Application of the p-Median Problem in School Allocation

This paper focus on solving the problem of optimizing students’ orientation. After four years spent in secondary school, pupils take exams and are assigned to the high school. The main difficulty of Education Department Inspection (EDI) of Dakar lies in the allocation of pupils in the suburbs. In this paper we propose an allocation model using the p-median problem. The model takes into account the distance of the standards imposed by international organizations between pupil’s home and school. The p-median problem is a location-allocation problem that takes into account the average (total) distance between demand points (pupil’s home) and facility (pupil’s school). The p-median problem is used to determine the best location to place a limited number of schools. The model has been enhanced and applied to a wide range of school location problems in suburbs. After collecting necessary numerical data to each EDI, a formulation is presented and computational results are carried out.


Introduction
Every year, the Education Department Inspection (EDI) of Dakar has difficulties for assigning pupils to high schools of the suburbs.After four years spent in secondary school, pupils take exams and are assigned to high schools.The EDI of Dakar must take several important decisions such as network design, pupil assignments, construction of new high schools, etc. Nowadays, due to the fact that the number of pupils increases each year, problems related to pupil assignments have become so complex that rudimentary techniques seam unsuitable.Thus, the need to use analytical tools to deal locationallocation problems has come out to the attention of EDI decision makers.With some processing techniques, we solve instances for the EDI of Guediewaye, Thiaroye and Keur Massar to find the optimal configuration, to install new high schools in order to attend the demand of a population.It is possible to build a high school inside some colleges that have space or in sites (vacant land) of the area.Location-allocation problems deal with decisions of finding the best (or optimal) configuration for the installation of one or more facilities in order to attend the demand of a population [1].
Nooradelena and Noraida [2] used the p-median problem to determine the best location to place a limited number of emergency medical services taking into account the uncertainty in the demand for the emergency services.The p-median problem is a location-allocation problem that has been extensively studied.The reason for the interest in the p-median problem is that it has practical applications in a wide variety of planning problems: locating telephone switching centers [3], school districting [4], and bank location [5].Chardaire and Lutton [6] extended the p-median problem to account for different types of located facilities that interact such as telecommunication terminals that are connected to concentrators, which are connected to each other.Densham and Rushton [7] worked on a p-median problem where every facility needs a minimum workload.
In this paper, the school location for pupils from secondary school to high school is presented.An allocation model using the p-median problem is proposed.More details concerning the p-median problem formulation can be found in Church and ReVelle [8].We solve the mixedinteger 0 -1 linear program using the IBM-CPLEX's MILP solver [9]; and show how the simulation techniques can be applied successfully to practical decision support of Dakar's EDIs.For each EDI in Dakar: Guediewaye, Thiaroye and Keur Massar, all schools (or demand points) are listed, with i the number of pupils who pass exam in each school with .Selected colleges will be expanded into high schools and free surfaces are built for high schools.Facilities or facility locations are places that are suitable to establish schools (service facilities).Service facilities cannot be erected outside facility locations.A configuration is a set of us- ually p candidates that are proposed to get a service facility.However, during the optimization process the configuration might consist of a different number.Every configuration that has exactly p facilities is a solution for the p-median problem.The integer number p must be chosen to specify how many facilities are to be located.The optimal solution is a configuration that has the minimal distance.The objective function is the total distance from every point of demand to the closest facility of the configuration.The distance is weighted or multiplied by the population of the site.The paper is organized as follows: In Section 2, we present the allocation model using the p-median problem formulation.Section 3 is devoted to the numerical simulation results for EDI of Guediewaye, Thiaroye and Keur Massar.Finally, summary and conclusions are presented in Section 4.

Problem Formulation
The p-median model formulation is based on the following notations: 1, , facility locations for facilities.the maximum distance to be traveled by pupil.number of school to be established.
The objective function minimizing the total weighted distance of assignment.

Min
(1)  and the associated weighted distance j ij a d is i in the weighted distance sum.Constraint (2) ensures that each pupil must be assigned to at least one school.Constraint (3) restricts the placement or allocation of facilities to exactly p, where p is the number of high schools to be established.Constraint (4) maintains that any such assignment must be made to only those locations that have been selected for a facility.Constraint (5) imposes a maximum distance to be covered by each pupil.Constraint (6)

EDI of Guediawaye
Guediawaye is located on southeast of Pikine.It covers 550 square kilometers.In 2002, Guediawaye became the fourth department of the Dakar region, such as Dakar, Pikine and Rufisque.It is divided into five district municipalities: Golf Sud, Medina Gounass, Ndiarme Limamoulaye, Sahm Notaire and Wakhinane Nimzatt.There is in the EDI of Guediawaye ten secondary schools and two high schools.The large number of pupils requires significant renovation decisions or reconstruction of new locations.After exam, pupils are assigned to the nearest high schools.The question is which site to choose, which site is to expand, how many locations will be rebuilt and where to place such sites so as to minimize the distance traveled by pupils.The option is to compare the current situation (in number of pupils and schools) in simulations where the same number of pupils is assigned to different schools in reduced number and location. It begins with an analysis stic).Regularly registered pupils are listed on a map based on their domicile;  Then we group the 15 de ites using the following crie land, in process, of being public or subool and for school.In total, less than 2 kilometers. Next we build a map of: a) location of potential s teria: i) availabl ject to expropriation.ii) no nuisances by the sch this gives 8 points provides (including existing).b) defined points of demand and supply; we can run a model and compare its results with those of the current situation.
Table 1 shows weights for the each demand; and Table 2 indicates distances between schools and sites.

S or in process
The numerical tests were performed by using the commercial MILP solver, namely IBM-CPLEX V12.3 [9].The numerical experiments were executed on a computer: 2×Intel(R) Core(TM) 2 Duo CPU 2.00 GHz, 4.0 GB of RAM, under UNIX system.We apply the p-median model for the targeted sites by setting 4 values for p.We have done tests for p = 5, 6, 7 and 8.We give here more details for the case p = 5.The procedure is the same for others (p = 6, 7 and 8).
Table 3 shows the facility locations for this scenario.Site 1, Site 2 and Site 3 are available land or in process or subject to expropriation, thus the population is put to 1. Pupils in Unit 5, LPk and LL do not live in these locations, they leave their home to come to these schools.
With the given 5 sites and from 8 facility locations, the optimal locations are Joseph Correa A tion 1), the Islamic Bank cation 2), Ndiarka D nit 5 (location 6) and the Site S2 (location 7).The dend it is po 3) Pupils from Ndiarka Diagne, Teachers' City and (loca (lo iagne (location 3), U termination in choosing which pupils allocated at these facility locations depends on the distance between demand points and facility locations (see Table 3).
1) Pupils from Joseph Correa A, Joseph Correa B, Pikine East A, Pikine East B and nearby Site 3 will be assigned to the "High School Joseph Correa A".A ssible to assigned pupils from Limamoulaye if necessary.
2) Pupils from Islamic Bank are assigned to their new "High School Islamic Bank".
nearby Site S1 will be assigned to the "High School Ndiar go Diop are assigned to th km of the Site 2 will be assigned to the "S necessary.

EDI o
Keur Massar is one of the six Pikine.There are thirteen sec of Keur Massar when four can be enlarged in order to build high schools; and four sites where new high school can be built.With the given 5 locations and from 8 facility locations, the optimal locations are Malika, Keur Momar Marme Diop, Yembeul secondary school, Site 7 and Site 8.In this case, the maximum distance that allows us to obtain results using the IBM-CPLEX's MILP solver is

Conclu
In this paper, we pupils from second tion model using the p-median problem is proposed.We have solved the related mixed-integer 0 -1 linear program; and have shown how the simulation techniques can be applied successfully to practical decision support of the EDI of Dakar.All facility locations are connected   to the demand points according to the sho etween them.

i
population (number of pupils) at the demand point .ij shortest distance between location and location .
represents the domain of variables.The model is solved using three EDIs in Dakar city ncluded : ED mber of demand points with demand values ility locations.These are the ocan at demand point i .The population 3. Simulation Results the coast of Dakar in the of the demand (here inelamands points separated by I of Guediawaye, EDI of Thiaroye and EDI of Keur Massar.We present in more detail only the results of Guediawaye's EDI.The input data to solve this p-median problem are:  a finite nu to specify the location and amount of demand that needs to be satisfied. a finite number of fac only positions where facilities can be established.Several values of p are tested for the three EDI. distances between demand points and facility l tions.Distances are calculated from the network of Dakar city. the populatio represents the number of pupil at school i .

Figure 1 .
Figure 1.Sites for Guediawaye's EDI.ite 1, Site 2 and Site 3 are available land Thiaroye is a district in Dakar, Senegal.It is located in the southern department of Pikine.There are twelve DI of Thiaroye, where six can a site where a new high school f Keur Massar teen boroughs in the city of ondary schools in the EDI secondary schools in the E become high schools and can be built.With the given 4 sites and from 7 facility locations the optimal locations are Thiaroye Azur, Mbao Kambe, Thiaroye secondary school and Mbao secondary school.The numerical results for 4 p  are shown in Figure 3.

2 K
 km, the model does not offer a solution.The numerical results for 4 p  are sh sion have presented school locations for ary school to high school.An alloca-own in Figure 4.