Geographical position #

For production units and service enterprises, there are always differences in spatial distribution between demand and supply, therefore, the selection and determination of Geographic Location of Institutional Facilities have a vital impact on their economic benefits and their own development.

The location evaluation and optimization of the facility is to optimize the spatial location distribution model by analyzing the interaction between the supply and demand of a facility or a facility network. Institutional facility location evaluation is an evaluation of the spatial location distribution pattern of existing service facilities, the optimization of institutional facility location is the search for its optimal location. Kristaqin’s central theory provides a basic framework for the theoretical research and application of empirical methods for the supply location optimization model, the market area size in this theory is determined by the scope of supply and service, and demand and supply, the relationship between the two is based on distance minimization and profit maximization. However, the central theory in practical applications, whether it is a theoretical method or its empirical methods, needs to be specifically analyzed according to specific problems. For example, contrary to the theory of the settlement world, the concentration and dispersion of the population in the region will have an impact on the size of the region; similarly, the choice of consumer behavior will also have an impact on the functional scope of a service center. Therefore, although the central theory provides a main framework for describing and interpreting the spatial interaction between the cargo service supply system and the demand behavior, it serves as a reference framework and needs to be expanded and adjusted for special situations. Figure 10-16 shows the conceptual framework for the application of central theory.

The classical model of Central Place Theory conducts spatial analysis and simulation of homogeneous market functional regions that contain a single objective and where all internal demands are directed toward this central objective. In contrast, the modern model of Central Place Theory addresses spatial analysis and simulation of market functional regions in the real world characterized by uncertainty, involving multi-objective consumer behaviors and complex supply behaviors.

Based on a more realistic study of market function behavior, many models have been developed for spatial analysis and simulation of real-world market functional areas, such as Reily’s Retail Gravitation, Batty’s Crack, Breakpoint Equation, Tobler’s Price Field and Interaction Wind, and numerous corrections. However, it is very difficult to use the gravity model and the crack point equation to analyze the multi-point center system. Due to the lack of visibility and continuity, the mapping of the analysis results is also difficult. The Tobler method is a semi-quantitative description of the characteristics of the central city market. Below, we will focus on a spatial linear optimization model for positioning-configuration problem solving.

Definition of spatial optimization pattern #

The spatial optimization model is used to solve the location-allocation problem. Location-allocation problems arise when planning the location of important public facilities and their affiliated areas (public facilities, such as hospitals, kindergartens, playgrounds, nursing homes, schools, police stations, fire brigades, first aid stations, management facilities, etc., which belong to the infrastructure within the national budget).

A location-allocation problem can generally be expressed as follows:

• There are a certain number of residential concentration points, which are called demand points (or consumption points, residential areas), and the demand distribution of a certain number of supply points (some public facilities) and/or supply points is sought to achieve a certain planning purpose.

• If a demand point is set and a supply point is sought, the location or location problem is involved.

• If a supply point is set up and allocation is sought, it involves allocation or allocation.

• If both supply points and allocation are sought, the location-Allocation problem is involved.

Through the distribution between the demand point and the supply point, the subsidiary area of the supply point is also determined, as shown in Figure 10-17.

The basic structure of an optimization model consists of a set of constraints and one (or rarely, several) objective function(s). Under these constraints, the maximum or minimum value of the objective function is sought. The constraints represent the mandatory conditions that must be fulfilled to achieve the planning objectives, reflecting the fundamental evaluation of the functional requirements of the target planning area. The optimization objective function (i.e., finding the extremum of the objective function) represents a planning goal that aims to achieve the maximum possible potential. Therefore, the planning conditions reflected in the constraints hold primary significance, while the importance of the planning goal corresponding to the optimization objective function is relatively secondary. Introducing the maximization or minimization of the objective function aims to derive a definitive solution to a location-allocation problem (specifically, in cases where multiple feasible solutions exist for the objective function under certain constraints).

Classification of spatial optimization patterns #

According to the time range, problem space type, service mode of public facilities, road cost bearer and use type of public facilities, space optimization modes can be classified, as shown in Figure 11-18.

Mathematical formulation of the static-discrete spatial optimization model: Linear programming. #

The static-discrete location-allocation problem arises when planning the locations of important public facilities and their service areas. This static-discrete location-allocation problem must consider numerous potential supply points, with the determined locations and service areas remaining unchanged over an extended period.

In the space optimization process, if the objective function and the boundary conditions are linear, the mathematical tool used is linear programming [1]_ Linear Programming). Linear programming refers to the problem of finding the maximum or minimum of a linear function under the constraints of a set of linear equations and inequalities.

The general form of linear programming is as follows:

Objectives:

min c_1 x_1 + c_2 x_2 +… …+c_n x_n

Constraints:

a_11 x_1 + c_12 x_2 +… …+c_1n x_n ≤c_1

… …

a_m1 x_1 + c_m2 x_2 +… …+c_mn x_n ≤c_1

x_1, x_2… … x_n ≥0

The following example belongs to the division of production and marketing, which can be solved by linear programming, there are three production areas A, B and C, with output of 50, 30 and 20 respectively, and demand of sales areas I, II and III is 40, 40 and 20 respectively. Firstly, through the shortest path analysis, the freight as described in Table 10-6 is obtained.

Table 10-6: Freight table for resource allocation (∞ indicates no connection)

The question is: how to determine the supply relationship, so that the total freight rate is the least. Assume that x_11 represents the traffic from A to I, x_12 represents the traffic from A to II, and so on. Then the problem can be described as:

Object:

min``2x_11 +3x_12 +5x_13 +1.5x_22 +8x_23 +2x_31 +4x_32``

Constraint:

x_11 + x_12 +x_13 =50

x_22 +x_23 =30

x_31 +x_32 =20

x_11 +x_31 =40

x_12 +x_22 +x_32 =40

x_13 +x_23 =20

x_11 , x_12 , x_13 , x_22 , x_23 , x_31 , x_≥0

At present, there are many mature solutions to linear programming, such as solution method, simplex method and so on. The answer to the above-mentioned supply and marketing zoning is shown in Table 10-7, and the total freight is 255.

Table 10-7: Traffic volume of resource allocation

Genetic Algorithm:

Genetic algorithm is a recently proposed method for solving optimization problems, which optimizes the values of variables in a set to obtain a target that can quantitatively describe the pros and cons. The idea of quantifying the appropriateness of the target comes from Darwin’s theory of evolution, during the evolution process, the content and form of the genes of different generations of biological chromosomes are changing, making the organism more suitable for the living environment. In genetic algorithms, variables are changed by a combination of different ways and by modifying values, which is similar to the pairing and variation of organisms. By comparing the appropriate indicators, the results of the calculations after changing the variables are tested, the “favorable” combination is retained, and further modifications are made. An ideal result is that the retained changes are superior to other combinations and changes, and an iterative process can result in an optimized result. The success or failure of genetic algorithms depends on how the various forms of chromosomes are simulated and evolved to increase its chances of survival in the optimal solution.The surviving chromosomes generally correspond to a local extremum, which can lead to a global optimal solution.

In a genetic algorithm, the contents to be determined include:

1. Gene manipulation, such as pairing and mutation;

2. Strategic parameters: such as the probability of pairing and mutation;

3. Expression pattern: such as gene coding, etc.

In GIS, genetic algorithms can be applied to spatial interaction modeling, which can be applied to public facilities such as stores, hospitals, etc.