The author i w•sh to rhank Gyula a~ for final preparatiOn of all dii gr11ms. 4 2  VOL. 32, NUMBER 4, NOVEMBER 1980 413 a variety of optimization models to allocate students to appropriate schools [ 72]. The basic design of these models is to achieve some measure of racial balance at each school while minimizing the overall distance traveled by students. Several authors have formulated the student-assignment problem as the classical transportation problem. Yeates was the first to use a distance-minimizing approach to analyze high school attendance zones in a rural Wisconsin school district, but he made no attempt to achieve racial balance [ 78]. Franklin and Koenigsberg incorporated racial balance constraints by placing upper limits on black and white enrollments at each school [5]. McDaniel used their formulation to generate a desegregation plan for the Milwaukee school district [ 3 ]. These efficiency-conscious models can solve very large problems but may not be in keeping with the spirit of many of today's desegregation orders, which emphasize the elimination of racially identifiable schools. Racially identifiable schools are defined as those schools which vary from the racial composition of the whole district by more than a fixed percent. lower limits on enrollment as well as upper limits are needed to insure adherence to these guidelines. However, most studies incorporating both upper and lower limits on black and white enrollment use standard linear programming packages such as MP.S or OPHELIA to solve the resulting problem [4, 8 9, 11 75]. Consequently, these studies generally suffer computational limitations that restrict the problem size of any study. Belford and Ratliff have improved the computational efficiency of the upper-and lower-bounded models by using a network flow formulation [1 ]. They used the out-of-kilter algorithm to develop a desegregation plan for Gainesville, Florida. This study proposes an alternative solutional technique to eliminate racially identifiable schools from Columbus, Ohio. ackground Information The Columbus City School District was chosen for empirical analysis because it represents one of the largest school districts in the country that currently faces desegregation orders. In March 1977 U.S. District Court Judge Robert M. Duncan found the Columbus City School District (CCSD) willfully operating under segregative conditions [74]. In reaching his decision, Judge Duncan cited instances of attendance-zone gerrymandering, attendance-zone discontiguities, and optional attendance zones-among other t m~ - signed to preserve a system of racially imbalanced schools. Judge Duncan was particularly concerned with the large number of racially identifiable schools in Columbus. In this instance, he defined as racially identifiable those schools which vary from the racial composition of the whole district by more than 1 5 percent. For example, black elementary students comprised 32 percent of the total 1976 elementary enrollment in the CCSD, so that any elementary school with less than 17 percent black enrollment or more than a 47 percent black enrollment would be termed racially iden tifiable. According to provisions outlined in the decision, any remedial plan submitted to the court must essentially remove all racially identifiable schools from the system. Practical reasons limited this study to an analysis of high school attendance patterns. The census is the only easily available government report that maintains a reasonably accurate count of school-age students. However, census reports do not distinguish between elementary and junior high school populations, although they do categorize the high school population by census tract. Model Formulation and Variables To comply with the court's decision, the problem requires that students be assigned to schools such that: ( 1 each school is racially unidentifiable; 2) every student is assigned to a school; 3) total distance traveled by students is minimized; and 4) student assign-  414 THE PROFESSIONAL GEOGRAPHER ments to each school do not exceed permissable enrollment. The problem may be formulated as the following linear program: ,,. i j subject to: . Loa; : 10 L x• l O; Uoa; j • . i j 1 - u.la; ,;;; L X ;j ,;;; 1 - Lola; where Z = total student miles, vi. Vi Vi vi, j x•u = black student flows from tract j to school i, X u = white student flows from tract j to school i, m = number of schools, n = number of tracts, du = distance from trac:;t j to school i, a; =total enrollment of school i f = number of black students in tract j 8 '   = number of white students in tract j 1) 2) 3) 4) 5) (6) L• = minimum percent of black students to be enrolled at all schools (a constant), u. = maximum percent of black students to be enrolled at all schools (a constant). The objective function ( 1) mi.nimizes total distance traveled by all students in the system. Equations (2) and 3) are the demand requirements for black and white students respectively. Upper and lower bounds on black student enrollment at each school are specified in Equation 4), and bounds on white enrollments are given in Equation (5). Equation (6) is the standard non-negative flow condition. There are four basic parameters in this problem. These correspond to racial-mix criteria at each school, the enrollment of each school, student-population distribution by race by district subdivision, and measurements of transit effort between schools and subdivisions. Racial-mix criteria are the easiest to define in the Columbus case. In interpreting the court's decision, it is clear that each school may depart no more than 15 percent from the system wide racial coefficient, which in the case of high schools is 26.7 percent. Therefore, upper bounds for black enrollments were established at 41.7 percent of total school enrollment and lower bounds at 11.7 percent of school enrollment. School enrollments were defined as total enrollments in 1970 and were obtained from Plantiffs Exhibit 390 from the Columbus court proceedings. Perhaps the best method for identifying student residential locations would be to divide the school district into grid cells with student tallies in each cell [18). However, this method requires student-locator  VOL. 32, NUMBER 4, NOVEMBER, 1980 sc TltM TS OOU 1 3 1 J 1 a . B~ 2 J 8lii(IC a• I 4 1 a· ~ • Numbe r o black stu dents in census uact J ~• Nvtnber o whtte students' c.. sus tract BLB • llu lower bound on enrollment ot Khool 1 2 B• I WHIT J a· 4 a; IUI,-81.1, IUI 1 -tlll 1 IU8 1 -IL8 1 BUB • Blacll ulll)er bound on enrollnwnt ot Khooll WUB, • Whiteupperboundonenrol '-ot>ehooll Arbitrarily hiah trons tioft ca.t 415 BUB BUB 1 BUB 1 wua, wua, wua, Figure 1. The bounded transportation problem structure (four srcins by three destinations). maps, which were not available for the CCSD. Residential distribution data by race were obtained from the 1970 Columbus SMSA census tapes (17]. The tapes give high school population by tract directly, and there was no need to use a more cumbersome age-group method of computing source data [8). Reconciling tract boundaries with district boundaries did present some problems. Where part of a tract extended outside district boundaries, the entire tract was simply included in the analysis. Usually the outlying area was so small that this method would not significantly bias the results. Several methods of measuring transportation cost have been used in studies that apply linear programming techniques to desegregation problems. Transit-time measurements reflect a truer picture of the transportation network but are difficult to obtain unless a public transportation system is used or unless the problem is very small. Transit effort in this study was therefore defined simply as the linear distance between census tract centroid and school site. A conventional method found in the literature, the linear distance measure, is easier to compute and interpret than other distance metrics. Additionally, linear distances are sensitive to a squaring technique that makes the resulting attendance zones more contiguous and compact [10, 7 7). Solutional Technique The formulation above has bounded capacity requirements, therefore it is not, strictly speaking, a classical transportation problem. Solution by a standard linear programming package would involve excessive computational time. However, this formulation is a special case of the bounded transportation problem [7). By structuring the problem as shown in Figure 1 any transportation algorithm may be used to solve the problem efficiently [3,  6]. In Figure 1, two blocks of variables correspond to black and white alloca-  416 THE PROFESSIONAL GEOGRAPHER TABLE 1 ACTUAL VS. MODELED BLACK ENROLLMENTS Actu•l Ophm•l 1970 1970 Bl•ck ...,_ Bl ck En- rollmrnt rollrrwnt School I I> I I ll 1 Brookhaven 1 3 0 0 2 Central 33 3 33 6 3 East 98 1 89 9 4 Eastmoor 18 4 33 5 5 Linden-McKinley 62 2 56 5 6 Marion-Franklin 28 8 28 6 7 North 10 4 7 7 8 Northland 0 3 0 4 9 South 35 9 40 4 10 Walnut Ridge 0 5 2 1 11 West 14 1 13 9 12 Whetstone 0 3 0 0 13 Mohawk 66 4 63 0 tions. The cells of the tableau that have been marked out represent impossible black white) student allocations to white black) student enrollment at each school. These cells are assigned arbitrarily large transportation values so that these activities will not enter the solution. A third block of slack variables makes the bounded formulation capable of solution by the transportation algorithm. The demand requirement for each column in this block does not represent excess capacity as in the classical transportation problem but instead refers to the surplus between the lower limit of black white) student enrollment and the upper limit of black white) student enrollment at each school. A value of zero is assigned to the active two slack variables in each column. These slack values insure that there will be an even substitution of black for white students. If black enrollment is at its upper limit at a particular school, then the white slack variable in that column must absorb the entire surplus and white enrollment will be at the lower white limit The capacity for either racial group at each school is expressed as the upper limit of enrollment, forcing activity in the slack variables. Thus capacities are met, but the proportion of blacks to whites will vary only within the prescribed bounds at each school. nalysis To assess the effect of constraining enrollments racially within the CCSD, a simplified version of the model described above was used to allocate students without regard to race. This is easily accomplished by replacing Equations 4) and 5) with a simple constraint on school enrollment and solving the resulting transportation problem. As seen in Table 1, black enrollments in the optimal solution closely approximated actual black enrollments, although some variation is expected owing to the attendance-zone irregularities described by Judge Duncan. Eight of the thirteen high schools are clearly racially identifiable. The total cost of the optimal solution is 35,270.7 student-miles. A higher proportion of black students than white students requires transportation 83.4 percent to 77.3 percent), and blacks, on average, must travel slightly farther to school than whites 2.39 miles to 2.22 miles). These figures and others summarized in Table 2 provide a useful reference in comparing the racially constrained solutions. The racially constrained model, the formulation of which appears in the previous sec-
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