**S. H. Sanaei-Nejad**

Ferdowsi University of Mashhad, P.O.Box: 91775-1163,

Mashhad, Iran,

[email protected]

**H. A. Faraji-Sabokbar**

University of Mashhad, P.O.Box: 9177948883 Mashhad, Iran

[email protected]

**Introduction**

When there are some resources and many demands, there should be a function to propose the most optimized routes for accessing the resources. This is the problem of many business and government institutes (Lea and Simmons, 2000). This subject is of great interest in third world countries where there is large demand for Infrastructure development communication, drinking water, educational facilities, public distribution system, electricity, financial institutions, markets, medical facilities & public health, transport, veterinary services, recreation & tourism which is the major concern of developing nations. (Inter-graph, 2002)

The basis of the models can be explained when we consider that, there must be transportation or movement through the network which connect different resources and demands throughout an area. For example, when there are limited sites to collect and store agricultural products in a farming area or assignment of specified number of schools with limited seats for students in a residential area, we need to use “allocation models” to find the best solution. Different model functions are applied depending on type and aim of the allocation issue (Klinkenberg, 1997). First we discuss different models.

- Private sector allocation model
- Public sector allocation model
- Emergency service location model
- Distance limitation:

This model is used to minimize costs and maximize efficiency. In fact there is an option, which the model can minimize total distance traveled from all demand points access resources.

This model applied to private fair service with maximum efficiency. To achieve these, we need to maximize the assignment of demands to each center. In this case we consider a linear likelihood function for assignment. The total distance traveled is also minimized, where the distance measured according to a power function. We should also consider the minimum total distance ensuring that no further demand point is in the given distance.

In some cases we need to serve as many people as possible within a given distance from a center. Using this model we intend to provide a service with no limitation in covering demand points, except for those area which practical constraints exist.

We need to find a mathematical equation to structure the whole issues of allocation properly. Therefore, first the aims and constraints need to be settled and then the optimizing the factors must be optimized to solve the problem.

For instance, this model can be used to utilize and organize rural settlements, in order of allocating two schools in the rural points. In this case we need to find appropriate places for the schools, so that the costs and traveling are minimized. Therefore the constraints are:

Two schools are considered to be allocated and their capacity is specific number. Obviously, students will choose the closest school to attend

No student will travel to a school, if it is very far from his/her settlement when the constraints and the aim of the plan are determined. The problem can be solved by optimizing the parameters in the model. In this example, location of the school and allocating students to them will be outputs of the model

Geographic Information Systems (GIS) is a software and hardware tool applied to geographical data far integration, collection, storing, retrieving transforming and displaying spatial data for solving complex planning and management problems. (Sarangi, et al., 2001)

Indeed, location-allocation methods are one of the few modeling and spatial analysis tools offered in proprietary GISs today (Kim, Openshaw, 2002). Most of GIS software are capable to solve optimizing problems and run allocation models, such as ARC/INFO, ILWIS, ARCVIEW, IDRISI, and CARIS. In this papers we used ARC/INFO (ESRI, 1998) to run the allocation model for the specified data.

**Applying the models **

Some of the models in each of the private sector and public sector are selected to apply for the study area.

- Private sector model
- The utilities are limited
- Each demand node will be traveled to the nearest unity center

- Specifications of the model:
- Running the model:
- The results of running minimizing rural distance model for the district is shown in map 1 in figure 3.

Minimizing travel distance model: Using this model we want to determine a suitable location for a number of specific utilities, so that the total traveling distance is minimized.

The constraints in this model are as following:

In this model the utility center is located in the weighting center of the area, which is in the middle location of demands node. As the model minimizes traveling distance and traveling costs, it is suitable for private sector. It should be noticed that minimizing total traveling distance might be concluded a point, which is not the nearest one to the center.

A district called Pevejen from Mashhad country has been chosen to use it data for running the model. The case study is to determine the best location for building two schools.

**Public sector model**

- Maximizing of service covering model (MSCM):
- The constraints of this model are as following:
- There are only limited service center
- Each demands should go to the nearest service center
- The covering of service area is reduced as travel distance increased.

- The specifications of the model:
- Running the model:
- Maximum Attendance Model:
- The constraints of the model are as following:
- There are limited service centers available
- Each demand goes to the nearest service center.
- Attendance of demand facility change linearly with distance

- The specifications of the model:
- Running the model:
- Minidistance power function:
- Running the model:
- Running the model:

This model is used to find a location for service centers, so that maximum demands can be respond.

In this model the service centers are located where the demand density is high. MSCM model is planned to maximize demands and should be located somewhere in the area, so that probability of using the services is reduced as traveling distance increased. Therefore the more closer demand to the service center, the highest utilizing from the services .we need to determine a limitation for traveling distance which services will not be available outside.

The limitations of the friction of distance is defined as β and its value is derived from the equation (1)

β = 1 / d ——————-(1)

Where d is a distance beyond which there is no responsibility for giving services.

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Figure 1. The diagrams representing different β value

Figures 1 shows the diagrams representing β, likelihood of traveling to the service center and distance between facility and demand.

The result of this model is shown in map 2 in figure 3. As it can be seen in the map most of the villages (demands points) are covered by the north center, where the density of the demand points are high.

This model is used to allocate services center for maximizing attendance.

unlike the previous model maximizing attendance, load services to the demand points, which are further from the service center.

Map 3 shows the result of running the model.

In this model when distance from the center increases the exponent function exaggerate the effect of distance. Therefore by applying larger power function the distance that individual demand points should travel to their nearest facility will be equalized.

In the study area, most demand points are located in distance between 1 and 8 Km. If we consider one more demand point, x, which is 20 Km away. In this case the total distance traveled to this facility is 64 Km as optimized arrangement. When this point is determined as optimized by the model, it means that if this facility were moved in any direction, the total distance traveled would increase using power function in the model we exaggerate the distance. Let’s consider the exponent of 2 in this example.

Points range in distance between 1 and 64 Km and demand point x is 400 Km away. It should be noticed that 400 km is exaggerated by exponent, of 2.

Comparing the demand points distance, 1-64 Km, with 400 Km, shows that the current location is not optimized any more, and should be moved toward x point .if the facility center moves 5 Km toward x point, the effective distance will be 225 Km.. In this case though many demand points will have to travel slightly further, but the amount is relatively small compared to the savings produced by locating the facility closer to demand location x.

The previous area, called Pevejen was chosen to run this model and the results are shown in Figure 3. As map 4 in this figure shows, the utility centers are located in Soltan-Abod-Nomak and Avareshk, villages, based on the model.

Comparing this map and map 1 shows that utility centers are displaced. The displacement can be explained based on β coefficient. In this analysis β is calculated for a distance equal to 8 Km, which is 0/000125

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Figure 2. Comparing linear function and power function in allocation model

Comparing the diagrams in figure 2 shows that if we displace the center slightly, the power function will be changed considerably. Therefore if we change the location site to new optimal configuration, say, 15 Km from demand point x, the effective distance will be 225 Km. this displacement will cost only a slight change in traveling distance for other demand points. Map 3 in figure 3 shows the utility center allocation in the area. Displacement in the utility centers and the related demand points can be seen in comparing with map 4. Using this model, Eslam-Ghale and Gol-Baghra which are close together will be served by Bazeh-Hour center.

It can be seen when we use the power function model, the utility centers location is changed and the village covered by the centers are also differ from the previous modes. Eslam Ghale and Gol Baghra have moved to the north center which means the power function model increased the covered area for this center an the result all exponent all 2 for the distance

Minimizing distance (with constraints) model: The objective of this model is the same as minimizing distance model, in which the location of a given number of facilities are specified. The specified number is as many that the total distance traveled is minimized.

This model is similar to the minimizing distance model and the only difference is distance threshold, which considered by this model. In this study the threshold is considered to be 8 Km and the results are shown in map 5 in figure 3. As the map in shows, a considerable displacement has been occurred in the location of utility centers. Both of the centers are determined in the north of area. This is because the number of demand points is higher in north of the area than in the south.

**References**

- ESRI, 1998, ARC, Version 7.2.1, Manual User Reference.
- Inter-graph, 2002, https://www.intergraph.com/gis/
- Kim Y. and S. Openshaw, 2002, Comparison of alternative location-allocation algorithms in GIS School of Geography, University of Leeds LS2 9JT
- Klinkenberg B, 1997. Location-Alocation on networks,
- Lea A. and J. Simmons, 2000, Location-Allocation Models for Retail Site Selection CSCA, Ryerson Polytechnic University 350 Victoria Street, Toronto, Ontario, Canada

Map1

Map2

Map3

Map4

Map5

**Figure 5. Map 1 – 5** show the results of running different models for a rural area called Pivejan in Khorasan province of Iran (North East of the country). More details are discussed in the text.