CN106022534B - The method for solving the problems, such as aircraft landing based on modified difference algorithm - Google Patents

The method for solving the problems, such as aircraft landing based on modified difference algorithm Download PDF

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CN106022534B
CN106022534B CN201610367227.9A CN201610367227A CN106022534B CN 106022534 B CN106022534 B CN 106022534B CN 201610367227 A CN201610367227 A CN 201610367227A CN 106022534 B CN106022534 B CN 106022534B
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aircraft landing
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高�浩
徐飞易
王保云
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Nanjing Post and Telecommunication University
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Abstract

The present invention proposes a kind of method for solving the problems, such as aircraft landing based on modified difference algorithm, specifically includes: establishing aircraft landing model and fitness function；It generates population primary and calculates the fitness value of each population at individual；It made a variation, intersected, generate population of new generation；Select keeping optimization；Judge termination condition, exports optimal solution.The present invention accelerates convergence rate, improves optimization precision and efficiency, while solving stagnation problem, and reduce the economic punishment in aircraft landing problem, has general applicability in aircraft landing problem.

Description

The method for solving the problems, such as aircraft landing based on modified difference algorithm
Technical field
The invention belongs to aircraft scheduling fields, especially a kind of to solve the problems, such as aircraft landing based on modified difference algorithm Method.
Background technique
In past 20 years, demand of the people to air transportation is dramatically increased, this leads to airspace increasingly congestion, is caused Airport is unable to cope with all demands.Therefore, airport administrator provide High-effective Service, shift or change some aircraft landings or Huge challenge is faced on the departure time, the poor efficiency that these variations may cause Airport Resources uses and reduce customer service, flies Machine landing problems play pivotal player on determining the landing times for reaching aircraft, therefore carry out the optimization of aircraft landing problem Calculating has great function to the cost reduction in practice.Aircraft landing problem includes building a set of aircraft landing time Table makes every airplane be dispensed on a specific time in this way and lands in a specific runway, while ensuring all problems Reach constraint condition with safety, it is therefore an objective to keep aircraft landing more early than the predetermined time to or it is late arrive caused by economic punishment minimize.
Aircraft landing problem is difficult nondeterministic polynomial (NP-hard) problem.For this reason that can connect In the time range received, heuristic and meta-heuristic algorithm is widely used for finding the solution of high quality instead of exact algorithm.To the greatest extent Pipe exact algorithm can provide best solution, but their calculating time is often as the increase of problem scale is at multiplication Long, this makes them be only suitable for middle-size and small-size problem.Although basic at present to solve the problems, such as that aircraft landing proposes numerous algorithms None algorithm is proved to be an effective solution problem method in all examples, and their performance usually with Example, which becomes larger, to be gradually reduced.
Differential evolution algorithm is a very famous evolution algorithm, highly effective when solving continuous optimization problems, can Cherish, differential evolution algorithm is not so good in the utilization of combinatorial optimization problem, wherein most important disadvantage is it Very high calculating is spent, and especially when population scale is larger, convergence rate declines rapidly, greatly reduces optimization Efficiency.Differential evolution algorithm is improved, just seems significant to make up the slow problem of convergence rate.
Summary of the invention
Technical problem solved by the invention is to provide a kind of solution aircraft landing based on modified difference algorithm and asks The method of topic introduces variation in difference algorithm, intersects, selection, accelerates convergence rate, improve optimization precision and efficiency, And reduce the economic punishment in aircraft landing problem.
The technical solution for realizing the aim of the invention is as follows:
The method for solving the problems, such as aircraft landing based on modified difference algorithm, comprising the following steps:
Step 1: the constraint condition and maximum number of iterations G of given aircraft landing problemmax, establish aircraft landing model and Fitness function；
Step 2: population primary, population scale NP being generated according to aircraft landing model, wherein each population at individual represents A kind of aircraft landing scheme；
Step 3: according to fitness function, calculating the fitness value of each population at individual；
Step 4: to the target solution of each population at individualMutation operation is carried out, mutation solution is generatedFollowing iterative formula It makes a variation for one-dimensional in target solution, i.e.,
Wherein, target solutionIndicate the set of the landing plans of all aircrafts；What n was represented is aircraft The sum of aircraft in landing problems, j indicate the number number of current aircraft,Indicate jth airplane in i-th set of landing plans Landing plans arrangement；F=rand (0.1,1.5), F indicate the mutagenic factor in mutation operation；WithIt is random respectively The solution chosen from current population, and
Step 5: to target solutionIt is solved with mutationCrossover operation is carried out, population of new generation is generated
Step 6: calculating population of new generationFitness value, and withFitness value compare, carry out selection generation Fresh target solution
Step 7: judging whether to reach maximum number of iterations Gmax, if so, outputOtherwise, step 3 is gone to.
Further, the method for the invention that solve the problems, such as aircraft landing based on modified difference algorithm, in step 1, The constraint condition of aircraft landing problem are as follows:
Wherein, xiFor the landing times of No. i-th aircraft, xjFor the landing times of jth aircraft, i=1,2 ..., n, j= 1,2 ..., n, i ≠ j, n are number of aircraft, EiFor earliest landing times as defined in No. i-th aircraft, LiFor as defined in No. i-th aircraft Landing times the latest, sijFor the regulation landing separations time of land No. i-th aircraft and jth aircraft on same runway.
Further, the method for the invention that solve the problems, such as aircraft landing based on modified difference algorithm, in step 1, Fitness function are as follows:
Wherein, f is fitness function, TiIndicate the target arrival time of No. i-th aircraft, C1iIndicate that No. i-th aircraft is opposite Target evening arrival time to and generate per unit time in economic punishment, C2iWhen indicating that No. i-th aircraft relative target reaches Between early arrive and generate per unit time in economic punishment, i=1,2 ..., n, n is number of aircraft.
Further, the method for the invention that solve the problems, such as aircraft landing based on modified difference algorithm, in step 2, Aircraft landing scheme includes the landing runway and landing times of every airplane.
Further, the method for the invention that solve the problems, such as aircraft landing based on modified difference algorithm, in step 5, Population of new generationAre as follows:
Wherein,CR ∈ [0,1] is cross rate, Rand (j) ∈ [0,1] Indicate that the aircraft for corresponding to jth number randomly selects a number, Rnd (i) ∈ { 1,2 ..., n } indicates the aircraft number randomly selected Number.
Further, the method for the invention that solve the problems, such as aircraft landing based on modified difference algorithm, in step 6, Fresh target solutionAre as follows:
Wherein,F () is fitness function.
The invention adopts the above technical scheme compared with prior art, has following technical effect that
1, method fast convergence rate of the invention, while solving stagnation problem, improve optimization efficiency；
2, method of the invention aircraft landing problem with during increase the diversity of population, optimize precision Height greatly reduces economic punishment cost；
3, method of the invention has general applicability in aircraft landing problem.
Detailed description of the invention
Fig. 1 is the method flow diagram of the invention that solve the problems, such as aircraft landing based on modified difference algorithm.
Specific embodiment
Embodiments of the present invention are described below in detail, the example of the embodiment is shown in the accompanying drawings, wherein from beginning Same or similar element or element with the same or similar functions are indicated to same or similar label eventually.Below by ginseng The embodiment for examining attached drawing description is exemplary, and for explaining only the invention, and is not construed as limiting the claims.
Aircraft landing problem is described as follows:
(1) every airplane must be assigned on a determining runway；
(2) the same time on same runway can only have lands no more than an airplane；
(3) landing times of every airplane all should be within the scope of time window predetermined；
(4) interval time of the interplane to land on same runway must reach the interval time of safety standard.
The parameter system of aircraft landing problem is initially set up, as shown in the table:
n Reach the number of aircraft
m The number of runway
Sij Aircraft i and aircraft j lands required interval time on same runway
Ti Aircraft i the set goal landing times
Ei The earliest landing times of aircraft i
Li The landing times the latest of aircraft i
C1i Per unit time interior economic punishment of aircraft i evening relative target time to generation
C2i The aircraft i relative target time early arrives the economic punishment interior per unit time of generation
xi The landing times of aircraft i
Method proposed by the present invention that solve the problems, such as aircraft landing based on modified difference algorithm, method flow such as Fig. 1 It is shown, specifically includes the following steps:
Step 1: the constraint condition and maximum number of iterations G of given aircraft landing problemmax, establish aircraft landing model and Fitness function.
The constraint condition of aircraft landing problem are as follows:
Wherein, xiFor the landing times of No. i-th aircraft, xjFor the landing times of jth aircraft, i=1,2 ..., n, j= 1,2 ..., n, i ≠ j, n are number of aircraft, EiFor earliest landing times as defined in No. i-th aircraft, LiFor as defined in No. i-th aircraft Landing times the latest, sijFor the regulation landing separations time of land No. i-th aircraft and jth aircraft on same runway.
Fitness function are as follows:
Wherein, f is fitness function, TiIndicate the target arrival time of No. i-th aircraft, C1iIndicate that No. i-th aircraft is opposite Target evening arrival time to and generate per unit time in economic punishment, C2iWhen indicating that No. i-th aircraft relative target reaches Between early arrive and generate per unit time in economic punishment, i=1,2 ..., n, n is number of aircraft.
Step 2: population primary being generated according to aircraft landing model, population scale takes NP=5, wherein each population at individual Represent a kind of aircraft landing scheme.The population scale that this place takes is smaller, and referred to as micro- differential evolution can effectively promote difference Divide the convergence rate of evolution algorithm, but will increase the risk for stagnation problem occur.
Aircraft landing scheme solves representation are as follows:
The landing plans of one airplane include the landing runway and landing times of aircraft, with the real number of a mixed decimal come table Show landing plans, integer part indicates the runway to land, and fractional part indicates the landing times on the runway.
The aircraft landing scheme of the present embodiment is as shown in the table:
Aircraft number 1 2 3 4 5 6
Landing plans 2.4 1.45 1.3 3.7 3.55 2.2
Here runway sum is 3, so the value of integer part respectively represents three runways, fractional part in [1,3] Dividing indicates landing times, on same runway, the priority in respect of time of mode in ascending order.Such as No. 3 fly The landing plans 1.3 of machine indicate are as follows: landing runway is runway 1, and (aircraft is default in 1 landing times of runway-runway 1 by 0.3= Earliest can landing times)/(runway 1 it is default the latest can landing times-runway 1 is default earliest can landing times).
Step 3: according to fitness function, calculating the fitness value of each population at individual.
Step 4: to the target solution of each population at individualMutation operation is carried out, mutation solution is generatedFollowing iterative formula It makes a variation for one-dimensional in target solution, i.e.,
Wherein, target solutionIndicate the set of the landing plans of all aircrafts；What n was represented is aircraft The sum of aircraft in landing problems, j indicate the number number of current aircraft,Indicate jth airplane in i-th set of landing plans Scheme arrangement；F=rand (0.1,1.5), F indicate the mutagenic factor in mutation operation；WithBe respectively at random from The solution chosen in current population, and
Here mutagenic factor F be not only randomly select it is primary, but when carry out mutation operation calculating every time all Again F=rand (0.1,1.5) operation is done, the diversity that can greatly increase population is done so, to efficiently solve contracting Small Population scale bring stagnation problem.
Step 5: to target solutionIt is solved with mutationCrossover operation is carried out, population of new generation is generated
Population of new generationIt indicates are as follows:
Wherein,CR ∈ [0,1] is cross rate, Rand (j) ∈ [0,1] Indicate that the aircraft for corresponding to jth number randomly selects a number, Rnd (i) ∈ { 1,2 ..., n } indicates the aircraft number randomly selected Number, Rnd (i) operation ensure thatIt at least can be fromOne variable of middle acquirement.
Step 6: calculating population of new generationFitness value, and withFitness value compare, carry out selection generation Fresh target solution
Fresh target solutionIt indicates are as follows:
Wherein,F () is fitness function.
Step 7: judging whether to reach maximum number of iterations Gmax, if so, outputOtherwise, step 3 is gone to.
So far, the method whole process for solving the problems, such as aircraft landing based on modified differential evolution algorithm terminates, and takes State process acquisition optimal solution, we it follows that
This method is generally applicable to aircraft landing problem first, and gained optimal solution reaches current existing algorithm optimization Level substantially reduces economic punishment and cost by optimization, moreover, it is excellent in solution combination to solve conventional differential evolution algorithm The problem of slow convergence in change problem, improve the speed and efficiency of optimization.
The above is only some embodiments of the invention, it is noted that for the ordinary skill people of the art For member, without departing from the principle of the present invention, several improvement can also be made, these improvement should be regarded as guarantor of the invention Protect range.

Claims (1)

1. the method for solving the problems, such as aircraft landing based on modified difference algorithm, which comprises the following steps:
Step 1: the constraint condition and maximum number of iterations G of given aircraft landing problemmax, establish aircraft landing model and adaptation Spend function；
Step 2: population primary, population scale NP, wherein each population at individual represents one kind are generated according to aircraft landing model Aircraft landing scheme；
Step 3: according to fitness function, calculating the fitness value of each population at individual；
Step 4: to the target solution of each population at individualMutation operation is carried out, mutation solution is generatedFollowing iterative formula is directed to One-dimensional in target solution makes a variation, i.e.,
Wherein, target solutionIndicate the set of the landing plans of all aircrafts；What n was represented is aircraft landing The sum of aircraft in problem, j indicate the number number of current aircraft,Indicate the landing of jth airplane in i-th set of landing plans Scheme arrangement；F=rand (0.1,1.5), F indicate the mutagenic factor in mutation operation；WithIt is at random from working as respectively The solution chosen in preceding population, and
Step 5: to target solutionIt is solved with mutationCrossover operation is carried out, population of new generation is generated
Step 6: calculating population of new generationFitness value, and withFitness value compare, carry out selection and generate new mesh Mark solution
Step 7: judging whether to reach maximum number of iterations Gmax, if so, outputOtherwise, step 3 is gone to；
In step 1, the constraint condition of aircraft landing problem are as follows:
Wherein, xiFor the landing times of No. i-th aircraft, xjFor the landing times of jth aircraft, i=1,2 ..., n, j=1, 2 ..., n, i ≠ j, n are number of aircraft, EiFor earliest landing times as defined in No. i-th aircraft, LiFor as defined in No. i-th aircraft most Late landing times, sijFor the regulation landing separations time of land No. i-th aircraft and jth aircraft on same runway；
In step 1, fitness function are as follows:
Wherein, f is fitness function, TiIndicate the target arrival time of No. i-th aircraft, C1iIndicate No. i-th aircraft relative target Evening arrival time to and generate per unit time in economic punishment, C2iIndicate that No. i-th aircraft relative target arrival time is early To and generate per unit time in economic punishment, i=1,2 ..., n, n is number of aircraft；
In step 2, aircraft landing scheme includes the landing runway and landing times of every airplane；
In step 5, population of new generationAre as follows:
Wherein,CR ∈ [0,1] is cross rate, and Rand (j) ∈ [0,1] is indicated Aircraft corresponding to jth number randomly selects a number, and Rnd (i) ∈ { 1,2 ..., n } indicates the aircraft number number randomly selected；
In step 6, fresh target solutionAre as follows:
Wherein,F () is fitness function.
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