So&-Econ. Plon Scl. Vol IO, pp 223-226 Pergamon Press 1976 Printed m Great Bntam

A STUDY IN USING LINEAR PROGRAMMING TO DESIGN NON-URBAN ATTENDANCE AREASI-

PATRICK McKEOWN School of Business, State University of New York at Albany. Albany, N.Y., U.S.A.

and

BRIAN WORKMAN The First National City Bank, New York. N.Y., U.S.A.

(Received 26 January 1976; revised 4 June 1976)

Abstract-A great deal of recent work has been directed toward using mathematical programming to achieve racial balance in urban schools. However, with Ihe ever increasing cost of fuel for buses, there also appears to be a need to apply these techniques in non-urban school districts to reduce the use of fuel. This paper describes the development of a linear programming model to assign students to schools in such a way, as to minimize student-miles. Using data gathered in a non-urban school district, this model was tested on a large scale linear programming package with encouraging results. The resulting computational results are presented and discussed along with comments on the additional work necessary to implement this model.

I. INTRODUCTION

Numerous recent papers have dealt with the problem of achieving racial balance in predominately urban school districts[l, 21. These models have used various mathemat- ical programming procedures to devise plans for assigning students and/or designing bussing policies. In contrast, there does not seem to be a great deal of literature dealing with the problems of the predominately non-urban school district. Though these types of districts also have some racial balance problems, a major concern tends to be one of efficiency of transportation. This is especially true in many of the consolidated school districts of upstate New York. These districts usually have several elementary schools and often cover large geographical areas. It is also a policy of these school districts to use bussing to transport the great majority of the students. For these reasons, a major point of interest for the school administrators is the rising fuel costs. Of secondary interest is the falling attendance levels and the resulting need to efficiently use the available classrooms or to decide on schools to close.

This paper describes a study into using a large scale linear programming routine to assign students to schools in such a way that the total student-miles traveled in the district were minimized. A suburban upstate district cooperated in this study by providing data on their students. The results of this study were extremely promising. However, due to data collection problems, the results are not conclusive. Also, due to the lack of outside funding, it was not possible to run a follow-up study and to verify and possibly implement these results. This study is being used as the basis for grant proposals to enlarge and to evaluate these preliminary results.

Section II of the paper will discuss the mathematical formulation and background of the linear programming technique used here. A discussion of computational methods is given in Section III, while results are given and

tThe research which this paper is based on was a part of Mr. Workman’s MBA Management Science Field Project and was directed by Professor McKeown at the School of Business, State University of New York at Albany.

discussed in Section IV of the paper. Finally, conclusions and suggestions for future research are given in Section V.

II. MATHKMATICAL F’ORhfULATION AND BACKGROUND

The problem of efficiently assigning students to schools can be considered to be a case of the public location problem discussed by Revelle and Swain[4]. They proposed the following formulation which we will term Pl to minimize distance traveled by clients to K service centers by determining the optimal choice of the centers among the n(> K) total centers of demand.

Min 2 2 a&X, ,=1 ,=,

s.t. 2 x, = 1, i=l,2 ,...,n, (Pl) ,=1

X,,ZX,,, i,j=l,2 ,..., n, i#j,

2 X, = K, ,=I

X,, 20, i,j = 1,2,. . , n.

where a, = the demand population at node i, d,, = the shortest time (distance) from node i to node j, X,, = 1 if node i assigns to a center at j, 0 otherwise, n = the number of nodes of the network j and K = the number of central facilities desired (where nodes refer to centers of demand).

The first constraints force each center to assign all of its population to some service center while the second set of constraints insure that population demand is assigned only to “open” service centers. Finally, the last constraint restricts the number of service centers. Revelle and Marks state that this LP problem has yielded the required binary values of X,, with “great regularity”. Note, however, that there will be n* - n constraints of the second type. Hence, for large scale problems, i.e. n E 50, the problem tends to become difficult to solve on the large scale linear programming packages, i.e. IBM’s MPS/360. This is caused by a combination of size and the high

223

224 PATRICK MCKEOWN and BRIAN WORKMAN

degree of degeneracy present in this type of problem. To get around this problem, Evans[3] suggested some

constraint reductions in his study of jail location in North Carolina. He did this by defining a set P, as follows: P, = fi: location j can serve host for location i}. He did this by reasoning that one can safely assume that not all possible centers should be considered as possible servers for each demand center. Two criteria were used to determine Pi. Criteria 1 says that if the n possible centers that can act as a host for center i are ranked in nondecreasing order in terms of cost (a,d,,), then F, = IHI, El,. . . , [n -k + 11). This simply states that center i need not be served by any center with “cost” worse than that of the (n - k + l),, such center. This will save nk constraints. For the second criteria, we note that the costs are of the form

C, = ad,

where d,, = unit cost to go from i to j and a, = total demand located at i. In Evans’ study, the d, were geographical distances and the a,‘s measure number of individuals to be moved. Hence, it can be assumed that the counties of the study (population centers) could be grouped in such a way that no county is considered to be served by another county over 100 miles away. It is possible but extremely improbable that such a procedure could exclude the optimum location using this criteria. The use of the two criteria reduced the number constraints for n = 100 from the needed by Pl of n* - n + n + 1 = 10,001 to less than 700. The resulting formulation is given below.

Min 2 c aid& ,=1 ,EP,

s.t. $X,,=l, i=l,..., n, ,=I

J&Z&, jEP,-{i}, i=l,..., n, (P2)

2 X,, = K: ,=I

X,20, jEP,, i=l,..., n.

It is a modification of the P2 formulation which we will - use in our student assignment model. In this case there are a set number of predetermined servers (i.e. schools) which must be used. Hence, the last constraint which insures that K servers are chosen from the n centers may be dropped. However, we need to add an additional constraint that limits the number of students assigned to a given school in each grade. This was done with the realization that the integrality of the X1,% may be violated. Previous computational results with Pl and P2 have shown this not to be a problem in those cases[3,4]. In our case, if we redefine X,, to be the fraction of demand at center i served by j then the X,‘s need not necessarily be integral for the model to use as a first approximation.

For any given grade, the model which we used is given below. The model was run grade-by-grade. The problems inherent in doing this are discussed later:

Min 2 c SF& i=l ,EP,

s.t. xX,,==, i=l,..., n, IEP,

x,-X, 20, jE{P,}, i = 1,. . . ,n,

2 S,X, 5 b,, j = 1, . . , m, ,=I

(P3)

X,,ZO, jEP,, i=l,..., n.

where S, = the number of students at bus stop i, F,, = the distance from the bus stop i to school j, b, = the number of students allowed at school j, and X,, = the fraction of students at bus stop i who attend school j.

Several items should be noted in this formulation. First, the value of b, for each grade is a function of maximum section size, i.e. 22 for kindergarten and 30 for grades l-5, and the number of sections at school j. Secondly, we are using bus stops as centers of population. In many cases, we clustered bus stops to reduce the number of constraints. By doing so and using the P, concept of Evans model p2, we are able to reduce the 190,000 constraints that would be needed for the 440 bus stops under PI to a manageable level. Thirdly, a fractional X, would indicate that a bus stop would be split between schools. This obviously is not a practical situation and would have to be remedied by some sort of heuristic procedure. In our case, 31 out of the 440 bus stops were split between schools. Finally, since we are using the model on a grade-by-grade basis, it could easily happen that different grades at the same bus stop would go to different schools. This would also call for a postoptimal heuristical trade-off procedure or possibly a revision of the procedure to consider grades, i.e. an X,ti variable. Future work will look at the latter possibility.

rIL USING THE MODEL

With z formulation as a model, we proceeded to gather data in the district under consideration. This was done by first locating and clustering bus stops. These clustered bus stops were then pinpointed on a map and distances were determined to the nearest two or three schools (out of six possible schools). This gave us the F,, values. It was then necessary to determine how many students in each grade were located at each bus stop. This data was taken by the bus drivers as they made each stop on their respective routes. As we shall see later, this data gathering procedure proved to be less than desirable. This gave us the S, values.

With this information, we then ran the model without the class-size constraints in order to determine where the natural groupings for each school fell. Using these natural groupings and the maximum section size, sections of each grade were assigned to each school. Using these assignments along with administratively set maximum class sizes, it gave us the b, values for each school for each grade. An alternative procedure here would have been to use administratively assigned sections. However, this was not deemed necessary in this case. With the class size constraints now in place, we then ran the model P3 - for each grade.

rvv. RESULTS

In order to compare our results with the present status of the school district under consideration, we will present data on the present situation first. The district has six elementary schools, which we term schools A-F. Table 1 gives the present enrollment per grade by school while Table 2 gives the present number of classes required. Finally, Table 3 shows the unused capacity in existing classes.

A study in using linear programming to design non-urban attendance areas 225

Table 1. Present assignments of students by grade

Grade

School K I 2 3 4 5

1. A 29 18 24 22 15 26

2. B Li 0 44 43 35 37

3. c 34 46 0 0 0 0

4. D 55 45 51 41 24 0

5. E 53 39 49 51 86 I69

6. F 45 55 47 55 55 0

Totals 216 203 215 212 215 232

Table 2. Present number of sections for each grade level

Grade

school K I 2 3 4 5

1. A 2 I 1 1 I I

2. B 0 0 2 2 2

3. C 2 2 0 0 0 0

4. D 3 2 2 2 1 0

5. E 3 2 2 3 6

6. F ? 2 2 2 0

13 9 9 9 9 9

Total 58

Table 3. Unused capacity in existing classes

Grade

School K 1 2 3 4 5

1. A 15 12 6 8 15 4

2. B 0 0 16 17 25 23

3. c 10 14 0 0 0 0

4. D 21 15 9 19 6 0

5. E 13 21 II 9 4 II

6. F 21 5 13 5 5 0

Totals 80 67 55 58 55 38

As mentioned earlier, the data on number of students

per grade at each bus stop was taken by the individual bus drivers. This method of data gathering proved to be something less than satisfactory. In grades K and 1, we found that we were missing 37 and 21% of the students, respectively. In the upper grades, we were missing, on the average, 12% of the students. This was felt to be a low enough rate to be attributed to absenteeism and for this reason, we shall concentrate our discussion on grades 2-5.

As mentioned earlier, the first run of the model was made without the class size constraints. The results of that run are shown in Table 4.

Using the Table 4 results, the grade sections were heuristically assigned to each school and the class size constraints put into the model. The results of running the model with class size constraints in place are shown in Table 5. The values in parentheses are the number of sections for that school for each grade.

Table 4. Unconstrained assignments

Grade

School K 1 2 3 4 5

1. A 24 22 33 25 32 28

2. B 13 23 20 28 19 20

3. C 1 7 7 9 3 3

4. D 32 32 39 27 36 56

5. E 52 50 66 63 67 66

6. F 14 26 23 29 31 37

Totals 136 160 188 181 187 210

Table 5. Class-size constrained assignments

Grade

School K 1 2 3 4 5

1. A 22 (1) 23 (1) 30 (1) 25 (1) 30 (1) 30 (1)

2. B 14 (1) 30 (1) 29 (1) 25 (1) 21 (1) 30 (1)

3. C 0 (0) 0 (0) 0 (0) 9 (1) 0 (0) 0 (0)

4. D 34 (2) 17 (1) 40 (2) 29 11) 26 (1) 60 (2)

5. E 44 (2) 54 (2) 60 (2) 60 (2) 80 (3) 60 (2)

6. P 22 (1) 26 (1) 29 (1) 30 (1) 30 (1) 30 (1)

Totals 136 (7) 160 (6) 188 (9) 181 (7) 187 (7) 210 (7)

Given these class assignments, we may see that the number of sections required for grades 2-5 is 28 as compared to 36 at present. The only grade which is at capacity under the model is grade 5. This would indicate the need for extra sections in one or more schools to handle the 22 students not included in the data collection. Even with these additional sections, it would still appear that the model has reduced the number of sections required substantially for these four grades. We also have an indication that school C should be closed since the model assigns only nine students there. In times of decreasing enrollment, this might be a method determin- ing which schools to close.

In terms of student-miles traveled, the present data indicates that the students covered in the survey generate 907.57 student-miles. By the use of model. we found that the student-miles generated would be 460.14. In both cases, the distance was measured along the shortest road route to the assigned school.

V. CONCLUSIONS AND DIRECTIONS FOR FUTURE WORK

Due to the missing data in some grades, it is not possible at this time to state any firm conclusions. However, there definitely does seem to be some indications that a model such as the one discussed here could be useful for planning by school administrators. We realize that this model does not take the political constraints into account and for that reason, it might be considered as being “impractical”. Our response to this would be that with the increasing resistance to new taxes, administrators are going to be pressed into using the most cost efficient plan regardless of the political problems involved. In addition, a mathematical model such as the one discussed here could serve as the starting point from which a search could be begun to find an alternate solution which would more closely satisfy constraints other than those specified in the model. Such a procedure would have a higher probability of arriving at a solution that would yield lower costs while still satisfying political constraints than a procedure which was not based on a mathematical procedure.

Three areas of future research present themselves immediately. First, an expanded and improved data collection procedure should be used to do a follow up study on this same school district and possibly other such districts. A grant has been requested for this purpose. Secondly. the resulting assignment scheme should be modified to yield an implementable assignment plan. Finally, the model itself should be revised to involve variables which consider the grades as well as schools, i.e. an X,,, variable.

226 PATRICK MCKEOWN and BRIAN WORKMAN

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2. S. Clarke and J. Surkis, An operations research approach to 4. C. Revelle and R. Swain, Central facilities location, Geogr.

racial resegregation of school systems, Sot. Econ. Plan. Sci. 1, Anal. 2. 30-42 (1970).

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