CN108921379B - Job-based scheduling problem heterogeneity analysis method driven by fitness terrain - Google Patents
Job-based scheduling problem heterogeneity analysis method driven by fitness terrain Download PDFInfo
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Abstract
The invention discloses a method for analyzing heterogeneity of Job-based scheduling problems driven by a fitness terrain, and belongs to the field of Job-based scheduling problems. Selecting a problem example according to actual needs, and drawing a fitness terrain according to the corresponding relation between the solution and the fitness in the solution space; and (3) regarding the fitness terrain as discrete sequence points, and analyzing the fitness terrain features from the angle of the amplitude spectrum by utilizing discrete time Fourier transform. Calculating a main evaluation index and an auxiliary evaluation index of the fitness terrain; and analyzing and comparing the numerical value and the change rule of each evaluation index to obtain the similarity and the difference of each problem example. The adaptability terrain characteristic parameters provided by the invention have strong adaptability and generality, and can be used for different scale examples of the same problem and the characteristics of longitudinal analysis problems, and also can be used for the same scale examples of different problems and the dissimilarity of different problems in transverse analysis.
Description
Technical Field
The invention belongs to the field of Job-based scheduling problems, and particularly relates to a method for analyzing heterogeneity of Job-based scheduling problems driven by a fitness terrain.
Background
In recent years, the scheduling problem is widely applied to various fields such as manufacturing industry, service industry, cloud computing and internet of things. The Job-based scheduling problem is a large branch of the scheduling field and comprises flexible workshop scheduling (FJSP), pipeline scheduling problem (FSSP), Test Task Scheduling (TTSP) and the like. The scheduling problem is composed of a series of tasks which are executed sequentially or in parallel, and under the condition that various constraint conditions are met, reasonable execution sequences and efficient resource allocation modes are arranged for the tasks, so that the method is beneficial to obtaining larger economic benefits with smaller time and resource cost, and plays an important role in improving task execution efficiency in related fields and optimizing resource allocation.
The Job-based scheduling problem can be abstracted as a combination of an allocation problem and a sequencing problem, namely a combination optimization problem under a plurality of constraint conditions, so that the Job-based scheduling problem has certain relevance. In terms of problem characteristics, there is a "combinatorial explosion" effect, which is an NP-hard problem from the computational time complexity point of view. In terms of optimizing the objective, the objective function is one or more of completion time, work error time, cost or average load, and the like, and can generally be converted into a minimization problem. From the solution, the scheduling problem mostly goes through the evolution process of an accurate algorithm, a heuristic algorithm, a meta-heuristic algorithm and a hybrid algorithm. The precise algorithm is only suitable for the early small-scale problem, and currently, the heuristic algorithm and the meta-heuristic algorithm are mostly fused so as to comprehensively consider the global search capability and the local search capability of the algorithm and improve the solution performance.
Various Job-based scheduling problems have strong relevance in the aspects of problem characteristics, scheduling targets, solving methods, development processes and the like, but the current research on the scheduling problems does not utilize solution space priori knowledge, and does not analyze the problem characteristics and relevance, and the research mode which is relatively isolated and does not refer to each other is not beneficial to the theoretical research and development of the scheduling problems.
Fitness terrain analysis is a common means of obtaining solution space prior knowledge. Through the topographic analysis of the fitness, the distribution of the fitness value in the solution space is researched, the structural characteristics of the solution space are facilitated, and the problem characteristics and the change rule are further analyzed. Common fitness terrain evaluation parameters comprise ruggedness, fitness distance correlation degree, evolvability and the like, and establishing a corresponding parameter description system aiming at different fitness terrain characteristics is a main direction for the theoretical development of the fitness terrain.
And expanding and embodying certain characteristic evaluation parameters by combining the problem description of the scheduling problems, providing a basis for the heterogeneity analysis and algorithm design among the Job-based scheduling problems, facilitating the mutual reference among the scheduling problems and realizing the tight coupling of the problem characteristics and the algorithm design.
Disclosure of Invention
The invention adopts a fitness terrain analysis method to research Job-based scheduling problems, further explores the similarity and difference among the scheduling problems, and promotes the mutual reference and theoretical development among scheduling fields. In particular to a Job-based scheduling problem heterogeneity analysis method driven by a fitness terrain.
Comprises the following steps:
step one, in a Job-based task scheduling problem, selecting a problem instance according to actual needs;
secondly, drawing a fitness terrain according to the corresponding relation between the solution and the fitness in the solution space of the problem example;
the fitness terrain arranges all solutions or sampling solutions in a solution space in sequence as an abscissa, and uses the fitness of each solution as an ordinate, so that the distribution and change rule of the solution space are intuitively reflected.
And step three, regarding the fitness terrain as discrete sequence points, and analyzing the characteristics of the fitness terrain from the angle of the amplitude spectrum by utilizing discrete time Fourier transform.
Considering each fitness terrain as a discrete time series x (n):
n is the number of discrete points in the sequence, j is an imaginary number, and ω is the angular frequency.
Substituting the discrete time series X (n) into amplitude values | X (e) of different frequency components in the frequency spectrumjω) And in |, performing discrete time Fourier transform to obtain a corresponding frequency spectrum.
Calculating a main evaluation index and an auxiliary evaluation index of the fitness terrain;
the main evaluation indexes aim at two fitness terrains and comprise a similarity index and a sharpness index which are used for comparing the two fitness terrains; the auxiliary evaluation indexes comprise amplitude variation Stability (SAC) and periodicity of each fitness terrain;
the similarity index is: and describing the similarity of two fitness terrains on an external structure by using the dynamic time bending distance so as to reflect the similarity of solution space.
The concrete description is as follows:
firstly, two fitness terrains f1And f2Respectively regarded as discrete sequences, and labeledNormalizing the elimination of amplitude effects to obtain a sequence F1And F2；
Then constructing an adjacent matrix, and finding out the satisfaction by adopting a dynamic programming mode
The shortest curved path of (a); w is akIs the bending consumption of the kth lattice point in the path.
Finally, the degree of similarity of the terrain is characterized by the dynamic time warping distance as sim (f)1,f2)＝DTW(F1,F2). The smaller the index value, the higher the degree of similarity between the two fitness terrains.
The sharpness index reflects the mutation condition between solution space neighborhood solutions and describes the sharpness degree of the adaptability terrain;
the concrete description is as follows:
firstly, aiming at the condition that the two compared solution spaces have different scales, a large-scale solution space is set as A, a small-scale solution space is set as B, the adaptability terrain corresponding to the large-scale solution space A is divided into a plurality of sections, and the number of solutions in each section is the same as that of the solutions in the small-scale solution space B.
Then, aiming at the large-scale solution space A or the small-scale solution space B, the sizes of two adjacent solution fitness values in the solution space are sequentially compared to obtain a new data sequence.
Starting from a first solution fitness value with the large-scale solution space A or the small-scale solution space B, if a second solution fitness value is larger than the first solution fitness value, replacing the first solution fitness value and the second solution fitness value by 1, and otherwise, replacing the first solution fitness value and the second solution fitness value by-1; if two adjacent solution fitness values are equal, 0 is used to replace the two. By analogy, a new integer sequence {1, -1,0} is obtainedN-1。
Continuing, traversing the new data sequence from front to back, if 0 appears continuously, calculating the frequency of 0 appearing continuously and recording as aiI represents the number of times this condition occurs; if 1 or-1 appears continuously, the number of times 1 or-1 appears continuously is counted and is respectively marked as ciAnd di(ii) a If 1 and-1 alternate, count the number of times of + -1 alternate and note as ei(ii) a If it goes outIn other cases, use biAnd (6) recording. A is marked in an accumulative way at the corresponding coordinate position of the scatter diagrami,bi,ci,di,eiConnecting the scatter points according to the generation order to obtain an undirected graph, and connecting ai,bi,ci,di,eiThe last cumulative value is denoted as asum,bsum,csum,dsum,esum。
Finally, according to the contribution of each case to the sharpness, weights are assigned to the accumulated values and the sharpness is obtained:
keetd＝asum×pa+bsum×pb+csum×pc+dsum×pd+esum×pe
pais an accumulated value asumAssigned weight, pbIs an accumulated value bsumAssigned weight, pcIs an accumulated value csumAssigned weight, pdIs an accumulated value dsumAssigned weight, peIs an accumulated value esumThe assigned weight; the assignment of weights is obtained from empirical values. The greater the sharpness value, the sharper the fitness terrain.
The amplitude variation Stability (SAC) reflects the degree of variation of the side lobe with respect to the main lobe (amplitude at a frequency of 0) in the adaptive terrain spectrum, and reflects the shape of the adaptive terrain to some extent.
The calculation formula is as follows:
wherein A isside(i)Represents the ith side lobe amplitude in the amplitude spectrum; a. themainRepresenting the amplitude of the main lobe in the amplitude spectrum. n is the number of all side lobes. The larger the difference between the main lobe and the side lobe, the more drastic the change of the side lobe relative to the main lobe, and the larger the value of sta.
The periodicity of the fitness terrain is characterized by the distance between the main lobe and the first side lobe in the frequency spectrum.
The calculation formula is as follows:
fHis the frequency at the first side lobe, i.e. the lobe closest to frequency 0. f. ofCIs the frequency at the main lobe.
A terrain is approximately divided into several parts with similar shapes and lengths, and each part is a period. The value of per is used to reflect the length of the period.
And step five, analyzing and comparing the numerical value of each evaluation index and the change rule to obtain the similarity and difference of each problem example.
The invention has the advantages that:
(1) a Job-based scheduling problem heterogeneity analysis method driven by fitness terrain provides fitness terrain characteristic parameters with strong adaptability and generality, does not need to depend on specific priori knowledge, and has no special requirements on coding modes, solution space distribution, algorithm selection and the like.
(2) A fitness terrain-driven Job-based scheduling problem heterogeneity analysis method provides an effective means for obtaining solution space priori knowledge, is beneficial to understanding problem characteristics, and provides guidance for algorithm design and parameter adjustment.
(3) A method for analyzing the heterogeneity of Job-based scheduling problems driven by a fitness terrain can analyze the similarity and the heterogeneity of different Job-based scheduling problems, is beneficial to promoting mutual reference among different scheduling fields, and provides theoretical support for the design of a unified solution framework of the scheduling problems.
(4) The method is flexible to apply, can be used for different-scale examples of the same problem and characteristics of longitudinal analysis problems, and can also be used for the same-scale examples of different problems and the heterogeneity among different problems is transversely analyzed.
Drawings
FIG. 1 is a block diagram of a method for analyzing the heterogeneity of a Job-based scheduling problem driven by a fitness terrain according to the present invention.
FIG. 2 is a terrain evaluation system for fitness of the present invention.
FIG. 3 is a schematic diagram of a similarity evaluation index according to the present invention.
FIG. 4 is a schematic diagram of a sharpness evaluation index according to the present invention.
Fig. 5 is a fitness terrain map containing 3 examples of Job-based scheduling problems in an embodiment of the present invention.
Fig. 6 is a fitness terrain map comprising 2 solution spaces in an embodiment of the present invention.
Fig. 7 is an undirected graph obtained by calculating a sharpness index in an embodiment of the present invention.
Detailed Description
The invention will be further explained in detail with reference to the drawings and the embodiments.
The invention discloses a Job-based scheduling problem heterogeneity analysis method driven by fitness terrain time-frequency domain evaluation indexes. And then, dividing the indexes into a main index and an auxiliary index according to the capability and the importance degree of each index for reflecting the terrain features, and researching the problem characteristics and the similarities and the differences of the Job-based scheduling problems.
The process is shown in FIG. 1, and comprises the following steps:
step one, in a Job-based task scheduling problem, selecting a problem instance according to actual needs;
in each Job-based task scheduling problem such as flexible workshop scheduling, parallel machine scheduling or automatic test task, a plurality of problem examples with different scales exist. Their task numbers, resource numbers, processing times are different, resulting in some differences in their solution spaces, but the same problem descriptions and constraints, in turn, determine that their solution spaces are likely to have some consistent characteristics. In addition, the Job-based scheduling problems belong to the combined optimization problem of the ordering problem and the distribution problem, and certain common points must exist among the problems, which are reflected in the similarity of the external structure of the solution space. In order to explore the problem characteristics of a specific problem or the heterogeneity among several scheduling problems, an appropriate problem instance can be selected according to actual needs.
Secondly, drawing a fitness terrain according to the corresponding relation between the solution and the fitness in the solution space of the problem example;
the fitness terrain is a reflection and abstraction of the solution space, which visually describes the correspondence of feasible solutions in the solution space to fitness values. All solutions or sampling solutions in a solution space are sequentially arranged according to a certain mode to be used as an abscissa, and the fitness of each solution is used as an ordinate, so that the distribution and change rule of the solution space are visually reflected.
For example, enumeration of the solution space may be performed through traversal of the full permutation of the task sequence and the solution combination, and then the fitness terrain may be generated in the order in which the solutions were produced. The complete arrangement method of the task sequence has various choices, such as a dictionary ordering method, an incremental carry method, a decremental carry method, an adjacent position exchange method, a recursive algorithm and the like, and the traversal of the scheme combination can adopt an arbitrary system addition counter to sequentially increase the scheme number of each task. Along with the increase of the number of tasks and the sharp increase of the solution space scale, a reasonable sampling mode can be adopted, for example, after equal-interval sampling, a fitness terrain is drawn.
And step three, regarding the fitness terrain as discrete sequence points, and analyzing the characteristics of the fitness terrain from the angle of the amplitude spectrum by utilizing discrete time Fourier transform.
The discrete-time fourier transform and its inverse are shown below:
each fitness terrain is considered as a discrete time series x (n), n is the number of discrete points in the series, j is an imaginary number, and ω is an angular frequency.
DTFT decomposes the sequence x (n) into a series of complex exponential sequences that differ in angular frequency. I X (e)jω) I represents the amplitude values of different frequency components in the frequency spectrum; it may reflect the sequence characteristics from a frequency domain perspective. The method is used for the fitness terrain analysis of the scheduling problem, the fitness terrain is regarded as a discrete sequence point, and the feature of the fitness terrain is analyzed from the angle of a magnitude spectrum by adopting DTFT.
Calculating a main evaluation index and an auxiliary evaluation index of the fitness terrain;
as shown in fig. 2, the fitness terrain evaluation system quantitatively evaluates certain characteristics of the fitness terrain from the perspective of time domain and frequency domain. The time domain indexes comprise similarity, sharpness and the like, and respectively reflect the similarity degree of the external structure of the terrain and the mutation degree between neighborhood solutions. The frequency domain indexes comprise amplitude change periodicity, stability and the like, and respectively reflect the period length and the fluctuation degree of the fitness terrain.
According to the capability and the importance degree of the evaluation indexes for reflecting the problem characteristics, the evaluation indexes can be divided into main indexes and auxiliary indexes for carrying out heterogeneity analysis; the main evaluation indexes aim at two fitness terrains and comprise a similarity index and a sharpness index which are used for comparing the two fitness terrains; the auxiliary evaluation indexes comprise amplitude variation Stability (SAC) and periodicity of each fitness terrain;
the similarity index is: and describing the similarity of two fitness terrains on an external structure by using the dynamic time bending distance so as to reflect the similarity of solution space.
As shown in fig. 3, the calculation process includes, for two fitness terrains to be compared, respectively performing terrain normalization and establishing an adjacency matrix to obtain similar point matching for dynamic planning, and calculating a shortest curved path and a dynamic curved distance DTW to obtain the similarity between the two fitness terrains. The concrete description is as follows:
firstly, two fitness terrains f1And f2Respectively regarded as discrete sequences, and normalized to eliminate amplitude influence to obtain a sequence F1And F2；
According to the formula
Normalizing the fitness terrain, wherein f (x)i) Is the solution x in the terrainiFitness value of (d), f' (x)i) Is solving for xiN is the number of solutions. The normalized fitness terrain for the two problem instances is treated as two sequences and is represented as:
F1＝f1'(x1),f1'(x2),...,f1'(xn)
F2＝f2'(x1),f2'(x2),...,f2'(xm)
then n m construct a contiguous matrix, the elements (i, j) in the matrix representing the point qiAnd point cjLocal distance d (q) therebetweeni,cj) The calculation formula is as follows: d (q)i,cj)＝(qi-cj)2(ii) a And (3) calculating the shortest curved path by adopting a dynamic programming mode:
γ(i,j)＝d(qi,cj)+min{γ(i-1,j-1),γ(i-1,j),γ(i,j-1)}
γ (i, j) represents the cumulative distance, which consists of two parts: the first part is the local distance d (q) of the current grid pointi,cj) The second part is the minimum cumulative distance to an adjacent grid point that can reach the current grid point.
The dynamic time warping distance may be expressed as DTW (F)1,F2) γ (n, m), and then the similarity sim (f) of two fitness terrains is obtained1,f2)＝DTW(F1,F2) (ii) a The smaller the index value, the higher the degree of similarity between the two fitness terrains.
The sharpness index reflects the mutation condition between solution space neighborhood solutions and describes the sharpness degree of the adaptability terrain;
the key steps are as shown in fig. 4, firstly, the adjacent adaptability values in the solution space of two adaptability terrains are compared, through sequence conversion and trend classification, coordinate points are calibrated through statistical classification, line segments are connected in sequence to draw an undirected graph, and therefore the sharpness index of the adaptability terrains is calculated.
The concrete description is as follows:
firstly, aiming at the condition that the two compared solution spaces have different scales, a large-scale solution space is set as A, a small-scale solution space is set as B, the adaptability terrain corresponding to the large-scale solution space A is divided into a plurality of sections, and the number of solutions in each section is the same as that of the solutions in the small-scale solution space B.
Then, aiming at the large-scale solution space A or the small-scale solution space B, the sizes of two adjacent solution fitness values in the solution space are sequentially compared to obtain a new data sequence.
Starting from a first solution fitness value of which the large-scale solution space is A or the small-scale solution space is B, if an adjacent second solution fitness value is larger than the first solution fitness value, replacing the first solution fitness value and the second solution fitness value by 1, and otherwise, replacing the first solution fitness value and the second solution fitness value by-1; if two adjacent solution fitness values are equal, 0 is used to replace the two. By analogy, a new integer sequence {1, -1,0} is obtainedN-1Replacing the original fitness terrain.
Continuing, traversing the new data sequence from front to back, if 0 appears continuously, calculating the frequency of 0 appearing continuously and recording as aiI represents the number of times this condition occurs; if 1 or-1 appears continuously, the number of times 1 or-1 appears continuously is counted and is respectively marked as ciAnd di(ii) a If 1 and-1 alternate, count the number of times of + -1 alternate and note as ei(ii) a If other conditions occur, use biAnd (6) recording. A is marked in an accumulative way at the corresponding coordinate position of the scatter diagrami,bi,ci,di,eiConnecting the scatter points according to the generation order to obtain an undirected graph, and connecting ai,bi,ci,di,eiThe last cumulative value is denoted as asum,bsum,csum,dsum,esum。
Finally, based on the contribution to sharpness for each case, weights are assigned to the accumulated values and sharpness kee is obtainedtd：
keetd＝asum×pa+bsum×pb+csum×pc+dsum×pd+esum×pe
paIs an accumulated value asumAssigned weight, pbIs an accumulated value bsumAssigned weight, pcIs an accumulated value csumAssigned weight, pdIs an accumulated value dsumAssigned weight, peIs an accumulated value esumThe assigned weight; the distribution of the weights is obtained from empirical values and can be adjusted slightly to achieve better results. The greater the sharpness value, the sharper the fitness terrain.
Assume that solution space 1 has 32 feasible solutions, and the fitness values of these solutions constitute a fitness vector [ 63468555647764352639561010101057491212 ]. The fitness terrain of the solution space is shown in fig. 6(a), and the sharpness index is calculated according to the steps:
1) the first step is skipped since there is only one solution space and the scale is small.
2) Traversing the solution space from the first one, 3 is less than 6 so they are replaced by-1, followed by 4 is greater than 3 so they are replaced by 1, and so on to replace the remaining solutions and get the new sequence [ -1111-.
3) The first two numbers in the new sequence belong to the case where 1 and-1 occur alternately, so e12-1+ 1-2. The 2 nd to 4 th numbers belong to the case of 1 occurring consecutively, whereby c14-2+ 1-3. The 4 th and 5 th numbers are again in the case of alternating 1 and-1, so that e25-4+ 1-2. In this way, the new sequence is traversed and other values b are obtained1＝2,a1＝2,b2＝2,e3＝3,b3＝3,d1＝3,e4＝8,c2＝2,b4＝2,a2＝3,b5＝2,e5＝4,c3＝2,b 62. Plotting these points at the corresponding positions in fig. 7(a) in an accumulative manner yields a scatter diagram. Scatter points are connected according to the generation sequence, and the accumulated value at each abscissa, a, can be obtainedsum＝a1+a2＝5,bsum＝b1+b2+b3+b4+b5+b6＝13,csum＝c1+c2+c3＝7,dsum＝d1＝3,esum＝e1+e2+e3+e4+e519. The connection process is shown in fig. 7 (a).
4) Finally, the sharpness value, p, is calculated according to the formulaa，pb，pc，pd，peSet to-1, -0.6, -0.2, 1:
kee1＝5*(-1)+13*(-0.6)+7*(-0.2)+3*(-0.2)+19*1＝4.2
assume that solution space 2 also has 32 feasible solutions, with a vector of fitness values of [ 666685556477645555555610101010574999 ]]. The suitability terrain of the solution space is shown in fig. 6(b), and an undirected graph is drawn in the same manner to obtain fig. 7(b), and the sharpness value is calculated to obtain kee2-17.2. Due to kee1＞kee2Therefore, the sharpness degree of the solution space 1 is greater than that of the solution space 2, which corresponds to the actual situation.
The amplitude variation Stability (SAC) reflects the degree of variation of the side lobe with respect to the main lobe (amplitude at a frequency of 0) in the adaptive terrain spectrum, and reflects the shape of the adaptive terrain to some extent.
The calculation formula is as follows:
wherein A isside(i)Represents the ith side lobe amplitude in the amplitude spectrum; a. themainRepresenting the amplitude of the main lobe in the amplitude spectrum. n is the number of all side lobes. The larger the difference between the main lobe and the side lobe, the more drastic the change of the side lobe relative to the main lobe, and the value of staThe larger.
The periodicity of the fitness terrain is characterized by the distance between the main lobe and the first side lobe in the frequency spectrum.
The calculation formula is as follows:
fHis the frequency at the first side lobe, i.e. the lobe closest to frequency 0. f. ofCIs the frequency at the main lobe.
If a terrain is divided approximately into several parts of similar shape and length, each part is a period. The value of per is used to reflect the length of the period.
And step five, analyzing and comparing the numerical value and the change rule of each evaluation index to obtain the characteristics and the similarities and the differences of the solution space of each problem example.
Through index analysis and rule summarization, the problem characteristics of the same problem with different scales and the similarities and differences among different problems can be searched.
The invention respectively selects small-scale problem examples in three fields to illustrate the specific implementation process.
First, three problem instances were selected: the method is taken from the fields of flexible workshop scheduling, parallel machine scheduling and automatic test task scheduling, and the conditions of the number of tasks, scheme selection, processing time and the like of each example are shown in table 1.
TABLE 1
Then, when the adaptability terrain is obtained, the problem solution space is enumerated through the full arrangement of the task sequence and the traversal of the scheme combination. Here, the incremental carry preparation method is selected as a full-permutation algorithm of the task sequence, and the intermediary number is used as an intermediate link for calculating the permutation. And for a certain task sequence, adopting an arbitrary system addition counter to increment the scheme sequence number of each task until all possibilities of scheme combination are traversed. And sequentially arranging all solutions according to the generation sequence to be used as a horizontal coordinate, and using the fitness value of each solution as a vertical coordinate to obtain the fitness terrain. The fitness topography for each example in table 1 is shown in fig. 5.
Continuing to perform discrete time Fourier transform, regarding each fitness terrain as a discrete time sequence x (n), and substituting into a formula
And carrying out discrete time Fourier transform to obtain a corresponding frequency spectrum.
And finally, calculating a terrain evaluation index: similarity among the three problem examples in the table 1 is calculated pairwise, sharpness is calculated on 3 fitness terrains in the graph 5 according to a small-scale solution space calculation mode, and calculation results of main indexes among different scheduling problems are recorded in the table 2.
TABLE 2
Respectively calculating 0-frequency amplitude A in frequency spectrums of 3 fitness terrainsmainAnd side lobe amplitude Aside(i)Substituting into the formula
The amplitude variation stability is obtained and the mean value characteristic is calculated from the 0-frequency amplitude.
Finding main lobe frequency spectrum f in frequency spectrum of 3 fitness terrains respectivelyCAnd frequency f at the first side lobeHSubstituting into the formula
Resulting in a periodicity of the three terrains.
The above secondary index calculation results between different scheduling problems are recorded in table 3.
TABLE 3
And finally, analyzing the numerical value and the change rule of each index in the table to obtain the similarity and difference of each problem in the aspects of external structure, fluctuation degree, periodicity, sharpness and the like.
Claims (5)
1. A Job-based scheduling problem heterogeneity analysis method driven by fitness terrain is characterized by comprising the following steps:
step one, in a Job-based task scheduling problem, selecting a problem instance according to actual needs;
secondly, drawing a fitness terrain according to the corresponding relation between the solution and the fitness in the solution space of the problem example;
step three, regarding the fitness terrain as discrete sequence points, and analyzing the characteristics of the fitness terrain from the angle of an amplitude spectrum by utilizing discrete time Fourier transform;
considering each fitness terrain as a discrete time series x (n):
n is the number of discrete points in the sequence, j is an imaginary number, and omega is the angular frequency;
substituting the discrete time series X (n) into amplitude values | X (e) of different frequency components in the frequency spectrumjω) In |, performing discrete time Fourier transform to obtain a corresponding frequency spectrum;
calculating a main evaluation index and an auxiliary evaluation index of the fitness terrain;
the main evaluation indexes aim at two fitness terrains and comprise a similarity index and a sharpness index which are used for comparing the two fitness terrains; the auxiliary evaluation index comprises the stability and periodicity of amplitude change of each fitness terrain;
the sharpness index reflects the mutation condition between solution space neighborhood solutions and describes the sharpness degree of the adaptability terrain; the concrete description is as follows:
firstly, aiming at the condition that the two compared solution spaces have different scales, setting a large-scale solution space as A and a small-scale solution space as B, dividing the adaptability terrain corresponding to the large-scale solution space A into a plurality of sections, wherein the number of solutions in each section is the same as that of the solutions in the small-scale solution space B;
then, aiming at the large-scale solution space A or the small-scale solution space B, sequentially comparing the sizes of two adjacent solution fitness values in the solution space to obtain a new data sequence;
starting from a first solution fitness value with the large-scale solution space A or the small-scale solution space B, if a second solution fitness value is larger than the first solution fitness value, replacing the first solution fitness value and the second solution fitness value by 1, and otherwise, replacing the first solution fitness value and the second solution fitness value by-1; if two adjacent solution fitness values are equal, 0 is used for replacing the two solution fitness values; by analogy, a new integer sequence {1, -1,0} is obtainedN-1；
Continuing, traversing the new data sequence from front to back, if 0 appears continuously, calculating the frequency of 0 appearing continuously and recording as aiI represents the number of times this condition occurs; if 1 or-1 appears continuously, the number of times 1 or-1 appears continuously is counted and is respectively marked as ciAnd di(ii) a If 1 and-1 alternate, count the number of times of + -1 alternate and note as ei(ii) a If other conditions occur, use biRecording; a is marked in an accumulative way at the corresponding coordinate position of the scatter diagrami,bi,ci,di,eiConnecting the scatter points according to the generation order to obtain an undirected graph, and connecting ai,bi,ci,di,eiThe last cumulative value is denoted as asum,bsum,csum,dsum,esum；
Finally, according to the contribution of each case to the sharpness, weights are assigned to the accumulated values and the sharpness is obtained:
keetd＝asum×pa+bsum×pb+csum×pc+dsum×pd+esum×pe
pais an accumulated value asumAssigned weight, pbIs an accumulated value bsumAssigned weight, pcIs an accumulated value csumAssigned weight, pdIs an accumulated value dsumAssigned weight, peIs an accumulated value esumThe assigned weight; the distribution of the weights is obtained from empirical values; the larger the sharpness value is, the sharper the adaptability terrain is;
and step five, analyzing and comparing the numerical value of each evaluation index and the change rule to obtain the similarity and difference of each problem example.
2. The method for analyzing the heterogeneity of Job-based scheduling problems driven by the fitness terrain of claim 1, wherein the fitness terrain in the second step is that all solutions or sampled solutions in the solution space are arranged in sequence as abscissa, and the fitness of each solution is used as ordinate, so as to visually reflect the distribution and change rules of the solution space.
3. The method for analyzing the heterogeneity of a Job-based scheduling problem driven by a fitness terrain according to claim 1, wherein the similarity index in the fourth step is: describing the similarity of two fitness landforms on an external structure by using the dynamic time bending distance so as to reflect the similarity degree of a solution space; the concrete description is as follows:
firstly, two fitness terrains f1And f2Respectively regarded as discrete sequences, and normalized to eliminate amplitude influence to obtain a sequence F1And F2；
Then constructing an adjacent matrix, and finding out the satisfaction by adopting a dynamic programming mode
The shortest curved path of (a); w is akThe bending consumption of the kth lattice point in the path;
finally, the degree of similarity of the terrain is characterized by the dynamic time warping distancesim(f1,f2)＝DTW(F1,F2) (ii) a The smaller the index value, the higher the degree of similarity between the two fitness terrains.
4. The method for analyzing the heterogeneity of Job-based scheduling problems driven by the adaptive terrain according to claim 1, wherein the amplitude variation stability in the step four reflects the shape of the adaptive terrain to some extent, relative to the variation degree of the main lobe, of the side lobe in the adaptive terrain spectrum;
the calculation formula is as follows:
wherein A isside(i)Represents the ith side lobe amplitude in the amplitude spectrum; a. themainRepresenting the amplitude of the main lobe in the amplitude spectrum; n is the number of all side lobes; the larger the difference between the main lobe and the side lobe, the more drastic the change of the side lobe relative to the main lobe, and the larger the value of sta.
5. The method for analyzing the heterogeneity of a fitness terrain-driven Job-based scheduling problem of claim 1, wherein the periodicity of the fitness terrain in the fourth step is characterized by a distance between a main lobe and a first side lobe in a frequency spectrum;
the calculation formula is as follows:
fHis the frequency at the first side lobe, i.e. the frequency at the lobe closest to frequency 0; f. ofCIs the frequency at the main lobe;
approximately dividing a terrain into a plurality of parts with similar shapes and lengths, wherein each part is a period; the value of per is used to reflect the length of the period.
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