CN102930340A - Adaptive genetic annealing calculation method for solving zero-one knapsack problem - Google Patents
Adaptive genetic annealing calculation method for solving zero-one knapsack problem Download PDFInfo
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Abstract
The invention relates to an adaptive genetic annealing calculation method for solving a zero-one knapsack problem. The algorithm adopts a selection mechanism combining roulette and an optimal storage strategy, so that the current optimal individuals are always kept in a group; by the adaptive intersection and variation probability, the group search range is expanded, then a simulated annealing algorithm is introduced, and the convergence speed of an iterative later-stage algorithm is increased; and finally, the improved adaptive genetic annealing algorithm is applied to a zero/one knapsack. An experiment result indicates that by the adaptive genetic annealing algorithm, a more satisfactory effect than that of a standard genetic algorithm and that of the conventional adaptive genetic algorithm can be achieved. The adaptive genetic annealing calculation method has the advantages of high convergence speed, high optimizing capacity and high stability, and is particularly suitable for achieving an effect of high-dimension constraint optimization.
Description
Technical field
The present invention relates to a kind of, particularly a kind of.
Background technology
The 0-1 knapsack: a given bag that dead weight capacity is w, n article, its weight is w
i, be worth and be v
i, 1＜=i＜=n requires: the article knapsack of packing into, and make in the bag Item Value maximum.In the 0-1 knapsack problem, object or be loaded into knapsack perhaps is not loaded into knapsack, only has two kinds of selections.Find the solution which article of taking-up and can make the most costly knapsack capacity that do not exceed again of the article that fill in the knapsack.
0-1 knapsack problem (Zero-one Knapsack Problem, be called for short ZKP) be typical NP(Non.deterministic Polynomia in the operational research) complete problem, it is the uncertain problems of polynomial expression complexity, np complete problem is the most difficult problem in the NP class, as long as its connotation is to have a np complete problem to have polynomial time algorithm, then all there is polynomial time algorithm in whole NP class problem.
At present, the conventional method that solves 0/1 knapsack comprises the method for exhaustion, dynamic programming and recursive backtracking method etc., but can only process on a small scale knapsack problem.Heuritic approach is the New Algorithm of simulating nature circle and biological behavior, has a model flexible, finds the solution the series of advantages such as quality height that speed is fast, separate, thereby obtained widespread use.But the optimized algorithm speed of convergence such as genetic algorithm, ant group algorithm, differential evolution algorithm are slow, global convergence is poor, and simulated annealing (Simulated Annealing Algorithm, be called for short SA) can improve the defective that is absorbed in locally optimal solution, make algorithm converge on rapidly globally optimal solution.Therefore, be necessary to carry out Adaptive Genetic annealing algorithm (Adaptive Genetic Annealing Algorithm, abbreviation AGAA) research, being about to simulated annealing SA and Genetic Algorithms combines, and improvement cross and variation strategy, so that the algorithm after improving had both kept the high-speed parallel of GA, kept again SA to jump out the ability of local optimum.
Summary of the invention
The present invention be directed to the easy Premature Convergence of standard genetic algorithm and the slow problem of speed of convergence, a kind of Adaptive Genetic annealing computing method of the 0-1 of solution knapsack problem have been proposed, have advantages of that speed of convergence, optimizing ability and stability are high, be particularly suitable for solving higher-dimension constrained optimization problem.
Technical scheme of the present invention is: a kind of Adaptive Genetic annealing computing method that solve the 0-1 knapsack problem specifically comprise the steps:
1) set algorithm parameter comprises population size popsize, chromosome length chromlong, and annealing initial temperature T0 moves back warm coefficient k etc.;
2) produce initial population pop (0);
3) according to fitness function each individual fitness value in the colony is made an appraisal, and judge whether it meets Optimality Criteria, if meet, the optimum solution of output optimized individual and representative thereof, and finish to calculate; Otherwise execution following steps;
A) carry out genetic manipulation, the selection that selection strategy adopts roulette and optimum conversation strategy to combine is machine-processed, and the optimum number of preserving is made as 2, and intersection, mutation operation adopt self-adaptation intersection, variation probability, produce SA initial population sa-pop;
Can b) simulated annealing operation, acceptable conditions be adopted the Metropolis criterion and are differentiated and enter colony of future generation; Carry out popsize iteration and produce the pop of colony of future generation (i+1);
C) the temperature operation is moved back in execution, and makes i=i+1; Judge whether to reach end condition, the words that are then stop, otherwise return the 3rd) step.
The selection mechanism that adopts roulette and optimum conversation strategy to combine in the described step a), to keep current optimum chromosome, namely keeping 2 optimum dyeing bodies from current population directly copies in the population of future generation, remaining population is selected to enter in the population of future generation by the roulette method, to improve convergence.
Intersection, mutation operation adopt self-adaptation intersection, variation probability in the described step a), and crossover probability Pc and the probability P m that makes a variation carry out the self-adaptation adjustment by following formula correspondence:
In the formula
P c1
=0.9,
P c2
=0.6,
P m1
=0.1,
P m2
=0.001,
f Max -be fitness value maximum in the population;
f Avg Average fitness value for per generation population;
f 1Be larger fitness value in two individualities of needs intersection;
fBe the individual fitness value of needs variation.
Beneficial effect of the present invention is: the present invention solves the Adaptive Genetic annealing computing method of 0-1 knapsack problem, introduce the genetic algorithm (AGAA) of self-adaptation intersection, variation and Annealing Strategy, added according to the competition between the chromosome, improved speed of convergence, enlarge simultaneously the hunting zone of population, strengthen the ability that population jumps out local optimum, be specially adapted to solve higher-dimension constrained optimization problem, had preferably search efficiency.
Description of drawings
Fig. 1 is the Adaptive Genetic annealing computing method schematic flow sheet that the present invention solves the 0-1 knapsack problem;
Fig. 2 is the best fitness value convergence curve comparison diagram of the present invention when solving embodiment scheme N=20 in the Adaptive Genetic annealing computing method of 0-1 knapsack problem;
Fig. 3 is the best fitness value convergence curve comparison diagram of the present invention when solving embodiment scheme N=50 in the Adaptive Genetic annealing computing method of 0-1 knapsack problem;
Fig. 4 is the present invention's independent operating average fitness convergence curve comparison diagram when solving in the Adaptive Genetic annealing computing method of 0-1 knapsack problem embodiment scheme N=20;
Fig. 5 is the present invention's independent operating average fitness convergence curve comparison diagram when solving in the Adaptive Genetic annealing computing method of 0-1 knapsack problem embodiment scheme N=50.
Embodiment
Solve as shown in Figure 1 the Adaptive Genetic annealing computing method schematic flow sheet of 0-1 knapsack problem, may further comprise the steps:
1) set algorithm parameter comprises population size popsize, chromosome length chromlong, and annealing initial temperature T0 moves back warm coefficient k etc.;
2) produce initial population pop (0);
3) according to fitness function each individual fitness value in the colony is made an appraisal, and judge whether it meets Optimality Criteria, if meet, the optimum solution of output optimized individual and representative thereof, and finish to calculate; Otherwise execution following steps;
A) carry out genetic manipulation, selection strategy adopts roulette and optimum conversation strategy, and the optimum number of preserving is made as 2, and intersection, mutation operation employing self-adaptation are intersected, the variation probability, produce SA initial population sa-pop;
For fear of the situation that is caused destroying optimum solution or approximate optimal solution by intersection, mutation operation, the selection mechanism that adopts roulette and optimum conversation strategy to combine, to keep current optimum chromosome, namely keeping 2 optimum dyeing bodies from current population directly copies in the population of future generation, remaining population is selected to enter in the population of future generation by the roulette method, to improve convergence.
In the whole genetic evolution process, the selection of crossover probability Pc and variation probability P m is the crux that affects genetic algorithm behavior and performance, in order to keep population diversity, Pc is larger, the new individual speed that produces is faster, when crossover probability was excessive, the destroyed possibility of hereditary pattern was larger, has the destroyed possibility of the high individual configurations of fitness larger; When crossover probability was too small, the possibility that good pattern keeps was larger, and population diversity is difficult to guarantee, search stagnation, easily Premature Convergence.For variation probability P m, if Pm is too small, just be difficult for producing new individual configurations; If the Pm value is excessive, genetic algorithm has just become pure random search algorithm so.For optimization problem, need repeatedly experiment to determine the value of intersection, variation probability, and be difficult to find the optimum solution that adapts to each problem.In order to overcome in the traditional genetic algorithm the fixedly On The Choice of crossover and mutation probability, self-adaptation is adjusted
P c With
P m Be incorporated in the traditional genetic algorithm, adopt following self-adaptation intersection outline and variation probability,
P c With
P m Carry out the self-adaptation adjustment by following formula correspondence:
Rule of thumb, in the formula
P c1
=0.9,
P c2
=0.6,
P m1
=0.1,
P m2
=0.001.
Wherein:
f Max The fitness value of maximum in the--population;
f Avg -per generation population the average fitness value;
f 1Larger fitness value in two individualities that-needs intersect;
f-need to the individual fitness value of variation.
From
P c With
P m The self-adaptation adjustment formula can find out, each individual fitness value of population reaches unanimity or when being tending towards local optimum,
P c ,
P m Corresponding increase; When each individual fitness value relatively disperses in the population,
P c ,
P m Correspondingly reduce.Simultaneously, be higher than the individuality of population average fitness value when fitness value, corresponding lower
P c ,
P m , so that the high individuality of fitness enters the next generation; And be lower than the individuality of average fitness value, correspond to higher
P c ,
P m The time, so that this solution is eliminated.Therefore, adaptive
Pc,
PmCan provide certain individual best
P c ,
P m
Can b) simulated annealing operation, acceptable conditions be adopted the Metropolis criterion and are differentiated and enter colony of future generation; Carry out popsize iteration and produce the pop of colony of future generation (i+1);
The first given original state that characterizes with the particle relative position
i, as the current shape of solid, the energy of this state is
E i Then make the displacement of certain particle of choosing at random produce randomly a subtle change with the perturbation device, obtain a new state
j, the energy of new state
EjIf
Ej＜
E i , then this new state is just as " important " state.If
Ej E i , then consider the impact of thermal motion, whether this new state " important " state, judge according to the probability that solid is in this state.Solid is in state
iWith
jProbability ratio, namely
Wherein
rBe one and produce one [0,1] interval random number less than 1 number with random generator, if
r, then new state j is as important state, otherwise casts out.From following formula as can be known, can accept under the high temperature can differ from larger new state with current state is important state, is important state and can only accept at low temperatures can differ from less new state with current state, when temperature goes to zero, just can not accept arbitrary
Ej E i New state j.
C) the temperature operation is moved back in execution, and makes i=i+1; Judge whether to reach end condition, the words that are then stop, otherwise return the 3rd) step.
Embodiment: the performance of finding the solution the 0-1 knapsack problem in order to investigate AGAA, the program of AGAA algorithm that the below has used the matlab language compilation, write simultaneously the program of GA genetic algorithm, AGA self-adapted genetic algorithm two kinds of algorithms, the program of three kinds of algorithms is Intel cole T6570 at CPU, in save as 2GB, operating system is on the computing machine of Window XP four knapsack examples to be studied.
In the experiment, the knapsack number is elected respectively 10,20,50,100 as and is test case, and two kinds of algorithms that participate in contrast are respectively AGA, GA.In order to reduce randomness to the impact of algorithm performance, each algorithm is tested example independent operating 20 times to each, in order to analyze their statistical nature.In the experiment, the fitness formula of three kinds of algorithms is all identical, and the population scale of AGAA, AGA, three kinds of algorithms of GA all elects 100 as, and intersection, aberration rate all are set as
P c1
=0.9,
P c2
=0.6,
P m1
=0.1,
P m2
=0.001, initial annealing temperature all is set as 800 in the AGAA algorithm, decay factor is set as K=0.95, and three kinds of algorithms are all take maximum iteration time as stopping criterion, and namely to set respectively maximum iteration time be 50,80,250,300 to knapsack number 10,20,50,100.
The performance standard of comparison is the optimal-adaptive degree value V that algorithm obtains in 20 independent operatings
Max, the poorest fitness value V
Min, average fitness value V
Avg, average stability bandwidth δ, minimum iterations T
Min, mean iterative number of time T
AvgAnd the number of times Z of the feasible optimum solution that obtains.δ=(V
opitimation-V
avg)/V
opitimation。
The statistical value that is obtained for 20 times by three kinds of algorithm independent operatings of table 1 more as can be known, for 10 number purpose knapsacks, the optimum solution number of times that AGA algorithm and GA algorithm obtain is obviously inferior to AGAA, average stability bandwidth is all greater than 0, shows that AGA, two kinds of algorithms of GA have at least once to have obtained approximate optimal solution in 20 independent operatings; For 20 number purpose knapsacks, the optimal value number that AGAA obtains is all many than other two kinds of algorithms, and on average stability bandwidth is 0.143%, is starkly lower than other two kinds of algorithms, shows that the stability of AGAA algorithm is better; For 30 number purpose knapsacks, GA, two kinds of algorithms of AGA all do not obtain optimal value, and AGAA has obtained 3 suboptimum solutions, as seen, when problem scale progressively increases, participate in two kinds of algorithm search relatively to the optimal value difficult, and average stability bandwidth also further increases, though the average stability bandwidth of AGAA algorithm has reached 0.370%, but still less than other two kinds of algorithms, and then show that the stability of AGAA is preferably; For 100 number purpose knapsacks, three kinds of algorithms all do not converge to optimal value, just obtained approximate optimal solution in the space, field of separating, but the average fitness value that AGAA obtains are maximum, and are higher than document, and as seen the optimizing ability of this algorithm is best.See on the whole, knapsack problem for different numbers, AGAA convergence of algorithm algebraically obviously is less than other two kinds of convergence of algorithm algebraically, average stability bandwidth also is better than other two kinds of algorithms, thereby show, the AGAA algorithm is when finding the solution extensive knapsack problem, and its speed of convergence, optimizing ability and stability are more satisfied than other two kinds of algorithms.
Table 1
Wherein, V
OpitimationThe theoretical optimal value that represents different knapsack numbers, N represents different knapsack numbers, does not carry out the statistics of this numerical result in "---" expression document, T
Min, T
AvgSpeed of convergence (time complexity) that can measure algorithm, V
Max, V
Min, V
Avg, the optimizing ability that Z can measure algorithm, the degree of stability that δ can measure algorithm.
Fig. 2～3rd, N is respectively the convergence curve figure of 20,50 o'clock best fitness value, as can be seen from Figure 2, three kinds of algorithms all can optimizing arrive desirable optimal value, and AGAA convergence of algorithm speed is obviously greater than participating in two kinds of algorithms relatively, as can be seen from Fig., only AGAA has converged to desirable optimal value, and speed of convergence is very fast.Fig. 4～5th, N is respectively 20,50 o'clock average fitness value, as can be seen from the figure, the convergence in mean value of AGAA is large than other two kinds of algorithms, and it is close to participate in two kinds of algorithm average search effects relatively, is difficult to demonstrate the superiority that participates in two kinds of algorithms relatively.
The selection mechanism that this algorithm adopts roulette and optimum conversation strategy to combine, so that current optimum individual remains in the population, and combining adaptive intersects, the variation probability, enlarges the hunting zone of population, then introduce simulated annealing, accelerate iteration later stage convergence of algorithm speed.At last, the Adaptive Genetic annealing algorithm after improving is applied in 0/1 knapsack, experimental result shows that the Adaptive Genetic annealing algorithm can obtain than standard genetic algorithm, the more satisfied effect of self-adapted genetic algorithm.
Claims (3)
1. Adaptive Genetic annealing computing method that solve the 0-1 knapsack problem is characterized in that, specifically comprise the steps:
1) set algorithm parameter comprises population size popsize, chromosome length chromlong, and annealing initial temperature T0 moves back warm coefficient k etc.;
2) produce initial population pop (0);
3) according to fitness function each individual fitness value in the colony is made an appraisal, and judge whether it meets Optimality Criteria, if meet, the optimum solution of output optimized individual and representative thereof, and finish to calculate; Otherwise execution following steps;
A) carry out genetic manipulation, the selection that selection strategy adopts roulette and optimum conversation strategy to combine is machine-processed, and the optimum number of preserving is made as 2, and intersection, mutation operation adopt self-adaptation intersection, variation probability, produce SA initial population sa-pop;
Can b) simulated annealing operation, acceptable conditions be adopted the Metropolis criterion and are differentiated and enter colony of future generation; Carry out popsize iteration and produce the pop of colony of future generation (i+1);
C) the temperature operation is moved back in execution, and makes i=i+1; Judge whether to reach end condition, the words that are then stop, otherwise return the 3rd) step.
2. the Adaptive Genetic of described solution 0-1 knapsack problem annealing computing method according to claim 1, it is characterized in that, the selection mechanism that adopts roulette and optimum conversation strategy to combine in the described step a), to keep current optimum chromosome, namely keeping 2 optimum dyeing bodies from current population directly copies in the population of future generation, remaining population is selected to enter in the population of future generation by the roulette method, to improve convergence.
3. the Adaptive Genetic of described solution 0-1 knapsack problem annealing computing method according to claim 1, it is characterized in that, intersection, mutation operation adopt self-adaptation intersection, variation probability in the described step a), and crossover probability Pc and the probability P m that makes a variation carry out the self-adaptation adjustment by following formula correspondence:
In the formula
P c1
=0.9,
P c2
=0.6,
P m1
=0.1,
P m2
=0.001,
f Max -be fitness value maximum in the population;
f Avg Average fitness value for per generation population;
f 1Be larger fitness value in two individualities of needs intersection;
fBe the individual fitness value of needs variation.
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| CN105515196A (en) * | 2016-01-05 | 2016-04-20 | 深圳合纵能源技术有限公司 | Region power grid dispatching system and method based on general demand side response |
| CN106526338A (en) * | 2016-10-19 | 2017-03-22 | 天津大学 | Indoor ray tracing parameter correction method based on simulated annealing |
| CN106021859B (en) * | 2016-05-09 | 2018-10-16 | 吉林大学 | The controlled-source audiomagnetotellurics method one-dimensional inversion method of improved adaptive GA-IAGA |
| CN112330023A (en) * | 2020-11-05 | 2021-02-05 | 安庆师范大学 | Method for solving 0-1 knapsack problem considering subjective demand and electronic equipment |
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| CN104820859A (en) * | 2015-04-27 | 2015-08-05 | 宁夏大学 | Preprocessing and adaptive genetic simulated annealing algorithm-based phase unwrapping method |
| CN104820859B (en) * | 2015-04-27 | 2017-09-05 | 宁夏大学 | Phase unwrapping method based on pretreatment and Adaptive Genetic simulated annealing |
| CN105515196A (en) * | 2016-01-05 | 2016-04-20 | 深圳合纵能源技术有限公司 | Region power grid dispatching system and method based on general demand side response |
| CN106021859B (en) * | 2016-05-09 | 2018-10-16 | 吉林大学 | The controlled-source audiomagnetotellurics method one-dimensional inversion method of improved adaptive GA-IAGA |
| CN106526338A (en) * | 2016-10-19 | 2017-03-22 | 天津大学 | Indoor ray tracing parameter correction method based on simulated annealing |
| WO2021027295A1 (en) * | 2019-08-12 | 2021-02-18 | 岭南师范学院 | Human body composition prediction method based on improved adaptive genetic algorithm |
| CN112330023A (en) * | 2020-11-05 | 2021-02-05 | 安庆师范大学 | Method for solving 0-1 knapsack problem considering subjective demand and electronic equipment |
| CN112700190A (en) * | 2020-12-29 | 2021-04-23 | 中国电子科技集团公司第十五研究所 | Improved method for distributing tray materials by scanning method and genetic simulation annealing method |
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Application publication date: 20130213 |