A multiperiod school location planning approach with

Environment and Planning A 2009, volume 41, pages 2929 ^ 2945

doi:10.1068/a40285

A multiperiod school location planning approach with free school choice Sven Mu«ller, Knut Haase, Sascha Kless

Transport and Economics, Technische Universita«t Dresden, Andreas-Schubert-Strasse 23, 01069 Dresden, Germany; e-mail: [email protected], [email protected], sascha [email protected] Received 24 October 2007; in revised form 20 February 2009; published online 19 October 2009

Abstract. The subject of this paper is a new approach to multiperiod school location planning in urban areas. Most of the existing approaches in the field do not consider free school choice nor are they able to consider substitution effects between school locations. We minimize the location and transport costs with respect to students choosing the school with the highest utility. Since these school choice probabilities (determined by a mixed multinominal logit model) depend on the available schools, we have to consider two steps. First, we generate for each period a set of scenarios indicating which school is open and which is not. For each scenario we allocate the students to available schools according to capacity and utility. Second, we select for each period one scenario in order to minimize total costs. Problems with a long planning horizon and a large number of demand points are solvable. We apply this approach to schools of the City of Dresden, Germany.

1 Introduction Due to low fertility rates the population has started to decline in Germany over the last decade. Particularly in the eastern parts of Germany this process has been intensified by negative net migration. Hence, there is a strong need for long-term and demandmeeting planning of public service facilities. In the City of Dresden (in the far east of Germany) the decreasing number of students has forced the education authority to close schools because of underuse. However, it is now expected that the number of students will increase in the near future. While making a decision about which school has to be closed or opened at a certain point in time, authorities should be aware of spatial ^ temporal and socioeconomic consequences. As there are numerous possible solutions for a given number of schools, demand points (blocks, for example), and periods, it will be nearly impossible to find manually an optimal solution related to a certain objective. So, sophisticated mathematical programming approaches are required. Therefore, we present a cost-minimizing approach toward school location planning with respect to dynamic number of students, choice-related probabilistic student allocation, and commuting mode choice. We refer to a probabilistic student allocation because German students are free to choose the secondary school to enroll at. That means enrollment is not mandatory as determined by the location of students. Although the literature mostly ignores free school choice (Lemberg and Church, 2000), it is evident that in recent years there has also been an emerging need for free school choice in countries other than Germany (Burgess and Briggs, 2006; Hoxby, 2003). The remainder of this paper is organized as follows. Section 2 is a literature review. Section 3 contains the construction of the model. The application of our approach and the data for the City of Dresden as well as the solution process and the computational results can be found in the fourth section. The paper finishes with some final remarks and comments on possible future research.

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S Mu«ller, K Haase, S Kless

2 Literature review 2.1 General facility location models

For a general overview of different classes of location problems see Drezner and Hamacher (2004) and ReVelle and Eiselt (2005). Drezner and Eiselt (2002) differentiate between location-allocation models and location-choice models. Within locationallocation models the demand is allocated to the facilities compulsorily (schools in the United States or fire stations, for example). The most prominent models of this class are the p-median problem (Hakimi, 1964; 1965), covering models (Current et al, 2002; Schilling et al, 1993), and centre models (Owen and Daskin, 1998). In locationchoice models customers choose the location themselves (colleges in Germany or supermarkets, for example). Eiselt et al (1993) discuss deterministic location-choice models, on the one hand, and probabilistic location-choice models, on the other hand. The former assume that the whole demand of a given demand point is totally satisfied by only one facility. The latter consider a distribution of the demand to more than one facility. Both classes of model consider a utility function (this could be linear or nonlinear, additive or multiplicative). Although probabilistic locationchoice models seem to be applicable to school location planning in Germany, these models are basically designed for a competitive framework (maximizing market share, for example). But in the narrower sense, competition is not usual for public service facility location planning (Plastria, 2001). Furthermore, facility location models for public service planning usually do not consider probabilistic demand (Drezner and Hamacher, 2004). Our approach to school location planning in Germany considers costs minimization and probabilistic demand as well as closing and opening of facilities over a finite horizon. Hence, neither probabilistic location-choice models nor facility location models for public service planning seem to be directly applicable to school location planning in Germany. 2.2 School location models

In the following discussion of school location planning approaches we do not consider school districting problems, because they do not enable us to decide whether to open or close a school. The interested reader is referred to Koenigsberg (1968), Holloway et al (1975), Lemberg and Church (2000), and Caro et al (2004), for example. Trifon and Livnat (1973) made an attempt to model decisions on minimum-cost school locations over a finite horizon. They use a linear programme so the decision whether a school should be opened at a potential site is implicitly given by capacity expansion. Henig and Gerchak (1986) present a dynamic programming approach. This model is suggested as a core of a procedure to determine the optimal number of classrooms and schools per period. On the basis of age cohorts they enumerate assignment alternatives. The model captures the dynamics of the expansion process caused by immigration. Moreover, the model is able to handle uncertain demand in the future. Therefore, the numbers of classrooms per period are considered to be random variables with a given probability (density) function. Greenleaf and Harrison (1987) used a binary programme to develop a minimum-cost configuration of open/close decisions on schools and on the assignment of students to schools over a given number of periods. Their approach considers schools to be sold and a deviation of student numbers from the capacity bounds. The model is applied to a region with dramatic decline in student numbers. Diamond and Wright (1987) regard different objectives like distance, compactness, safety considerations, utilization, and student dislocation. For a given number of schools the model determines the schools to be opened and the assignment of students (or rather demand points) to these schools. Concerning the assignment decision they apply a specific constraint in order to ensure contiguity

Multiperiod school location planning

2931

of school districts. Church and Murray (1993) presented a reformulation of this model to remove inconsistency in minimizing the deviations in capacity utilization. The dynamic modular capacitated facility location problem (DMCFLP) by Antunes and Peeters (2000) minimizes fixed and variable location costs as well as costs for capacity expansion/reduction over a finite horizon. Students are assigned to schools according to capacity and attendance costs (including transport costs). Capacity expansion/ reduction is limited to a set of predefined standards (a given number of classrooms, for example). Moreover, the state of a facility will not change more than once over all periods. They present a simulated annealing approach to solve the DMCFLP. None of the school location models discussed so far considers probabilistic demand (location choice models) and/or free school choice. In contrast, the following contributions consider the choice of a school location. Koroglu (1992) has developed a probabilistic approach to identify optimal locations for new universities in Turkey. On the basis of existing facilities the author considers combinations of new facilities. For a given combination the probabilistic demand (generated by a variation of the classic gravity model) for facilities is determined. For each combination this yields the expected market share of each facility. The objective is to select the combination with maximum demand. The work of Church and Schoepfle (1993) allows for free school choice. They do not consider demand points but individual students. The three most preferred schools per student are given. The objective of this single-period, multipleknapsack problem is to maximize the total preference values and to minimize the penalties associated with negotiated assignments of students. This is constrained by capacity, utilization, and racial balance. Antunes et al (2004) present an approach based on two mixed-integer problems which maximizes school accessibility under distancesensitive demands. The weighted distances to a given number of opened schools are minimized given a maximum utilization of schools. The objective and capacity constraints contain demand-decay factors to express the decrease in demand that occurs when the distance to facilities increases. However, this entails some students not enrolling at all. 3 A new multiperiod school location planning problem Our approach minimizes the cost of a net of schoolsöwhich is the set of open schools in a given area (a city, for example)öover a given time horizon while it accounts for maximum student utility for the resulting net of schools. In order to consider utility-maximizing school choice behaviour, we assume that a student k will choose the school s with the highest utility, Uks , such that Uks > Uk~s , 8 s~ 6 s, with Uks Vks eks . Vks is the deterministic part of utility, containing observable variables like distance to school and schools' authority, and eks is the stochastic part of utility. The resulting choice probabilities, Pks , are determined by a random utility model (in particular, a discrete choice mode). Note that other approaches, like the Huff model (Huff, 1963; 1964) and the multiplicative competitive interaction model (Nakanishi and Cooper, 1974), yield a probabilistic student allocation as well. To our knowledge only discrete choice models (for example, multinomial logit model öMNL) incorporate individual utility-maximization behaviour. Moreover, a disaggregate approach (discrete choice models) has several advantages (in transferability and efficiency) over the aggregate approach (gravity models). For more details on this issue see Koppelman and Bhat (2006). In a spatial choice context one has to be aware of (spatial) substitution patterns between alternatives and hence the ìndependence of irrelevant alternatives' property of the MNL is not valid. Thus, more sophisticated discrete choice models like the mixed multinomial logit model (MMNL) have to be applied.

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In particular, the MMNL allows for flexible substitution patterns and random taste variation within a spatial choice context (Hunt et al, 2004). Mu«ller and Haase (2009) and Hastings et al (2005) have shown that the MMNL is an appropriate approach to model the behaviour of school choice. The choice probabilities determined by the MMNL are the integrals of the well-known logit probabilities over a density of parameters (Bhat, 2000). expb T qks Pks y fb jy db , (1) X expb T qk~s s~

where y are the parameters of the density function f(.) to be estimated by maximum simulated likelihood (Train, 2003), qks is a vector of observed variables such as the distance to school or the authority (public or private), and b is a vector of parameters. The probabilities Pks will be interpreted as the empirically expected share of students located at a given demand point attending available schools. However, this leads to two problems. First, there is no explicit capacity constraint within the MMNL. Within a forecasting framework this means that as many students will attend a certain school as determined by the MMNL. Thereby, very attractive schools would be overcrowded. Second, the choice probabilities could not be implemented in the school location planning problem directly, because the decision of students of which school to choose depends on the choice set (available schools) which in turn is a result of the school location planning problem itself. Therefore, a two-step approach is required. First, for each period of the planning horizon the enrollment has to be computed for all possible combinations of open (available) and closed (nonavailable) schools with respect to capacity. We call the combination of open and closed schools the scenario. For a given scenario the MMNL has to be applied in order to determine the expected share of students attending available schools. The possible deviation between the expected share of students and the computed proportional enrollment is minimized. The result of the first step is the enrollment of students in schools according to the expected share of students and schools' specific capacity. Note that we are considering only newly enrolled students (freshers). Once the decision has been made which school to enroll at, this decision is permanent for the whole primary or secondary school career. We model the enrollment decisions and deduce the number of students and the related costs from enrollment numbers. In general, it is very unusual for students to switch from one school to another school on the same educational level: for example, from one college to another college. Around 5% of the students switch from college to high school from grade 5 to 6 while around 5% of high school students enroll at a college after grade 10. Thus, the total number of students attending a given school is known by multiplying the number of 1st or 5th grade students by the number of grades. For the elements of the German educational system see figure 1 (for example, the number of grades and student age at school entry). This approach does not allow students to change between schools due to closings during school career. For example: school A is closed in period 3. So, no enrollment for 1st grade students is allowed in school A from period 3. However, 1st grade students of periods 2 and 1 still attend this school until the end of their school career. If a school career lasts four years, school A will finally be closed in period 6. At this point in time the last student cohort has graduated and the school could be sold, for example. Unfortunately, this seems to be a limiting assumption, since in the case of a school closure a remarkable number of students (100 to 500) is forced to switch to a different school (usually the students of grades 10, 11, and 12). This problem has to be tackled in future models. The first step (student allocation) computes enrollment numbers


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Figure 1. Main features of the educational system of Germany.

according to student utility and school capacity. This is a small and easy to solve problem, but it has to be solved several times. In a second step for each period one scenario is selected. The costs of each scenario are based on the number of students, location costs and transport costs. Furthermore, the second step (scenario selection) provides information about which school is to be closed or opened in a given period. The costs of the entire net of schools over the planning horizon are minimized. This large problem has to be solved only once. 3.1 Student allocation

Let t 2 T be the periods, s 2 S the schools (this also includes potential school locations in the future), i 2 I the demand points, and j 2 J the scenarios. Sj denotes all available schools s embedded in scenario j. Jt is the set of all feasible scenarios j in period t with regard to the total number of students in period t and the total capacity of all schools s 2 Sj . The number of students located in demand point i is given by ni ; pis is the empirical expected share of students located in demand point i attending an available school s according to the MMNL; g and l are parameters which have to be specified by the decision maker in order to weight the two parts of the objective (capacity expansion and student rejection). The maximum number of students per school s is denoted by Gs , which is the product of maximum number of students per class and maximum number of classes per grade. Since we consider only newly enrolled students in this paper, school capacity is based on 1st or 5th grade students. To identify the need for capacity expansion we consider the variable ws , which is the number of students exceeding the capacity of school S. Hs denotes the accepted number of students above the capacity of school s. The variable xis represents the proportional enrollment of students located in demand point i at school s. The discrepancies between xis and pis related to capacity are denoted by zis (above) and zisÿ (below). The problem of student allocation can be formulated as follows: X ws 2 X 2 minimize g l zis zisÿ , (2) G s s2S i 2 I, s 2 S subject to xis ÿ zis zisÿ 1 , pis X xis ni ÿ ws 4 Gs , i2I

j

i 2 I, s 2 Sj ,

(3)

s 2 Sj ,

(4)

2934

and

X


xis 1 ,

i 2 I ,

(5)

xis 5 0 ,

i 2 I, s 2 Sj ,

(6)

zis , zisÿ 5 0 ,

i 2 I, S 2 Sj ,

(7)

ws 2 0, Hs ,

s 2 Sj .

(8)

s 2 Sj

The first part of the objective (2) of this quadratic constrained problem minimizes the number of students exceeding the schools' capacity relative to the schools' capacity. The second part minimizes the deviation from the empirical expected student shares. Constraint (3) determines the enrollment and the relative deviation from the empirical expected student shares. Inequality (4) in interaction with constraints (8) states that the capacity of school s must not be exceeded by more than Hs . For each demand point the share of students has to sum up to one over all schools available [constraint (5)]. The domains are given by constraints (6), (7), (8). Note, that the model has to be solved for each combination of t 2 T and j 2 Jt . If no solution exists for j 2 Jt , then j is removed from Jt . To view constraint (3) in the proper perspective, consider the following example. There are two demand points with 150 students each and three schools with a capacity of 100 students each. Thus, total demand is met. For reasons of simplicity we neglect capacity expansion ws . All specifications of parameters and variables due to the student allocation model can be found in table 1. Table 1. Coherences of student allocation: parameter settings and variable outcomes. School A

B

C

100

100

100

300

p1s p1s n1 x1s x1s n1 z1s z1sÿ p1s ÿ x1s

0.10 15 0.18 27 0.77 0.00 ÿ0.08

0.10 15 0.16 24 0.56 0.00 ÿ0.06

0.80 120 0.66 99 0.00 0.17 0.14

1 150 1 150

p2s p2s n2 x2s x2s n2 z2s z2sÿ p2s ÿ x2s X pis ni

0.30 45 0.49 73 0.63 0.00 ÿ0.19

0.50 75 0.51 76 0.02 0.00 ÿ0.01

0.20 30 0.01 1 0.00 0.95 0.19

1 150 1 150

Gs Demand point 1 (n1 150)

2 (n2 150)

X i

ni 300

Total

60

90

150

300

100

100

100

300

i

X i

xis ni


2935

The school of major interest is school C since its capacity constraint is not met by empirically determined market share. The question is: how do we reduce the student number of school C by 50 students? Basically, there are two points of view subject to the enrollment policy of the (local) education authority. First, the probability of being rejected is the same for all 150 students. Hence, the number of students of demand points 1 and 2 expected to enroll at school C is reduced by the same proportion (1/3). Second, the reduction is based on a weighting factor. For example, it could be more likely to reject students who are located far away from the school under consideration (which is the actual policy in Dresden, Germany). For reasons of simplicity we consider the distance dis between demand point i and school s to be the only element of Vis and d2C > d1C . However, school C is the empirically expected first choice school for 30 students of demand point 2 (U2C > U2s 8 s 6 C). Nevertheless, V2C < V1C and thus p1C > p2C . Accordingly, the proportion of rejected students of demand point 2 is larger than that of demand point 1 (z1sÿ < z2sÿ ) due to the weighting of the difference between pis and xis by pis [see equation (3)]. If capacity constraints are not met by empirically expected student shares, our approach primarily intends to reject students from demand points located far away (low pis ). By a proper specification of Vis the distinction of schools could be attractive/unattractive rather than near/far. Note that this problem of student allocation does not consider decisions on an individual basis: for example, which particular student is allowed to enroll at a given school and which is not. Instead, this problem approximates the market share measured in student numbers of each school. Finally we should mention that this approach is just one way of allocating students to schools. A promising approach to be tracked in future research is to modify Vis in such a way that the capacity constraints of the schools are met. 3.2 Scenario selection

Let Jst be the subset of all scenarios j 2 Jt which contain school s. Moreover, the costs of each scenario j 2 Jt for period t are given by ctj. This includes: (a) the fixed location costs of each available school, (b) the variable costs of each available school, and (c) the transport costs for all students. The fixed location costs (a) are costs related to rent, maintenance, and so forth. The variable costs (b) reflect the cost of 1st or 5th grade students for the whole of their primary or secondary school career. These costs include the penalty costs due to violating minimum and maximum capacity. The transport costs (c) for each scenario j 2 Jt and period t depend on the transport mode chosen for the commute to school. Since the allocation of students from demand point i to school s is known for each period t and scenario j 2 Jt , it is straightforward to compute the share of students choosing a certain transport mode. Therefore, we use an MNL determining the choice probabilities of the commuting modes (walking, cycling, public transport, and car/ motorcycle). Additionally, we have to define the costs of closing or opening a school s in period t by cst and cstÿ , respectively. Note, the costs of closing a school could include the gain of selling or letting the school. The binary variable ytj equals one, if scenario j 2 Jt is selected in period t (0, otherwise); xst denotes whether a school s is opened in period t ( 1) or not ( 0); and xstÿ equals one for a school closure (0, otherwise). The scenarios are selected as follows: X X minimize ctj ytj cst xst cstÿ xstÿ , (9) t 2 T, j 2 Jt

s 2 S, t 2 T

2936


subject to X ytj 1 ,

t 2 T ,

(10)

s 2 S, t 2 T ,

(11)

s 2 S ,

(12)

ytj 2 f0, 1g ,

j 2 Jt , t 2 T ,

(13)

xst , xstÿ 2 f0, 1g ,

s 2 S, t 2 T .

(14)

j 2 Jt

X

ytj ÿ

j 2 Jst

X j2J

yt ÿ 1, j ÿ xst xstÿ 0 ,

s, t ÿ 1

X xst xstÿ 4 1 , t 2T

The objective (9) of this combinatorial problem minimizes the total costs over the planning horizon. Constraint (10) guarantees that exactly one scenario is selected per period. Constraint (11) forces a school s to be opened or closed, if two different scenarios are chosen in successive periods and only one of the two scenarios contains school s. Constraint (12) ensures that a school could be opened or closed only once over all periods. A closed school could not be reopened and a opened school could not be closed in following periods. In interaction, constraint (11) and (12) decrease the number of possible different scenarios selected over the planning horizon. The domains are constraints given by (13) and (14). To view constraint (11) in the proper perspective, consider the following cases for a given school s and period t: (a) In periods t and t ÿ 1 scenarios j 2 Jst and j 0 2 Js, t ÿ 1 are selected. Hence, ytj 1 and yt ÿ 1, j 1 and in combination with the objective function (9) xst and xstÿ are zero since cst , cstÿ > 0. (b) In periods P t and t ÿ 1 scenario j with j 2 = Jst and j 2 = Js, t ÿ 1 is selected. Hence, P ÿ y and y do not exist and therefore x st and xst have to be zero tj t ÿ 1 j , j 2 Jst j2j 0

,

s tÿ1

since the corresponding coefficients of the objective function (9) cst , cstÿ > 0. (c) In period t ÿ 1 scenario j 2 Js, t ÿ 1 is selected and in period t scenario j 02 = Jst is P ÿ selected. Consequently, j 2 J ytj does not exist. Hence xst 1 (school closure). (d) In period t ÿ 1 scenarioPj 2 = Js, t ÿ 1 is selected and in period t scenario j 0 2 Jst is selected. Consequently, yt ÿ 1, j does not exist. Hence, xst 1 (school j2J , opening). 0

st

0

s tÿ1

4 Application In the aftermath of German reunification in 1990 a remarkable east ^ west migration has taken place. In particular, younger, highly educated families migrated from the former German Democratic Republic to the western parts of Germany. At the same time the fertility rate declined. These two facts led to the problem that particularly in the following decade college student numbers decreased dramatically in eastern parts of Germany, like Saxony. Even urban regions faced this phenomenon. However, in some prospering regions (mostly urban areas) the student numbers have increased after periods of decline. Here, we take Dresden in the far east of Germany (south of Berlin, east of Leipzig) close to the Polish and Czech border as an example of an urban region with a temporary remarkable decline in student numbers.


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4.1 Data

Dresden consists of 400 wards (demand points) and covers an area of about 328 km2. We have computed the street network distances from each ward to 26 college locations (schools). The number of 5th grade college students in each ward per year are based on population statistics (1995 ^ 2004) and a forecast (2005 ^ 20) by the local statistic authority as well as an extensive survey of student mobility in Dresden in 2004 (Mu«ller et al, 2008). Figure 2 shows the development of the number of 5th grade college students from 1995 to 2020 in each district. For better visualization we have chosen districts (64), since the ward level (400) is too detailed for the figures. The maximum capacity of each college can be seen in figure 3. We depend on information on the fixed location costs per year of each college. Unfortunately, no exact information exists about these costs. Hence, we use the replacement value related to the remaining lifetime of a location to approximate fixed location costs. The investment costs per year for each school (modernization, for example) are added to the replacement value, yielding periodized fixed location costs for each college (see figure 3). The share of students to enroll at a given college is determined by the MMNL proposed by Mu«ller and Haase (2009) and Mu«ller (2008). The variable costs are set to 850 per student per year for all colleges (Gu«nther, 2005). This includes operating costs for water and electricity. Moreover, for each student exceeding the maximum capacity we assume costs of 1105 reflecting investment in capacity. The penalty costs per student below the minimum utilization are 1105. The weighting parameter g 1, and l 1=jIj. Our objective is to minimize the total costs incurred by the local education authority. The local education authority is not responsible for the funding of private schools, so we do not consider fixed

Number of students

5th grade college students 1995 ^ 2020 100

2

0

2

4

km

Figure 2. Number of college students (5th grade) in Dresden district level).

Period (1995 ± 2020)

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Capacity in 5th grade student numbers 150 75 15 Fixed costs per year ( 0.1 0.5 1.0

2

0

2

million)

4

km

Figure 3. Capacity provided for students located in Dresden and periodized fixed location costs for each college.

and variable costs for private schools. We do not consider individual transport costs and external diseconomies of transport. The local education authority has to pay compensation of 150 per student per year for students of grades 5 ^ 10 using public transport for distances over 3.5 km (Stadtrat der Landeshauptstadt Dresden, 1997). The transport mode choice probabilities are determined by the MNL described by Mu«ller et al (2008). We deflate all costs by 1.4% which is the average inflation rate of the period 1995 ^ 2004. Alternatively, the costs could be discounted (for example, by about 6% which corresponds to the interest rate of municipal bonds in Germany). We assume the costs for the opening of a school are 1 million. Therefore, no facilities will be built, but other facilities like high school buildings will be modified for use as colleges. Closing a school causes handling costs of approximately 100 000. Due to lack of information, we do not consider that facilities could be sold or let. Since we just want to model the decision making within the responsibility of the local education authority we fix private schools to be open over the whole planning horizon. Furthermore, we fix magnet schools which offer a special service (RomainRolland and Martin-Andersen-Nexo«). Locally very important colleges like Plauen, Cotta, Dreiko«nigschule, Carl-Maria-von-Weber, and Klotzsche are also fixed. We assume that it would be challenging within a political context to close these colleges. 4.2 Solution process

The fixing of schools yields a reduced number of possible scenarios and reduces computational effort. We have 26 colleges and 25 periods. Fixing none of the colleges


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yields 226 scenarios. Computing the student allocation (section 3.1) we would have to consider 2 26 25 possible combinations. Some of these combinations are not feasible for capacity reasons. Nevertheless, this is an immensely high number of combinations. We fix 11 colleges to be open over the whole planning horizon. This yields a maximum of 819 200 combinations of scenarios and periods. 4.3 Results

The student allocation (section 3.1) has been computed with GAMS/CPLEX 22.2 (GAMS, 2006) on a dual-core AMD Opteron 880 with 2.4 GHz and 8 GB memory. The computation time using two parallel threads was 7478 minutes with all feasible combinations of scenarios and periods solved to optimality. On the basis of the results, we consider four different alternatives for scenario selection (section 3.2): 1. No decision variables are fixed 2. Variable ytj is fixed to equal 1 for t 1995 and the scenario j 2 Jt which represents the historical scenario in 1995. 3. The same as 2, but additionally xst 0 for t < 2008 for the colleges Bu«hlau, Seidnitz, and Bu«rgerwiese. The local education authority decided in 2006 to open Bu«hlau in 2008 and the other two colleges later on. These locations were not available before 2008. 4. All ytj are fixed in such a way that all decisions on school closures and openings of the local education authority are reproduced. Here, no model has to be solved. All costs can be computed directly. Computation times with GAMS/CPLEX 22.2 for the three alternatives areöas expected övery different (see table 2). The results of alternatives 2 and 3 (computation time of alternative 3: 163 minutes) are exactly the same, so we consider only alternatives 1, 2, and 4. All alternatives are solved to optimality. An overview of the results is given in table 2. The deviance between the political solution (alternative 4) and the optimal solution (alternatives 1 and 2) yields the political costs. This is for the whole planning horizon between 1.731 million and 1.787 million per year. We consider the Table 2. Comparison of alternatives concerning total costs for different time frames, time of computation, and the number of closed colleges. Alternative

1995 ^ 2020 Total costs ( million) Relative deviation from alternative 1 Relative deviation from alternative 2 Absolute deviation from alternative 1 ( million per year) Absolute deviation from alternative 2 ( million per year) 1995 ^ 2010 Total costs ( million) Relative deviation from alternative 1 Relative deviation from alternative 2 Absolute deviation from alternative 1 ( million per year) Absolute deviation from alternative 2 ( million per year) Number of closed colleges CPU time in minutes

1

2

4

433.88

435.28 0.003

478.56 0.10 0.10 1.787

0.055

1.731

297.53

294.27 ÿ0.01 ÿ0.217

322.70 0.08 0.10 1.678 1.895

9 795

8 178

6 0

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time span from 1995 to 2010, since the future decisions of the local education authority are not foreseeable. In this case the political costs still range from 1.678 million to 1.895 million per year. Note that alternative 2 outperforms alternative 1 for the time span 1995 ^ 2010. This is due to the fixed location costs (see below). It is of great interest to find which schools are opened or closed at certain points in time for a given alternative. Figure 4 shows the resulting decisions on college openings and closings for all three alternatives. All schools which are not fixed are affected by opening or closure or even both (for different alternatives) except Vitzthum College, which is open over the whole planning horizon for all three alternatives. The decision of the local education authority (alternative 4) to close colleges seems to be related to the low number of students in the surrounding districts (see figure 2); the closing of Johann-Andreas-Schubert, Erich-Wustmann, and GroÞzschachwitz as well as the opening of Bu«hlau and Seidnitz seem to be predicated on this strategy. However, the optimal solution based on a historical starting scenario (alternative 2) shows that it is inadvisable to close a lot of colleges located in the sparsely populated outskirts. It seems to be appropriate to thin out the spatial cluster of colleges located near the city centre. This finding is confirmed by the results of alternative 1. However, at the same time it is suggested that certain colleges like Joseph-Haydn and Julius-AmbrosiusHu«lÞe should be closed due to their high fixed costs. Nevertheless, if these colleges are kept open they show relatively good utilization (see the online appendix, http:// dx.doi.org/10.1068/a40285). The optimal solutions (alternatives 1 and 2) provide better utilization than the political solution (alternative 4). This result is related to the strategy not to close facilities located at the outskirts, but to close locations within a spatial cluster of schools (city centre). The different costs of the alternatives can be seen College opened in period 2008 2010 2011 2014 College closed in period 1999 2000 2001 College open 1995 ^ 2020

2

0

2

4

km

(a) Figure 4. School locations of (a) alternative 1, (b) alternative 2, (c) alternative 4.


(b)

(c) Figure 4 (continued).

2941

2942


in figure 5. The total costs of the political alternative differ remarkably from the costs of the optimal alternatives for two reasons. First, the optimal alternatives feature more closed schools (see table 2). And, second, different decisions are made on the opening of colleges. This becomes more obvious if we take just the fixed costs into account. For 2007 onwards the fixed costs of alternative 4 are considerably above the fixed costs of the optimal solutions. The fixed costs of alternative 1 are higher than those of alternative 2 for the time span 1995 ^ 99. From 2000 to 2010 they are almost the same.

million)

14

25

Fixed costs (

Total costs (

million)

30

20 15

12

thousand)

million)

16

Variable costs (

10 18

Transport costs (

14

10 8 6 1995

1999

2003

2007

2011

2015

2019

Alternative 1 Alternative 2 Alternative 4

13 12 11 10 9 8 7

6 800 700 600 500 400 300 1995

1999

2003

2007

2011

2015

2019

Figure 5. Costs of the entire net of schools for alternatives 1, 2, and 4. 0.220

0.27 Proportion of students

0.26 0.25

0.210

0.24

0.205

0.23 0.22

0.200

0.21

0.195

(a) 0.62

0.19

0.190 0.0115

0.60

0.0110

Proportion of students

0.20

(c)

Alternative 1 Alternative 2 Alternative 4

0.215

0.58

(b)

0.0105

0.56 0.0100

0.54

0.0095

0.52 0.50 1995

1999

2003

2007

2011

2015

2019

0.0090 1995

(d)

1999

2003

2007

2011

2015

2019

Figure 6. Modal split of alternatives 1, 2, and 4: (a) walking, (b) cycling, (c) public transport, (d) car/motorcycle.


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Thus, alternative 2 outperforms alternative 1 for the planning horizon 1995 ^ 2010 (see table 2). The education authority decided to open Bu«hlau in 2008 and Bu«rgerwiese and Seidnitz in 2010. In contrast, the optimal solutions incorporate the opening of BertoldBrecht in 2010 as well as Hans-Erlwein in 2014 (alternative 1) and Bu«hlau in 2010, Seidnitz in 2011, and Bu«rgerwiese in 2014 (alternative 2). So, the peaks later than 2005 in total costs are related to the fixed costs of newly opened colleges as well as to the opening costs. Compared with the fixed costs and the total costs, the variable costs differ only slightly between the alternatives. This is because these costs depend on the number of attending students. Nevertheless, there is an observable discrepancy between alternative 2 and alternative 4 for the years 2000 ^ 15. Besides the costs for attending students there are penalty costs included in variable costs for violating minimum and maximum capacity constraints (section 4.1). The penalty costs of alternative 4 due to overcapacity preponderates over the penalty costs due to the undercapacity of alternative 2. Until 2000 the transport costs for alternative 1 are remarkably lower than the alternatives with a fixed (starting) scenario. The schools of alternative 1 are easier to access by cycling and walking than alternatives 2 and 4 (see figure 6). This net of schools is optimally adjusted to the spatial ^ temporal patterns of student location until the year 2000. After the period of school closing (1999 ^ 2001) we see a different picture. Now alternative 2 outperforms alternatives 1 and 4öin particular, when the student numbers are increasing again. Fewer colleges are closed for alternative 4 and generally the average distance to school is lower than is the case for alternative 1. So alternative 4 yields less transport costs. But the spatial ^ temporal pattern of the school locations for alternative 2 is better than that of alternative 4 although it contains fewer schools: in alternative 2 more students choose walking or cycling for the journey to school. These two modes are more appropriate for short travel distances and thus indicate a better spatial accessibility for the schools of alternative 2. 5 Conclusion We have seen that free school choice is rather neglected for multiperiod school location planning. Nevertheless, we suppose that free school choice will become more important in the near future. In this contribution we have presented a two-step planning approach for the multiperiod school location planning problem which incorporates free school choice. The first step allocates the students to available school locations according to schools' capacity and students' utility (determined by an MMNL). This allocation has to be done for every feasible combination of period and scenario (set of available schools). This yields the costs of each scenario for every feasible combination of scenario and period. Within a second step the scenarios are assigned to the periods of the planning horizon to minimize the total costs over all periods. The results show that with an improvement of at least 8% to 10% an amount of 1.5 to 2 million per year could be saved. We interpret this gap between the optimal solution found and the political solution as political costs. In other words, the community is willing to pay these costs in order to establish a desirable net of schools. Concerning the spatial patterns and economic consequences, the overall optimal solution guarantees a net of schools with minimum costs as well as best accessibility over time. One major shortcoming of our approach is inflexible capacity. It would be desirable to allow for modular capacity expansion and reduction. Another weak point is that no change between schools due to closure is allowed for students. Future research should focus on these items as well as on faster methods of student allocation and efficient

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