CN107133375B - Facility site selection optimization method based on Euclidean distance linear approximation - Google Patents

Facility site selection optimization method based on Euclidean distance linear approximation Download PDF

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CN107133375B
CN107133375B CN201710178131.2A CN201710178131A CN107133375B CN 107133375 B CN107133375 B CN 107133375B CN 201710178131 A CN201710178131 A CN 201710178131A CN 107133375 B CN107133375 B CN 107133375B
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左晓荣
肖依永
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Abstract

The invention provides a facility site selection optimization method based on Euclidean distance linear approximation, which comprises the following steps: step one, preparing data preprocessing; step two, establishing a linear mathematical programming model; step three, solving the model; step four, outputting results: determining the optimal position coordinates of the warehouse points; through the steps, the invention provides a global optimal site selection method for the warehouse point site selection problem of the logistics site selection, and the problem of the logistics facility site selection in the actual life is solved more efficiently.

Description

Facility site selection optimization method based on Euclidean distance linear approximation
One, belonging to the technical field
The invention provides a facility optimal site selection method based on Euclidean distance linearization, which can be used for urban public facility planning and solving optimal garbage warehouse point coordinate site selection and corresponding resident distribution aiming at the distribution of the existing urban residents so as to reduce the transportation cost between the residents and the garbage warehouses distributed by the residents, and belongs to the field of logistics facility planning and site selection.
Second, background Art
In the field of logistics facility planning, the problem of warehouse site selection is a key problem affecting transportation cost and efficiency. . In the course of economic development, it is often necessary to provide one or more "warehouse" points at which to distribute materials, transmit information, or perform certain services, in order to transmit information or provide services to nodes around it. For example, in city construction and planning of commercial centers, hospitals, fire stations, parking lots, garbage collection sites, and other public facilities, it is often necessary to consider where to click the centers in order to optimize the operation of the urban system. Thus, in practical terms, these addressing issues are issues relating to optimal location of facilities. How to select the address of the service site is a difficult problem in the field of logistics addressing, so that the total distance from all the customer points to the corresponding service points is shortest on the premise of ensuring that each customer point is allocated to a proper site for service.
The design problem is that the traditional k-means (namely k-means) is used for calculation at present, but the method needs to give an initial central point in advance, the calculation result is influenced by the selection of the initial central point, and the calculation central point of the averaging method of the method cannot be guaranteed to be the optimal distance center, so that the optimal solution cannot be calculated. The invention provides an optimal addressing method based on Euclidean distance linearization, which is a global optimization method, ensures that a calculation result can be controlled within an arbitrarily small error range, and avoids the influence of an initial solution on a final result.
Third, the invention
3.1 objects of the invention
The invention aims to provide a facility site selection optimization method based on Euclidean distance linear approximation, which is an efficient optimal scheme selection method for the optimal site selection problem of warehouse points for logistics site selection, so that the warehouse points meet the requirement of providing service for surrounding client points, and the total distance from all the client points to the distributed warehouse is shortest.
3.2 technical solution
The problem is first described in a specification: there are n scattered customers in a certain area, and the coordinates and needs of the customers are known. M warehouses (or service facilities) need to be established, each with a corresponding upper bound on total capacity/service capacity. The optimal warehouse coordinate addressing and customer allocation is solved such that the total distance from all customer points to their allocated warehouse/facilities is the shortest. The decision variable for the problem is what location should be selected to build the warehouse site and for which customer sites each is serving, so that the total distance of all customer sites to their assigned warehouse is the shortest.
The Euclidean distance between two points is expressed as
Figure BDA0001252859650000021
Wherein
Figure BDA0001252859650000022
And
Figure BDA0001252859650000023
representing the x and y axial distances between two points, respectively. Since the distance expression is non-linear, it is difficult to obtain an optimal solution to the warehouse location problem. The patent designs a method for approximating a group of tangent planes to replace a nonlinear Euclidean distance formula and ensure the replacement errorThe difference can be within an arbitrarily small predetermined error range epsilon. The expression for this set of tangent plane approximations is as follows:
Figure BDA0001252859650000024
where q is the number of tangent planes and θ is the angle of rotation of two adjacent tangent planes, both being constant, depending on a given fraction ε. When the value of ε is given, the equations for q and θ are as follows:
Figure BDA0001252859650000025
in the above formula, the character is calculated
Figure BDA0001252859650000026
This means that the smallest integer of x or more is taken. The partial relationship data of epsilon, theta and q are shown in the following table 1:
TABLE 1 relationship of ε, θ and q
Figure BDA0001252859650000027
In the following, a plurality of symbols are defined in a summary manner, so as to accurately describe the implementation steps of the patent:
Figure BDA0001252859650000031
based on the above-mentioned symbol definitions,
the invention discloses a facility site selection optimization method based on Euclidean distance linear approximation, which is completed by four steps, wherein the four steps are as follows:
step one, data preprocessing preparation
Firstly, numbering customer points in a two-dimensional plane, wherein the numbering value is from 1 to N; secondly, the coordinates of the customer point need to be given, namely XiAbscissa, Y, representing customer point iiRepresents the ordinate of the customer point i; then, giving the serial number of the warehouse/facility to be selected, wherein the serial number value is from 1 to K;finally, assigning values for a given positive integer q and a given angle radian theta;
step two, establishing a linear mathematical programming model
According to the idea of the optimal addressing problem, the coordinate of the finally selected warehouse point is calculated to minimize the transportation cost, namely the sum of the distances between the warehouse point and the served customer point is minimized, and the sum of the distances between the warehouse point and the served customer point is called as an objective function; establishing a linear mathematical programming model based on a facility site selection method of Euclidean distance linear approximation;
wherein, the linear mathematical programming model is built by the following steps:
(1) optimizing the sum of the distances between each client point and the subordinate warehouse points in the addressing problem, and defining the sum of the distances as an objective function; the objective function of the linear mathematical programming model of the problem is established by the distance d between each customer point i and the warehouse point k to which it belongsikThe summation is performed and the sum of the distances is made as small as possible. Based on this, the following objective function is established:
Figure BDA0001252859650000041
(2) in the invention, each customer point in the two-dimensional coordinate plane is a demand point, and the warehouse points aim to provide services for the customer points, so that the limitation condition needs to ensure that each customer point i has one warehouse point k to provide services for the customer point i, namely, for any customer point i, the customer point i can be and can only be allocated to one warehouse point k. Establishing constraint 1):
1)
Figure BDA0001252859650000042
(3) the distance constraint in the invention adopts Euclidean distance for constraint, so when a customer point i and a warehouse point k have a service relationship, the x/y axial distance between the customer point i and the warehouse point k to which the customer point i belongs is not less than the horizontal/vertical coordinate difference between the customer point i and the warehouse point k to which the customer point k belongs, and constraint conditions 2) -5) are established:
2)
Figure BDA0001252859650000043
3)
Figure BDA0001252859650000044
4)
Figure BDA0001252859650000045
5)
Figure BDA0001252859650000046
(4) d is obtained by defining the number q of tangent planes and the rotation angle theta of two adjacent tangent planes according to the above restrictions 2) to 5)ik xdik yCan be used to specify the Euclidean distance d between the customer point i and the warehouse point kikI.e. establishing the constraint 6):
6)
Figure BDA0001252859650000047
the linear mathematical programming model can therefore be summarized as follows:
an objective function:
Figure BDA0001252859650000048
constraint conditions are as follows:
1)
Figure BDA0001252859650000049
2)
Figure BDA00012528596500000410
3)
Figure BDA0001252859650000051
4)
Figure BDA0001252859650000052
5)
Figure BDA0001252859650000053
6)
Figure BDA0001252859650000054
step three, solving the model
Solving is carried out aiming at the linear mathematical programming model, and various solving modes are considered: (1) directly solving, namely solving the linear mathematical programming model by utilizing a simplex method, a branch and bound method and a secant plane method; (2) solving by using commercial software, such as Lingo, CPLEX and the like;
the mathematical programming model is linear and completely has the feasibility of solving the optimal solution;
wherein, the model refers to a linear mathematical programming model composed of the objective function established in the step two and the constraint conditions 1) to 6);
the 'solving model' selects to call a CPLEX solver solving model by using AMPL language, and the concrete method for solving is as follows:
(1) inputting data to be clustered and basic clustering parameters, and establishing an AMPL data file xxx.dat;
(2) establishing an AMPL model file xxx.mod, and establishing a linear mathematical programming model;
(3) establishing an AMPL batch file xxx.sh;
(4) using AMPL to call a batch file xxx.sh, and starting to solve;
step four, outputting results: determining optimal warehouse point location coordinates
According to the final calculation result, the decision variable x of the modelk、ykAnd uikThe coordinates of the warehouse points and corresponding customer affiliations (if u) may be determinedik1 means that customer point i is provided by warehouse/facility kService) to find the optimal objective function value (i.e., Total _ Dis), i.e., the Total distance between the warehouse and the customer.
Through the steps, the invention provides a global optimal site selection method for the warehouse point site selection problem of the logistics site selection, and the problem of the logistics facility site selection in the actual life is solved more efficiently.
3.3 efficacy and advantages of the invention
The invention has two advantages:
(1) compared with the traditional K-means, the method avoids the problem that the local optimal solution is influenced by the initial solution;
(2) the method adopts linear programming for optimization, and ensures that an output result is a global optimal solution.
Description of the drawings
FIG. 1 is a flow chart of the method of the present invention.
FIG. 2 is a diagram of an example result of a conventional k-means clustering method.
FIG. 3 is a graph based on the results of an example of the method of this patent.
The numbers, symbols and codes in the figures are explained as follows:
the small squares represent the coordinates of the warehouse points calculated by the new method
The circles represent known coordinates of the customer points in a two-dimensional plane
Triangles represent warehouse point coordinates calculated using the traditional k-means method
The line segment represents the calculated affiliation between the warehouse point and the customer point
Fifth, detailed description of the invention
The following is a description of specific embodiments of the process of the present invention using specific examples. The calculation example is as follows: in a two-dimensional coordinate plane, there are 10 clients whose coordinate points are known, and 3 service stations whose positions are to be determined need to be established and service relationships are allocated, so that the total distance is shortest, and the assumed service point capability is not constrained.
The invention discloses a facility site selection optimization method based on Euclidean distance linear approximation, which is shown in figure 1 and comprises the following specific implementation steps:
step one, data preprocessing
The coordinate data for 10 customer points in the two-dimensional plane are summarized in the following table:
TABLE 2 customer Point coordinate values
Figure BDA0001252859650000061
The warehouse point is then assigned a value of 3, and finally given a positive integer q of 18 and a fixed angle radian θ of 0.0895.
To fully demonstrate the feasibility of mathematical models, we solve using modeling language (i.e., AMPL)/mathematical programming model solver (i.e., CPLEX) solving software that describes and solves large-scale complex mathematical problems. AMPL is a powerful comprehensive algebraic language that can solve linear mathematical programming models. After the AMPL software reads in the model and the data file, the relevant solver is called to solve according to the execution strategy, and the solver used in the method is a CPLEX solver.
Step two, establishing a linear mathematical programming model
An objective function:
Figure BDA0001252859650000071
constraint conditions are as follows:
1)
Figure BDA0001252859650000072
2)
Figure BDA0001252859650000073
3)
Figure BDA0001252859650000074
4)
Figure BDA0001252859650000075
5)
Figure BDA0001252859650000076
6)
Figure BDA0001252859650000077
step three, solving the model
Based on the AMPL language, we create a data file p.dat for the above case:
Figure BDA0001252859650000078
Figure BDA0001252859650000081
the model file p.mod for the case is written according to a given mathematical model:
Figure BDA0001252859650000082
the mathematical model is linear, so that the optimal solution solving feasibility is realized. This example was solved using AMPL. Writing a corresponding batch file p.sh:
Figure BDA0001252859650000091
and finally, calling the batch processing file p.sh by using the AMPL to start solving.
Step four, outputting results
The final calculation results for the run with AMPL are as follows:
Figure BDA0001252859650000092
Figure BDA0001252859650000101
Figure BDA0001252859650000111
according to the final calculation result of the AMPL operation, the final warehouse point coordinates and the corresponding supply and demand relationship between the customer point and the warehouse can be obtained, as shown in fig. 2. The calculation result shows that the objective function value solved by the improved model, i.e. the sum of the distances between all the customer points and the warehouse points to which the customer points belong is 1.61633, while the objective function value obtained by the conventional k-means clustering method is 1.654592, and the final coordinates of the warehouse points and the corresponding supply and demand relationships between the customer points and the warehouse are shown in fig. 3. Finally, for the convenience of observation, the improved model and the distribution and belonging relationship graph of the traditional k-means calculation result warehouse and the customer points are drawn into a scatter diagram by using Tecplot drawing software.
Based on the same customer point, the calculation result is compared with the calculation result of the traditional k-means model, and it can be seen that the total distance obtained by the new method is smaller than that obtained by the traditional k-means method. The position coordinates of the warehouse calculated by the two methods are different from each other in the scatter diagram, and the method provided by the patent can find a better optimal site selection scheme.

Claims (1)

1. A facility site selection optimization method based on Euclidean distance linear approximation is characterized by comprising the following steps: the method comprises the following steps:
step one, data preprocessing preparation
Firstly, numbering customer points in a two-dimensional plane, wherein the numbering value is from 1 to N; secondly, the coordinates of the customer point need to be given, namely XiAbscissa, Y, representing customer point iiRepresents the ordinate of the customer point i; then, giving the serial number of the warehouse/facility to be selected, wherein the serial number value is from 1 to K; finally, assigning values for a given positive integer q and a given angle radian theta;
step two, establishing a linear mathematical programming model
According to the idea of the optimal addressing problem, the coordinate of the finally selected warehouse point is calculated to minimize the transportation cost, namely the sum of the distances between the warehouse point and the served customer point is minimized, and the sum of the distances between the warehouse point and the served customer point is called as an objective function; establishing a linear mathematical programming model based on a facility site selection method of Euclidean distance linear approximation;
step three, solving the model
Solving is carried out aiming at the linear mathematical programming model, and various solving modes are considered: (1) directly solving, namely solving the mathematical model by utilizing a simplex method, a branch and bound method and a secant plane method; (2) solving by using Lingo and CPLEX software;
the mathematical programming model is linear and completely has the feasibility of solving the optimal solution;
step four, outputting results: determining optimal warehouse point location coordinates
According to the final calculation result, the decision variable x of the modelk、ykAnd uikDetermining the coordinates of the warehouse points and the corresponding customer affiliations, if uikIf 1, the customer point i is served by the warehouse/facility k, so that an optimal objective function value, namely Total _ Dis, is obtained, namely the Total distance between the warehouse and the customer;
wherein, the establishment of the linear mathematical programming model in the step two is as follows:
(1) optimizing the sum of the distances between each client point and the subordinate warehouse points in the addressing problem, and defining the sum of the distances as an objective function; the objective function of the linear mathematical programming model of the problem is established by the distance d between each customer point i and the warehouse point k to which it belongsikSumming is performed and the sum of the distances is made as small as possible; based on this, the following objective function is established:
Figure FDA0002449971900000011
(2) each customer point in the two-dimensional coordinate plane is a demand point, and the warehouse points aim to provide services for the customer points, so that the constraint condition needs to ensure that each customer point i has one warehouse point k to provide services for the customer point i, that is, for any customer point i, it can be and can only be allocated to one warehouse point k, and the constraint condition 1 is established):
1)
Figure FDA0002449971900000021
(3) the distance constraint adopts Euclidean distance for constraint, so when a service relationship exists between a customer point i and a warehouse point k, the requirement that the x/y axial distance between the customer point i and the warehouse point k is not less than the horizontal/vertical coordinate difference between the customer point i and the warehouse point k is met, and constraint conditions 2) -5) are established:
2)
Figure FDA0002449971900000022
3)
Figure FDA0002449971900000023
4)
Figure FDA0002449971900000024
5)
Figure FDA0002449971900000025
(4) defining the number q of tangent planes and the rotation angle theta of two adjacent tangent planes, and obtaining d from the above limiting conditions 2) to 5)ik xdik yThereby specifying the euclidean distance d of the customer point i from the warehouse point kikI.e. establishing the constraint 6):
6)
Figure FDA0002449971900000026
therefore, the linear mathematical programming model is summarized as follows:
an objective function:
Figure FDA0002449971900000027
constraint conditions are as follows:
1)
Figure FDA0002449971900000028
2)
Figure FDA0002449971900000029
3)
Figure FDA00024499719000000210
4)
Figure FDA00024499719000000211
5)
Figure FDA00024499719000000212
6)
Figure FDA00024499719000000213
wherein N is a customer set, and N ═ {1,2,3, …, N }; k is the set of warehouses/facilities, K ═ {1,2,3, …, m }; m is a large number;
the model in the third step refers to a linear mathematical programming model composed of the objective function established in the second step and the constraint conditions 1) to 6); the "solving model" in the third step, the CPLEX solver solving model is called by using AMPL language, and the concrete method of solving is as follows:
(1) inputting data to be clustered and basic clustering parameters, and establishing an AMPL data file xxx.dat;
(2) establishing an AMPL model file xxx.mod, and establishing a linear mathematical programming model;
(3) establishing an AMPL batch file xxx.sh;
(4) sh, batch file xxx, was called using AMPL and the solution was started.
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