CN104809890B - The Traffic Signal Timing optimization method of Orthogonal Genetic Algorithm is improved based on principal component analysis and Local Search - Google Patents
The Traffic Signal Timing optimization method of Orthogonal Genetic Algorithm is improved based on principal component analysis and Local Search Download PDFInfo
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Abstract
The Traffic Signal Timing optimization method of Orthogonal Genetic Algorithm is improved based on principal component analysis and Local Search, this algorithm is proposed by analyzing the inner link between genetic algorithm and image procossing and pattern-recognition, can be used for solving various function optimization problems.This algorithm proposes a kind of improvement orthogonal crossover operator based on principal component analysis.The operator carries out PCA projections to prechiasmal population first, reduces individual lengths when intersecting, then implements Orthogonal crossover operator in projection domain；Again luv space is projected to after the completion of intersection, is reduced because of redundancy individual number and computing cost caused by intersection.In order to further improve convergence of algorithm speed, local searching strategy is also introduced.This algorithm is applied to Single Intersection signal timing optimization problem, by illustrating the versatility and validity of algorithm with existing algorithm progress test comparison, the effective timing time has been obtained, has reduced the queuing vehicle number before intersection.
Description
Technical field
The invention belongs to the optimization problem of municipal traffic control signal timing.Know with Orthogonal Genetic Algorithm and biological characteristic
The algorithm (being specifically related to Orthogonal Genetic Algorithm, principal component analysis PCA and local searching strategy) of other technology crossing domain is realized
City Single Intersection signal timing dial is controlled.
Background technology
With the fast development of China's economy, urbanization process is constantly accelerated, vehicle guaranteeding organic quantity also quick increase therewith,
Traffic trip amount increases significantly, and transportation supplies wretched insufficiency, disparities between supply and demand highlight.By taking Beijing as an example, Beijing motor vehicle at present
Recoverable amount has broken through 2,000,000, and urban road annual growth rate is 3%, and vehicle growth rate is 15%, and vehicle flowrate year increases
Speed is up to 18%.
As the important component of city traffic network, intersection is the bottleneck and traffic jam and thing of road passage capability
Therefore multiplely.The traffic congestion in city is due to largely the traffic capacity deficiency of intersection or does not make full use of and causes
, this causes wagon flow to be interrupted, accident increases, it is serious to be delayed.Motor vehicle in big city is in about three points of intown running time
One of be used for level-crossing；And U.S.'s traffic accident there are about more than half and occur in intersection.As can be seen here, to intersecting cause for gossip
The management of row science and the important subject that control is traffic control engineering, it is to ensure the traffic safety of intersection and fully send out
The important measures of the traffic capacity of intersection are waved, are the effective ways for solving urban transport problems.
At present, the signal controller of most domestic intersection derives from SCOOT (the Split Cycle and of Britain
Offset Optimization Teclmiquel) system, Australia SCAT (Sydney Coordinated
Adaptive Traffic) system and Japan the system of capital three, using timing controlled and Self Adaptive Control.These methods are passing through
It is widely used after crossing improvement.
At present, based on the control system of China's signal is controlled with single-point, so the signal timing dial research to Single Intersection has
A lot：He Zhao into et al. the multiphase traffic signal real-time control method, Zhang Cuicui et al. based on state demarcation that proposes use
Webster splits algorithm optimize control to model, Mu Haibo etc. proposes control method based on Petir nets etc..
Due to the non-linear of traffic, ambiguity and uncertainty, the optimization problem of intersection signal timing can typically be attributed to non-convex
Nonlinear problem, for traditional optimization method frequently with algebraic method and diagram method etc., it is global that these methods can not find it well
Optimal solution, and genetic algorithm is a kind of search technique based on natural selection and evolution, is widely used in optimization problem, therefore
Genetic algorithm is also widely used in the signal timing optimization problem of traffic control.Song Xue birch et al., which has been invented, is based on genetic algorithm
Single Intersection signal timing optimization method, the genetic algorithm used in the invention is standard genetic algorithm, wherein selection strategy
Optimum reserved strategy is added, but standard genetic algorithm local search ability is not strong, is easily trapped into early Convergent Phenomenon.
For this problem, the internal relation between present invention analysis genetic algorithm and the general principle of feature extraction, carry
A kind of orthogonal crossover operator based on principal component analysis is gone out, PCA projections first is carried out to population before being intersected, reduced and intersect
When individual lengths, project to luv space again after the completion of intersection, reduce because of redundancy individual number and calculating caused by intersection
Expense.In order to further improve convergence of algorithm speed, local searching strategy is also introduced.Solves city Single Intersection with this
Signal timing dial control problem.
The content of the invention
The purpose of the present invention is to propose to a kind of improvement Orthogonal Genetic Algorithm based on principal component analysis and Local Search
(HPOGA) it is used for the control of city Single Intersection signal timing dial, signal is carried out by optimization aim of the queuing vehicle number before crossing
Timing designing, realize the optimal control of traffic signals.
The improvement Orthogonal Genetic Algorithm (HPOGA) based on principal component analysis and Local Search of the present invention, it is characterised in that
PCA projections first are carried out to population before being intersected, reduces individual lengths when intersecting, original sky is projected to again after the completion of intersection
Between, reduce because of redundancy individual number and computing cost caused by intersection.In order to further improve convergence of algorithm speed, also introduce
Local searching strategy.
A kind of traffic signal optimization timing method that Orthogonal Genetic Algorithm is improved based on principal component analysis and Local Search, bag
Include following steps：
S1 carries out individual UVR exposure, initialization data, and setup parameter：
The individual represents the combination of green time；Use tiThe green time of i phases is represented, for offspring caused by holding
The validity of body, using 3 ageings, individual UVR exposure form is：＜ t1 t2 t3＞, decimally encoded；It is described to set
Determining parameter includes：Crossover probability Pc is set as 0.8, mutation probability Pm is 0.01, individual lengths 3；
S2 initialization of population：
According to following orthogonal n 3 individuals of initialization of population method generation, composition initial population P0；
S2.1 finds the s dimensions for meeting following formula；
If S2.2 solution spaces are [l, u], then solution space is divided into S sub-spaces in s Wei Chu
[l(1),u(1)],[l(2),u(2)]...[l(s),u(s)]；
Wherein, Is=[c1,j]1×N,
S3 generates interim population P ' gen
If gen is evolutionary generation, each individual in colony Pgen is selected with Probability p cross, adds interim group
Body P ' gen；
S4 crossover operations
If xiRepresent to elect the individual to be intersected according to crossover probability Pc, then X=[x1,x2...xn] represent
The population colony of Orthogonal crossover operator will be carried out, n represents population size, the specific steps of the orthogonal crossover operator based on PCA
It is as follows：
X is expressed as matrix Pop by S4.1n×m=[x1 T,x2 T...xn T]T, an each of which behavior individual, m expression dyeing
Body length.
S4.2 carries out PCA projections to Pop, obtains Pop'n×p=[y1 T,y2 T...yn T]T, the data dimension in p expression PCA domains,
yiIndividual x is represented respectivelyiThe data obtained after dimensionality reduction.
S4.3 carries out random pair, every a pair of individual y to the individual in colony Pop'iAnd yi+1Carry out Orthogonal crossover operator,
Orthogonal arrage construction step construction orthogonal arrage L in orthogonalM(QF)=[ai,j]M×p, p is yiAnd yi+1Dimension,
Utilize orthogonal arrage LM(QF) come to yiAnd yi+1Orthogonal is carried out, M data combination will be produced.
New caused M data combination is carried out PCA reflections and is mapped to original domain by S4.4, to obtained individual progress fitness
Evaluation, a best individual of fitness is selected to be added in colony Cgen.
S4.5 colonies Cgen is exactly that Pop' carries out new caused progeny population after crossover operation.
S5 Local Searches
Population P generates new population Lgen after clustering Local Search.Comprise the following steps that：
If S5.1 populations P scale is n, cluster is carried out according to individual similarity to the individual in P and is divided into some height kinds
Group, the individual amount in every sub- population is set as m (m=3 during this paper algorithms are set)；
S5.2 performs SPX operations to the every sub- population divided in S5.1, produces g offspring individual, i.e., each division
The progeny population (g=10 in this paper algorithms) of population；
S5.3 is added to each progeny population in population Lgen.
S6 mutation operations
To any individual pi=(pi, 1, pi, 2 ... pi, N) in population P', i ∈ { 1,2 ... n }, with probability
Pmutation participates in mutation operation：Produce a decimal r ∈ [0,1], and a random integers j ∈ [1, N]；Make pi, j=lj+
R* (uj-lj), row variation is entered to colony P ' gen and produces novel species colony Ggen.
S7 selection operations
In order to keep population diversity, first chosen from population (Pgen+Cgen+Lgen+Ggen)Individual fitness
It is worth best individual and adds population Pgen+1 of future generation, then from the remaining individual of population (Pgen+Cgen+Lgen+Ggen), with
Machine is chosenIndividual is added to population Pgen+1 [28] of future generation.
S8 end conditions judge
Reach and stop algebraically or continuous 50 generation optimal value is constant or find globally optimal solution, stop evolving and output result,
Otherwise S3 is turned.
Thus effective traffic signal optimization timing is obtained.
Compared with prior art, the present invention has the advantages that.
Improvement Orthogonal Genetic Algorithm (HPOGA) based on principal component analysis and Local Search, first to kind before being intersected
Group carries out PCA projections, reduces individual lengths when intersecting, luv space is projected to again after the completion of intersection, reduce and produced because intersecting
Raw redundancy individual number and computing cost.In order to further improve convergence of algorithm speed, local searching strategy is also introduced.
By the test in Single Intersection signal timing optimization problem, the versatility and validity of algorithm are demonstrated, is effectively matched somebody with somebody
When the time, reduce intersection before queuing vehicle number.
Brief description of the drawings
Fig. 1 is the wagon flow distribution map of Single Intersection；
Fig. 2 is the Signal Phase Design figure that Single Intersection is applied in the present invention, in figure：First phase is East and West direction straight trip and the right side
Turn；Second phase is turned left for thing；Third phase is that north-south is kept straight on and turned right；4th phase is turned left for north and south；
Fig. 3 is the flow chart of method therefor of the present invention.
Fig. 4 is principal component analysis (PCA) orthogonal crossover operator figure.
Specific implementation
The present invention will be further described with specific embodiment below in conjunction with the accompanying drawings.
The present invention uses four phases, and three lanes design method, the wagon flow distribution map of Single Intersection is as shown in figure 1, Characteristics for Single Staggered
The Signal Phase Design figure of mouth is as shown in Fig. 2 the orthogonal heredity of the improvement proposed by the present invention based on principal component analysis and Local Search is calculated
As shown in figure 3, whole flow process is the orthogonal crossover operator based on PCA, the population that will be intersected projects to the flow chart of method
PCA subspace, Orthogonal crossover operator then is carried out to population on PCA transform domains, caused individual passes through PCA again after intersection
Back projection carries out fitness evaluation to original domain.Wherein, principal component analysis (PCA) orthogonal crossover operator is as shown in Figure 4.
The implementation process of the present invention is described in detail with reference to Fig. 3.Embodiments of the invention are with the technology of the present invention
Implemented premised on scheme, give detailed embodiment and specific operating process, but protection scope of the present invention
It is not limited to following embodiments.
Embodiment from Single Intersection signal timing dial optimize problem to innovatory algorithm (HPOGA) presented herein and
Classical genetic algorithm (SGA), based on minimum spanning tree cluster genetic algorithm (CGA) and conventional method tested and compared.
By comparing it can be seen that algorithms of different is handling the performance of different problems.These algorithms use optimum maintaining strategy.
In this embodiment, by solving object function (with track of being let pass on clearance track after respective phase state
On queuing vehicle sum be performance indications) obtain the signal timing dial of current period, i.e., the green time under each phase, institute
With the combination of individual expression green time here.Use tiRepresent the green time of i phases, while in order to after caused by keeping
The validity of generation individual, using 3 ageings, individual UVR exposure form is：＜ t1 t2 t3＞, encoded with binary system.
The explanation of each detailed problem involved in the inventive technique scheme is provided in detail below：
S1 carries out individual UVR exposure, initialization data, and setup parameter：
The individual represents the combination of green time；Use tiThe green time of i phases is represented, for offspring caused by holding
The validity of body, using 3 ageings, individual UVR exposure form is：＜ t1 t2 t3＞, decimally encoded；It is described to set
Determining parameter includes：Crossover probability Pc is set as 0.8, mutation probability Pm is 0.01, individual lengths 3；
S2 initialization of population：
According to following orthogonal n 3 individuals of initialization of population method generation, composition initial population P0；
S2.1 finds the s dimensions for meeting following formula；
If S2.2 solution spaces are [l, u], then solution space is divided into S sub-spaces in s Wei Chu
[l(1),u(1)],[l(2),u(2)]...[l(s),u(s)]；
Wherein, Is=[c1,j]1×N,S refers to S sub-spaces, and s refers to s dimension.
S sub-spaces are may be divided into i.e. each s dimensions, are S sub-spaces on the whole
S3 generates interim population P ' gen
If gen is evolutionary generation, each individual in colony Pgen is selected with Probability p cross, adds interim group
Body P ' gen；
S4 crossover operations
If xiRepresent to elect the individual to be intersected according to crossover probability Pc, then X=[x1,x2...xn] represent
The population colony of Orthogonal crossover operator will be carried out, n represents population size, the specific steps of the orthogonal crossover operator based on PCA
It is as follows：
X is expressed as matrix Pop by S4.1n×m=[x1 T,x2 T...xn T]T, an each of which behavior individual, m expression dyeing
Body length.
S4.2 carries out PCA projections to Pop, obtains Pop'n×p=[y1 T,y2 T...yn T]T, the data dimension in p expression PCA domains,
yiIndividual x is represented respectivelyiThe data obtained after dimensionality reduction.
S4.3 carries out random pair, every a pair of individual y to the individual in colony Pop'iAnd yi+1Carry out Orthogonal crossover operator,
Orthogonal arrage construction step construction orthogonal arrage L in orthogonalM(QF)=[ai,j]M×p, p is yiAnd yi+1Dimension,
Utilize orthogonal arrage LM(QF) come to yiAnd yi+1Orthogonal is carried out, M data combination will be produced.
M individuals caused by new are carried out PCA reflections and are mapped to original domain by S4.4, and fitness evaluation is carried out to obtained individual,
One best individual of fitness of selection is added in colony Cgen.
S4.5Cgen is exactly that Pop carries out new caused progeny population after crossover operation.
S5 Local Searches
Population P generates new population Lgen after clustering Local Search.Comprise the following steps that：
S5.1 assumes that population P scale is n, and carrying out cluster according to individual similarity to the individual in P is divided into some height
Population, the individual amount in every sub- population are set as m (m=3 during this paper algorithms are set)；
S5.2 performs SPX operations to the every sub- population divided in Step1, produces g offspring individual, i.e., each division
The progeny population (g=10 in this paper algorithms) of population；
S5.3 is added to each progeny population in population Lgen.
S6 mutation operations
To any individual pi=(pi, 1, pi, 2 ... pi, N) in population P ' gen, i ∈ { 1,2 ... n }, with probability
Pmutation participates in mutation operation：Produce a decimal r ∈ [0,1], and a random integers j ∈ [1, N]；Make pi, j=lj+
R* (uj-lj), row variation is entered to colony P ' gen and produces new population Ggen.
S7 selection operations
In order to keep population diversity, first chosen from population (Pgen+Cgen+Lgen+Ggen)Individual fitness
It is worth best individual and adds population Pgen+1 of future generation, then from the remaining individual of population (Pgen+Cgen+Lgen+Ggen), with
Machine is chosenIndividual is added to population Pgen+1 [28] of future generation.
S8 end conditions judge
Reach and stop algebraically or continuous 50 generation optimal value is constant or find globally optimal solution, stop evolving and output result,
Otherwise S3 is turned.
The experimental result of the explanation present invention is explained in detail below：
In order to prove validity of the method for the invention in the control of Single Intersection signal timing dial, HPOGA is respectively adopted
(algorithm in the present invention), SGA (StandardGeneticAlgorithm, standard genetic algorithm), CGA are (based on minimum generation
Tree cluster genetic algorithm) and conventional method Single Intersection signal timing dial is optimized, each optimization process calculates 10 cycles
Timing.Experimental result is as shown in table 1.
Table 1HPOGA and the Comparative result of conventional method, SGA and CGA
As shown in Table 1, timing optimal solution can be found using the inventive method each cycle, and SGA and CGA are only able to find
3 near-optimal solutions, conventional method then can not find optimal solution.Absolutely prove that the queue length that the inventive method obtains not only is far smaller than
The queue length that conventional method obtains, also it is significantly less than the queue length obtained using SGA and CGA methods.Therefore, with existing skill
Art is compared, and the present invention can obtain the more effective timing time, is reduced the queuing vehicle number before intersection, is effectively improved Characteristics for Single Staggered
The traffic capacity of mouth.
Claims (1)
1. improving the Traffic Signal Timing optimization method of Orthogonal Genetic Algorithm based on principal component analysis and Local Search, its feature exists
In：Comprise the following steps,
S1 carries out individual UVR exposure, initialization data, and setup parameter；
The individual represents the combination of green time；Use tiThe green time of i phases is represented, for offspring individuals caused by holding
Validity, using 3 ageings, individual UVR exposure form is：＜ t1 t2 t3＞, decimally encoded；The setting ginseng
Number includes：Crossover probability Pc is set as 0.8, mutation probability Pm is 0.01, individual lengths 3；
S2 initialization of population,
According to following orthogonal n 3 individuals of initialization of population method generation, composition initial population P0；
S2.1 finds the s dimensions for meeting following formula；
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Wherein, Is=[c1,j]1×N,
S3 generates interim population P ' gen
If gen is evolutionary generation, each individual in colony Pgen is selected with Probability p cross, adds interim population P '
gen；
S4 crossover operations
If xiRepresent to elect the individual to be intersected according to crossover probability Pc, then X=[x1,x2...xn] represent will
The population colony of Orthogonal crossover operator is carried out, n represents population size, and the orthogonal crossover operator based on PCA comprises the following steps that：
X is expressed as matrix Pop by S4.1n×m=[x1 T,x2 T...xn T]T, an each of which behavior individual, m expression chromosome length
Degree；
S4.2 carries out PCA projections to Pop, obtains Pop'n×p=[y1 T,y2 T...yn T]T, the data dimension in p expression PCA domains, yiPoint
Individual x is not representediThe data obtained after dimensionality reduction；
S4.3 carries out random pair, every a pair of individual y to the individual in colony Pop'iAnd yi+1Orthogonal crossover operator is carried out, according to
Orthogonal arrage construction step construction orthogonal arrage L in orthogonalM(QF)=[ai,j]M×p, p is yiAnd yi+1Dimension, utilize
Orthogonal arrage LM(QF) come to yiAnd yi+1Orthogonal is carried out, M data combination will be produced；
New caused M data combination is carried out PCA and reflexes to original domain by S4.4, and fitness evaluation is carried out to obtained individual,
One best individual of fitness of selection is added in colony Cgen；
S4.5 colonies Cgen is exactly that Pop' carries out new caused progeny population after crossover operation；
S5 Local Searches
Population P generates new population Lgen after clustering Local Search；Comprise the following steps that：
If S5.1 populations P scale is n, cluster is carried out according to individual similarity to the individual in P and is divided into some sub- populations, often
Individual amount in individual sub- population is set as m；
S5.2 performs SPX operations to the every sub- population divided in S5.1, produces g offspring individual, i.e., each divides sub- population
Progeny population；
S5.3 is added to each progeny population in population Lgen；
S6 mutation operations
To any individual pi=(p in population P'i,1, pi,2…pi,N), i ∈ { 1,2 ... n }, participate in becoming with probability P mutation
ETTHER-OR operation：Produce a decimal r ∈ [0,1], and a random integers j ∈ [1, N]；Make pi, j=lj+r* (uj-lj), to group
Body P ' gen enter row variation and produce new colony Ggen；
S7 selection operations
In order to keep population diversity, first chosen from population (Pgen+Cgen+Lgen+Ggen)Individual fitness value is most
Good individual adds population Pgen+1 of future generation, then from the remaining individual of population (Pgen+Cgen+Lgen+Ggen), random choosing
TakeIndividual is added to population Pgen+1 [28] of future generation；
S8 end conditions judge
Reach and stop algebraically or continuous 50 generation optimal value is constant or find globally optimal solution, stop evolving and output result, otherwise
Turn S3；
Thus effective traffic signal optimization timing is obtained；
The method for calculating ideal adaptation angle value in population is as follows,
Queuing vehicle sum is used as optimization on the phase clearance track after being terminated using Single Intersection in a cycle per phase
Object function in target, i.e. genetic algorithm, and fitness function, its expression formula are
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</mrow>
<mn>3</mn>
</munderover>
<msub>
<mi>t</mi>
<mi>i</mi>
</msub>
</mrow>
<mo>)</mo>
<mo>-</mo>
<msub>
<mi>p</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
<mi>k</mi>
</mrow>
</msub>
<mo>*</mo>
<msub>
<mi>u</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
<mi>k</mi>
</mrow>
</msub>
<mo>*</mo>
<msub>
<mi>t</mi>
<mi>i</mi>
</msub>
<mo>-</mo>
<msub>
<mi>p</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
<mi>k</mi>
</mrow>
</msub>
<mo>*</mo>
<msub>
<mi>u</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
<mi>k</mi>
</mrow>
</msub>
<mo>*</mo>
<mo>(</mo>
<mrow>
<mi>T</mi>
<mo>-</mo>
<munderover>
<mi>&Sigma;</mi>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mn>3</mn>
</munderover>
<msub>
<mi>t</mi>
<mi>i</mi>
</msub>
</mrow>
<mo>)</mo>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mrow>
<mi>s</mi>
<mo>.</mo>
<mi>t</mi>
<mo>:</mo>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<msub>
<mi>t</mi>
<mn>1</mn>
</msub>
<mo>+</mo>
<msub>
<mi>t</mi>
<mn>2</mn>
</msub>
<mo>+</mo>
<msub>
<mi>t</mi>
<mn>3</mn>
</msub>
<mo>+</mo>
<msub>
<mi>t</mi>
<mn>4</mn>
</msub>
<mo>=</mo>
<mi>T</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>6</mn>
<mo>&le;</mo>
<msub>
<mi>t</mi>
<mi>i</mi>
</msub>
<mo>&le;</mo>
<mi>T</mi>
<mo>-</mo>
<mn>18</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>,</mo>
<mrow>
<mo>(</mo>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mn>2</mn>
<mo>,</mo>
<mn>3</mn>
<mo>,</mo>
<mn>4</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula, T is the Cycle Length of Single Intersection signal control；tiThe timing of the expression phase of crossing four, i=1,2,3,4；λijkTable
Show the vehicle arriving rate in the i-th phase j directions k tracks, j=1,2,3,4, represent respectively eastwards, westwards, southwards and northwards four enter
Mouth direction, k=1,2,3, represent respectively, three tracks of straight trip and right-hand rotation；uijkRepresent the car in the i-th phase j directions k tracks
Clearance rate；After the l cycles, the vehicle queue number in the i-th phase j directions k tracks, expression formula is
<mrow>
<msubsup>
<mi>s</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
<mi>k</mi>
</mrow>
<mi>l</mi>
</msubsup>
<mo>=</mo>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<msubsup>
<mi>s</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
<mi>k</mi>
</mrow>
<mrow>
<mi>l</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msubsup>
<mo>+</mo>
<msub>
<mi>&lambda;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
<mi>k</mi>
</mrow>
</msub>
<mo>*</mo>
<msub>
<mi>t</mi>
<mi>i</mi>
</msub>
<mo>-</mo>
<msub>
<mi>p</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
<mi>k</mi>
</mrow>
</msub>
<mo>*</mo>
<msub>
<mi>u</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
<mi>k</mi>
</mrow>
</msub>
<mo>*</mo>
<msub>
<mi>t</mi>
<mi>i</mi>
</msub>
<mo>,</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<msubsup>
<mi>s</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
<mi>k</mi>
</mrow>
<mrow>
<mi>l</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msubsup>
<mo>+</mo>
<msub>
<mi>&lambda;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
<mi>k</mi>
</mrow>
</msub>
<mo>*</mo>
<msub>
<mi>t</mi>
<mi>i</mi>
</msub>
<mo>&GreaterEqual;</mo>
<msub>
<mi>u</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
<mi>k</mi>
</mrow>
</msub>
<mo>*</mo>
<msub>
<mi>t</mi>
<mi>i</mi>
</msub>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>0</mn>
<mo>,</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<msubsup>
<mi>s</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
<mi>k</mi>
</mrow>
<mrow>
<mi>l</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msubsup>
<mo>+</mo>
<msub>
<mi>&lambda;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
<mi>k</mi>
</mrow>
</msub>
<mo>*</mo>
<msub>
<mi>t</mi>
<mi>i</mi>
</msub>
<mo><</mo>
<msub>
<mi>u</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
<mi>k</mi>
</mrow>
</msub>
<mo>*</mo>
<msub>
<mi>t</mi>
<mi>i</mi>
</msub>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
</mrow>
In formula, pijkRelease status matrix is represented, its expression formula is：
<mrow>
<msub>
<mi>p</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
<mi>k</mi>
</mrow>
</msub>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mfenced open = "{" close = "}">
<mtable>
<mtr>
<mtd>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo>,</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>,</mo>
<mn>0</mn>
<mo>,</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo>,</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>,</mo>
<mn>0</mn>
<mo>,</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
</mtd>
</mtr>
<mtr>
<mtd>
<mfenced open = "{" close = "}">
<mtable>
<mtr>
<mtd>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>,</mo>
<mn>0</mn>
<mo>,</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo>,</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>,</mo>
<mn>0</mn>
<mo>,</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo>,</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
</mtd>
</mtr>
<mtr>
<mtd>
<mfenced open = "{" close = "}">
<mtable>
<mtr>
<mtd>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>,</mo>
<mn>0</mn>
<mo>,</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>,</mo>
<mn>0</mn>
<mo>,</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>,</mo>
<mn>0</mn>
<mo>,</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>,</mo>
<mn>0</mn>
<mo>,</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
</mtd>
</mtr>
<mtr>
<mtd>
<mfenced open = "{" close = "}">
<mtable>
<mtr>
<mtd>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>,</mo>
<mn>0</mn>
<mo>,</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>,</mo>
<mn>0</mn>
<mo>,</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>,</mo>
<mn>0</mn>
<mo>,</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>,</mo>
<mn>0</mn>
<mo>,</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
</mtd>
</mtr>
</mtable>
</mfenced>
</mrow>
In formula, " 0 " represents that the corresponding track under respective phase is in and forbids release status, and " 1 " represents the correspondence under respective phase
Track is in release status.
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