CN113887122A - A hybrid leapfrog solution method for multi-objective knapsack problem - Google Patents
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Abstract
The invention discloses a mixed frog leap solving method of a multi-target knapsack problem, (1) reading problem information, wherein the problem information comprises value and weight information of each cargo of each knapsack and weight limit information of each knapsack; (2) initializing algorithm parameters; (3) calculating target values of all individuals in the population, and determining a non-dominated solution set to be put into an external memory; (4) entering a rapid convergence stage, dividing subgroups of the population according to a rapid non-dominated sorting result by using an S-type grouping mode, performing local search based on a discrete jump rule and greedy generation on each subgroup, shuffling each subgroup, updating an external memory, judging whether a target evaluation frequency meets a termination condition of the rapid convergence stage, if not, continuing iteration, and if so, entering the next stage; (5) entering an exploration expansion stage; (6) and entering an extremum mining stage. The method has the advantages of high searching speed, strong searching capability and high profit of the planned backpack.
Description
Technical Field
The invention relates to the technical field of multi-target knapsack problem solving, in particular to a hybrid frog jump solving method for a multi-target knapsack problem.
Background
The multi-objective combination optimization problem generally exists in various fields in production and life such as industrial design, information processing, transportation and the like. The backpack problem is to place items in different combinations into the backpack given a set of items and a weight-limited backpack so that the total weight of the items does not exceed the backpack weight limit while maximizing the total profit of the items in the backpack. The multi-objective knapback problem (MOKP) is a typical multi-objective combinatorial optimization problem, where given m backpacks and m groups of items, each backpack corresponds to one group of items, the number of each group of items is n, and each item in each group has unique attributes of profit and weight. All groups of articles with the same serial number are put into or not put into the corresponding backpack at the same time. For example, assuming that an item with the number 1 of the backpack 1 is put into the backpack 1, an item with the number 1 of the backpack j is also put into the backpack j (j is 1, 2. Under the premise of meeting the capacity constraint of each backpack, the sum of profits of the articles in each backpack is respectively maximized. In engineering practice, many practical problems can be converted into MOKP to be solved, such as three-dimensional packing, logistics freight loading, investment portfolio optimization, password portfolio, express distribution, advertisement putting and other engineering problems. However, the multi-objective knapsack problem has been demonstrated as an NP-hard problem, and as the scale of the problem increases, the feasible solution set also grows exponentially, and the phenomenon of combined explosion inevitably occurs. Therefore, the method has important theoretical scientific research value and social and economic benefits for the research of the MOKP solving algorithm.
The mixed frog leap algorithm (SLFA) is proposed in 2003 for solving the problem of combination optimization by Eusuff and Lansey, and also belongs to one of group intelligent algorithms. The frog leaping algorithm is used as a bionic swarm intelligence algorithm, the foraging behavior of the frog population in the wetland is simulated, namely a plurality of stones are distributed in the wetland, the frog jumps on the stones to find the most food points, each frog has different information, the frog population is divided into different groups, and information exchange is carried out through different individual frogs to realize local search; when the local search is performed to a certain extent, the subgroups are shuffled to realize global information exchange. The novel multi-target mixed frog leaping algorithm is an improved version based on a basic multi-target mixed frog leaping algorithm, and comprises the following basic steps: randomly generating an initial population by adopting an 0/1 coding mode, and calculating a target value after processing by using a relaxation constraint repair strategy; selecting all non-dominated liberations in the current population according to the target value and putting the selected non-dominated liberations into an external memory; in a rapid convergence stage, dividing subgroups of the population according to a rapid non-dominated sorting result by using an S-type grouping mode, randomly selecting an exclusive global optimal solution distributed to each subgroup from an external memory, generating a new solution by using a discrete skip rule and a greedy generation strategy as an individual generation mode in local search, shuffling all subgroups, and updating the external memory by using a dominated relationship; in an exploration extension stage, a guide set guiding the exploration of each subgroup is extracted from an external memory according to a target using a leading edge division strategy in sequence, the subgroup is divided into a plurality of subgroups according to the target in sequence, each subgroup is divided in an S-type grouping mode according to a quick non-dominated sorting result, an exclusive global optimal solution distributed to each subgroup by an individual is randomly selected from the guide set, a discrete jump rule and a greedy generation strategy are adopted in local search as individual generation modes, all subgroups are shuffled, the weight is reduced, and the external memory is updated; in an extreme value mining stage, a population is divided into a plurality of subgroups according to targets in sequence, subgroups are divided by using an S-type subgroup division mode according to a descending sorting result for each subgroup, a new solution is generated by using a discrete skip rule and a greedy generation strategy as individual generation modes in local search, all subgroups are shuffled, weight is reduced, and an external memory is updated. However, the problem of solving the multi-target backpack by adopting the traditional multi-target mixed frog leaping algorithm still has the following defects: the convergence speed is low, the convergence precision is poor, premature convergence is easy, the diversity of solution sets is poor, and the distributivity is poor. Therefore, it is necessary to provide a solution method with high convergence rate, high solution accuracy, good diversity and good distribution.
Disclosure of Invention
Aiming at the defects in the prior art, the invention provides a mixed frog leap solving method for the multi-target knapsack problem, which can greatly improve the convergence rate, solving precision, diversity and distribution of the algorithm, thereby providing various high-quality article selection schemes.
In order to achieve the purpose, the invention adopts the following technical scheme:
a mixed frog leap solving method for a multi-target backpack problem comprises the following steps:
s1, reading information input by the problem, defining an optimization target and setting constraint conditions;
s2, initializing a novel multi-target mixed frog leaping algorithm parameter; setting the scale of an evolution population of a novel mixed frog-jump algorithm as m × N, a subgroup number e, local search times L, a global maximum target evaluation time G, a segmentation percentage alpha%, an information receiving percentage beta%, a variation probability pm and an evaluation time counter t equal to 0;
s3, generating an initial population, calculating an optimized target value f after processing by using a relaxation constraint repair strategyj(ii) a Using 0/1 encoding, m × N individuals were randomly generated to form a population Pop, each individual representing an item to be placed in a respective backpack:
X＝{x1，x2，…，xn}
wherein x is1Indicating whether the 1 st item of each backpack is placed in the backpack; each time of calculationIndividual optimum target value fj(X)， t＝t+m*N；
S4, selecting all non-dominated liberations in the current population Pop according to the target value and putting the non-dominated liberations in an external memory A;
s5, entering a rapid convergence stage: dividing the population Pop into subgroups MPops according to the fast non-dominated sorting result by using an S-type grouping modei；
S6, searching each subgroup locally;
s7, shuffling all subgroups, updating the external memory A by using a domination relation, judging whether a termination condition of a rapid convergence stage is met, if so, terminating iteration of the rapid convergence stage, and entering the next stage, otherwise, turning to S4;
s8, entering the exploration expansion phase: according to the target f in turnjExtracting a bootstrap set A for bootstrap of each subgroup exploration from an external memory A using a leading edge partition strategyjIn turn according to the target fjDividing population Pop into m sub-populations CPopjFor each subgroup, the subgroup MPop is divided in an S-type grouping manner according to the fast non-dominated sorting resulti；
S9, for each subgroup CPopjEach subgroup MPop ofiLocal searching; from the boot set AjIn the random selection of w individuals to allocate to each subgroup MPopiIs given by the exclusive global optimal solution XigAdopting discrete jump rule and greedy generation strategy as individual generation mode, and updating local worst X according to the rule of high or lowiwEach subgroup is searched locally L times, t is t + m N;
s10, shuffling all subgroups, carrying out de-duplication operation on repeated solutions of the target vector, updating the external memory A by using a domination relation, judging whether a search expansion stage termination condition is met, if so, terminating search expansion stage iteration, entering the next stage, otherwise, turning to the step S8;
s11, entering an extremum mining stage: according to the target f in turnjDividing population Pop into m sub-populations CPopjFor each subgroup, the subgroup MPop is divided in an S-type grouping manner according to the descending sorting resulti；
S12, for each subgroup CPopjEach subgroup MPop ofiLocal search: adopting discrete jump rule and greedy generation strategy as individual generation mode, and updating local worst X according to the rule of high or lowiwEach subgroup is searched locally L times, t ═ t + m × N:
s13, shuffling all subgroups, carrying out weight reduction operation on repeated solutions, updating the external memory A by using a domination relation, judging whether an extreme value mining stage termination condition is met, if so, terminating iteration of the extreme value mining stage, and outputting the external memory A; otherwise, go to S11.
In order to optimize the technical scheme, the specific measures adopted further comprise:
further, in S1, the input information of the question includes value and weight information of each item of each backpack and a question scale n; the optimization objective is to maximize the total profit for the items in each backpack; the constraint is that each backpack has an upper weight limit, and the total weight of the items placed in each backpack cannot exceed the upper weight limit of the backpack.
Further, in S1, the reading of the information input by the problem and the definition of the optimization goal, and the process of setting the constraint condition includes the following steps:
setting the scale of the problem to represent the number n of articles which can be put into each backpack;
defining the optimization objective body as the size of the total profit for each knapsack, which is defined as:
wherein f isjRepresents the total profit for the jth backpack; x denotes a decision variable XiA set of (a); x is the number ofiIndicating whether the ith item in the backpack is put into each backpack or not; p is a radical ofijIndicates the interest of the ith item in the jth backpackMoistening; i.e. the larger the j target values of an individual are, the better;
defining constraints includes the following two:
(1) weight limit of jth backpack cjNamely:
wherein n is the number of items that can be placed in the jth backpack; w is aijIs the weight of the ith item that can be placed in the jth backpack;
(2) the total weight of the items placed in the jth backpack must not exceed the weight limit of the backpack, i.e.:
further, in S2, after a new individual is generated each time, if the new individual exceeds the constraint condition limit, a relaxed constraint repairing strategy is used for processing, and the infeasible solution is repaired to a feasible solution, which includes the following specific steps:
s21, calculating profit p of each item in each knapsackijW/weightij；
S22, calculating profit p of ith item in all backpacks in sequenceijW/weightijAverage value AK ofiRepeating S22 until the average value of the profit-to-weight ratio of each article in all the backpacks is calculated;
s23, for the articles in the backpack, AKiThe smallest one removed from the backpack;
s24, for the articles outside the backpack, in AKiRandomly selecting one of k articles with the largest value to be placed in a backpack; if the number of the articles outside the backpack is more than 5, making k equal to 5, otherwise, making k equal to the number of the articles outside the backpack;
and S25, calculating the total weight of the articles in each backpack, outputting a feasible solution if the weight limit of each backpack is met, and turning to S23 if the weight limit of each backpack is not met.
Further, in S5, the "S" type grouping manner specifically includes:
assuming that the sorted population Pop with the size of m × N is divided into e subgroups, each group of m × b individuals, satisfying m × N ═ e × m × b; the individuals from 1 to e are sequentially placed into the groups from 1 to e, the individuals from w +1 to 2 × e are placed into the groups from w to 1, the individuals from 2 × e +1 to 3 × e are sequentially placed into the groups from 1 to e, and so on.
Further, S6 includes randomly selecting w individuals from the external memory A to be allocated to each subgroup MPopiIs a proprietary global optimal solution XigAdopting discrete jump rule and greedy generation strategy as individual generation mode, and updating local worst X according to the rule of high or lowiwEach subgroup is searched locally L times, t is t + m N;
the discrete jump rule is implemented by the following steps:
s61, determining inferior solution X needing jumpingwAnd the learned optimal solution Xb；
S62, calculating optimal solution XbAnd poor solution XwThe hamming distance dis between the two sets, the gene bit indexes with different values of the superior solution and the inferior solution are stored in the set dif, and the gene bit indexes with the same value are stored in the set same;
s63, randomly selecting rounded up beta%. multidot.dis gene sites from the set dif, and solving the superior X on the corresponding gene sitesbImpartation of inferior solution XwAt the corresponding gene position;
s64, selecting a gene locus in the set same, and generating a [0, 1 ]]Random number rand in between, if rand < pm, inferior solution XwTurning over the value of the corresponding gene position, and repeating the step S64 until the gene position in the set same is selected;
s65, outputting inferior solution XwNew solution X improved by discrete skip rulewn。
Further, in S6, the greedy generation strategy specifically includes the following steps:
s66, determining a new solution X needing to be generatedwnWeight and profit information for all items in the backpack;
s67, selecting a Knapsack Knapack using the strategy of roulettejSelecting the weight and the profit information of the articles in the backpack;
s68, selecting the profit p of the ith itemijAnd weight wijInformation, will pij/wijThis ratio is mapped to a probability P using the sigmod functionijWhile generating one [0, 1 ]]Random number rand betweeniIf randi＜PijIf yes, putting the ith item into the backpack, otherwise, not putting the ith item into the backpack, and repeating the step S608 until all the n items are judged to be finished;
s69, outputting a new solution X generated by the greedy generation strategywn。
Further, in S8, the leading edge division strategy specifically includes the following steps:
s81, calculating the size of the external memory A, and obtaining the number num of the solutions to be selected through the rounded up alpha% size; alpha% is the segmentation percentage;
s82, sorting according to the jth target in descending order, selecting the first num individuals to be put into the guide set AjAnd repeating the step S82 until all the boot sets are generated and output.
Further, in S8, the target f is sequentially selectedjDividing population Pop into m sub-populations CPopjThe method comprises the following specific steps:
s83, determining the Pop of the population to be divided;
s84, sorting according to the jth target in descending order, selecting the first N individuals to be put into a subgroup CPopjAnd repeating S84 until m sub-groups CPopjAnd finishing and outputting.
Further, in S10, the specific steps of the weight reducing operation are as follows:
s101, calculating the profit p of the existing articles in each backpackijW/weightij；
S102, calculating profits p of ith item in all backpacks in sequenceijW/weightijAverage value AK ofiAnd repeating the step S102 until the average value of the profit-to-weight ratios of all the backpacks of each article is calculated；
S103, average value AKiAnd (5) performing ascending sorting, taking the first sorted article out of the backpack, and constructing a new solution.
The invention has the beneficial effects that: (1) the invention adopts a novel multi-target mixed frog-leaping algorithm to solve the multi-target knapsack problem, utilizes heuristic information of the multi-target knapsack problem, introduces strategies of increasing population diversity, enhancing population development capability, enhancing local search and the like, and promotes the performance of the algorithm to be superior to that of the traditional multi-target mixed frog-leaping algorithm.
(2) A staged optimization mechanism is designed, and the optimization mechanism is divided into three stages of rapid convergence, exploration and extension and extreme value mining, so that the different side points of the population in different stages are realized, and the convergence, diversity and distribution of the algorithm are effectively improved.
(3) An S-type grouping mode is provided, so that individuals in each subgroup are more uniform, the information of the individuals in the subgroups is effectively exchanged, and the solving precision of the algorithm is improved.
(4) The method provides a front edge division strategy, guides subgroups to perform local search by using a guide set focusing on different targets, explores different areas of a Pareto front edge, and improves diversity and distribution of solution sets.
(5) A discrete jumping rule is provided, aiming at the coding form of the multi-target knapsack problem, the coding strings of optimal solution and inferior solution are compared, and for the gene positions with different values, the inferior solution tracks the optimal solution so as to explore in a better direction in a decision space; for gene loci with the same value, the mutation of the basic locus is used to find more excellent solutions near the individual. The discrete jump rule directly learns the individual information in the discrete space, so that the utilization rate of the individual high-quality information is improved.
(6) A greedy generation strategy is provided, aiming at the problem that a traditional multi-target leapfrog algorithm random generation mode may generate a new solution with worse quality than an inferior solution with a high probability, heuristic information of a multi-target knapsack problem is introduced, and if the ratio of the profit to the weight of a certain object is larger, the probability that the certain object is selected is higher. The proposed strategy reduces the randomness and the search of an algorithm for an invalid decision space, improves the quality of generated solutions by using problem specific knowledge, and saves computing resources.
(7) A relaxation constraint repairing strategy is provided, and the traditional constraint repairing method is too greedy, so that the algorithm is easy to fall into premature convergence. The put articles are randomly selected from a plurality of better articles, the improved strategy relieves the excessive dependence of a constraint processing mechanism on problem heuristic information, enables a group to retain more effective information, and enhances the exploration capability of an algorithm.
Drawings
Fig. 1 is a main flow chart of a mixed frog-leaping solving method of the multi-target backpack problem of the present invention.
Fig. 2 is a diagram comparing the basic multi-target mixed frog-leaping algorithm with the present invention.
Detailed Description
The present invention will now be described in further detail with reference to the accompanying drawings.
It should be noted that the terms "upper", "lower", "left", "right", "front", "back", etc. used in the present invention are for clarity of description only, and are not intended to limit the scope of the present invention, and the relative relationship between the terms and the terms is not limited by the technical contents of the actual changes, but also should be regarded as the scope of the present invention.
In one embodiment of the present invention, sample 250 was selected with two backpacks, 50 items per backpack being selectable, each item having profit and weight attributes, as shown in Table 1.
TABLE 1
The example is solved by using the mixed frog leap solving method for the multi-target knapsack problem, the main flow is shown in figure 1, and the specific steps are as follows:
(1) and (5) initializing. Reading input information of instances, including
And S1, reading the information input by the problem, including the value and weight information (see table 1) of each item of each backpack and the problem scale n, giving definition of an optimization target and setting a constraint condition.
The optimization objective is to maximize the total profit for the items in each backpack, which is defined as:
wherein f isjRepresents the total profit for the jth backpack; x denotes a decision variable XiA set of (a); x is the number ofiIndicating whether the ith item in the backpack is put into each backpack or not; p is a radical ofijRepresents the profit for the ith item in the jth backpack;
the constraints are two:
constraint one: weight limitations of jth backpack, namely:
wherein n is the number of items that can be placed in the jth backpack; w is aijIs the weight of the ith item in the jth backpack;
constraint two: the total weight of the items placed in the jth backpack must not exceed the weight limit of the backpack, i.e.:
(2) initializing the parameters of the novel multi-target mixed frog leaping algorithm:
setting the scale of an evolution population of a novel mixed frog-jump algorithm to be m × N2 × 100, the subgroup number e to be 10, the local search frequency L to be 10, the global maximum target evaluation frequency G to be 100000, the segmentation percentage alpha% to be 50%, the information receiving percentage beta% to be 80%, the variation probability pm to be 0.01, and setting an evaluation frequency counter t to be 0;
(3) generating an initial population, calculating an optimized target value f after processing by using a relaxation constraint repair strategyj：
Using 0/1 encoding, m × N individuals were randomly generated to form a population Pop, each individual representing an item to be placed in a respective backpack:
X＝{x1，x2，…，xn}
wherein x is1Indicating whether the 1 st item of each backpack is placed in the backpack; calculating an optimized target value f for each individualj＝(X)，t＝t+m*N：
Wherein p isijRepresents the profit for the ith item in the jth backpack; i.e., larger individual j target values are better;
after a new individual is generated every time, if the new individual exceeds the constraint condition limit, a relaxation constraint repair strategy is used for processing, the infeasible solution is repaired into a feasible solution, and the local implementation steps of the relaxation constraint repair strategy are as follows:
(a) calculate profit p for each item in each backpackijW/weightii；
(b) Sequentially calculating the profit p of the ith item in all backpacksijW/weightijAverage value AK ofiRepeating the step S22 until the average value of the profit-to-weight ratios of all the backpacks of each article is calculated;
(c) for the objects in the backpackProduct, will AKiThe smallest one removed from the backpack;
(d) for articles outside the backpack, in AKiRandomly selecting one of the k articles with the largest value to be put into the backpack (if the number of the articles outside the backpack is more than 5, making k equal to 5, otherwise, making k equal to the number of the articles outside the backpack);
(e) and (4) calculating the total weight of the articles in each backpack, outputting a feasible solution if the weight limit of each backpack is met, and turning to the step (c) if the weight limit of each backpack is not met.
(4) Selecting all non-dominated releases in the current population Pop according to the target value and putting the non-dominated releases in an external memory A;
(5) entering a rapid convergence stage:
dividing subgroup MPops for population Pop by using S-type grouping mode according to fast non-dominated sorting resulti；
The local implementation steps of the S-type grouping mode are as follows:
(a) assume that the sorted population Pop is divided into e subgroups of m × N and m × b individuals in each group, satisfying m × N ═ e × m × b. The individuals from 1 to e are sequentially placed into the groups from 1 to e, the individuals from w +1 to 2 × e are placed into the groups from w to 1 in a reverse order, the individuals from 2 × e +1 to 3 × e are sequentially placed into the groups from 1 to e, and so on.
(6) Local search for each subgroup:
randomly selecting w individuals from the external memory A to be distributed into each subgroup MPopiIs given by the exclusive global optimal solution XigA discrete jump rule and a greedy generation strategy are adopted as individual generation modes, and the local worst X is updated according to the rule of high or lowiwEach subgroup is searched locally L times, t is t + m N;
the discrete jump rule local implementation steps are as follows:
(a) determining a poor solution X requiring a jumpwAnd the learned optimal solution Xb；
(b) Calculating optimal solution XbAnd poor solution XwThe hamming distance dis between the two sets, the gene bit indexes with different values of the superior solution and the inferior solution are stored in the set dif, and the gene bit indexes with the same value are stored in the set same;
(c) randomly selecting upward integrated beta%. multidot.dis gene positions from the set dif, and optimally solving X on the corresponding gene positionsbImpartation of inferior solution XwAt the corresponding gene position;
(d) selecting a gene position in the set same, and generating a [0, 1 ]]Random number rand in between, if rand < pm, inferior solution XwTurning over the value of the corresponding gene position, and repeating the step S604 until the gene position in the set same is selected;
(e) output of inferior solution XwNew solution X improved by discrete skip rulewn。
The greedy generation strategy is realized locally by the following steps:
(a) determining a new solution X that needs to be generatedwnWeight and profit information for all items in the backpack;
(b) strategy for using roulette to select a Knapsack knappackjSelecting weight and profit information for the backpack item at the same time;
(c) selecting profit p for ith itemijAnd weight wijInformation, will pij/wijThis ratio is mapped to a probability P using the sigmod functionijWhile generating one [0, 1 ]]Random number rand betweeniIf randi＜PijIf yes, putting the ith item into the backpack, otherwise, not putting the ith item into the backpack, and repeating the step (c) until all the n items are judged to be finished;
(d) outputting a new solution X generated by a greedy generation strategywn。
(7) Shuffling all subgroups, updating an external memory A by using a domination relation, if t is more than 0.2G, terminating the iteration of the fast convergence stage, entering the next stage, and otherwise, turning to the step (5);
(8) entering an exploration expansion phase:
according to the target f in turnjExtracting a bootstrap set A for bootstrap of each subgroup exploration from an external memory A using a leading edge partition strategyjAccording to the target f in turnjDividing population Pop into m sub-populations CPopjUsing "S" for each subgroup according to the fast non-dominated sorting result "Grouping mode division subgroup MPopi；
The partial implementation steps of the leading edge division strategy are as follows:
(a) calculating the size of the external memory A, and obtaining the number num of solutions to be selected through the rounded up alpha% size;
(b) sorting according to the jth target in descending order, selecting the first num individuals to be put into a guide set AjAnd (c) repeating the step (b) until all the boot sets are generated and output.
According to the target f in turnjDividing population Pop into m sub-populations CPopjThe local implementation steps are as follows:
(a) determining a population Pop to be divided;
(b) sorting according to the jth target in descending order, selecting the first N individuals to be put into a subgroup CPopjAnd (c) repeating the step (b) until m sub-groups CPopjAnd finishing and outputting.
(9) For each subgroup CPopjEach subgroup MPop ofiLocal search:
from the boot set AjIn the random selection of w individuals to be allocated to each subgroup MPopiIs given by the exclusive global optimal solution XigAdopting a dispersion jump rule and a greedy generation strategy as individual generation modes, and updating local worst X according to the rule of high or lowiwEach subgroup is searched locally for L times, t is t + m is N;
(10) shuffling all subgroups, performing a de-duplication operation on repeated solutions of the target vector, updating an external memory A by using a domination relation, if t is more than 0.6G, terminating the iteration of an exploration expansion stage, and entering the next stage, otherwise, turning to the step (8);
the local implementation steps of the weight reduction operation are as follows:
(a) calculate the profit p of the existing items in each backpackijW/weightij；
(b) Sequentially calculating the profit p of the ith item in all backpacksijW/weightijAverage value AK ofiRepeating the step (b) until the average value of the profit-to-weight ratios of all the backpacks of each article is calculated;
(c) average AKiAnd (5) performing ascending sorting, taking the first sorted article out of the backpack, and constructing a new solution.
(11) Entering an extremum mining stage:
according to the target f in turnjDividing population Pop into m sub-populations CPopjFor each subgroup, the subgroup MPop is divided in an S-type grouping manner according to the descending sorting resulti；
(12) For each subgroup CPopjEach subgroup MPop ofiLocal search:
adopting discrete jump rule and greedy generation strategy as individual generation mode, and updating local worst X according to the rule of high or lowiwEach subgroup is searched locally L times, t is t + m N;
(13) and (4) shuffling all the subgroups, carrying out a duplicate reduction operation on the repeated solutions, updating the external memory A by using a domination relation, if t is greater than G, terminating the iteration of the extreme value mining stage, and outputting the external memory A, otherwise, turning to the step (11).
The effect of the invention can be further illustrated by the following simulation experiment:
1. the experimental conditions are as follows:
the CPU is AMD Ryzen 54600U with 2.10GHz of radiation Graphics, memory 16GB, WINDOWS
Simulation was performed on 10 systems using Matlab2019 b.
2. The experimental contents are as follows:
sample 250 was selected with two backpacks, 50 items per backpack, each having profit and weight attributes, as shown in table 1.
3. Results of the experiment
The problem is solved by adopting the method and the existing basic multi-target mixed frog-leaping algorithm. The two methods were run independently 30 times each in the examples. Table 2 lists the mean values and standard deviations of solutions obtained by the two methods in 30 runs in three indexes of IGD, HV and Spread, respectively, and the smaller the IGD, the larger the HV, the better the convergence of the algorithm; the smaller the Spread, the wider and more uniform the distribution of the searched solution set. And simultaneously, Wilcoxon rank sum test with a significance level of 0.05 is introduced to carry out statistical test on the results of the embodiment, wherein, + -represents that the novel multi-target mixed frog leaping algorithm is significantly superior to the basic multi-target mixed frog leaping algorithm, and- — represents that the novel multi-target mixed frog leaping algorithm is significantly inferior to the basic multi-target mixed frog leaping algorithm.
As can be seen from the table 2, compared with the existing basic multi-target mixed frog leaping algorithm, the method can search individuals with better indexes, greatly reduce the project period and cost, and remarkably improve the project development efficiency.
TABLE 2
Fig. 2 shows a comparison graph of the optimal Pareto front searched by the basic multi-target mixed frog-leaping algorithm, so as to compare the convergence, diversity and distribution of the two methods. As can be seen from fig. 2, the solutions searched by the present invention can basically and completely branch the solutions of the basic multi-target mixed frog-leaping algorithm, which shows that the present invention is superior to the basic multi-target mixed frog-leaping algorithm in all three aspects of convergence, diversity and distribution. Therefore, the algorithm of the invention has higher convergence rate, solving precision, excellent diversity and distribution, and can find most of the theoretical optimal Pareto frontiers of the problem.
In summary, the mixed frog leap solving method for the multi-target backpack problem provided by the invention has the advantages that the optimization mechanism is staged on the basis of the basic multi-target mixed frog leap algorithm, different strategies and different modules are used in different stages, so that the algorithm realizes different functions, the convergence, the diversity and the distribution of the algorithm are improved, the designed front edge division strategy can well guide the algorithm to search different areas of the optimal Pareto front edge in the exploration and mining stage, and the diversity and the distribution of the algorithm are improved. The designed discrete jump rule directly learns the individual information in the discrete space, so that the utilization rate of the individual high-quality information is improved. The designed greedy avaricious generation strategy reduces the randomness and the search of the mixed frog-leaping frame on an invalid decision space, improves the quality of generated solutions by using problem specific knowledge, and saves computing resources. The designed relaxation constraint repairing strategy relieves the excessive dependence of a constraint processing mechanism on problem heuristic information, enables a group to retain more effective information and enhances the exploration capability of an algorithm.
In this disclosure, aspects of the present invention are described with reference to the accompanying drawings, in which a number of illustrative embodiments are shown. Embodiments of the present disclosure are not necessarily defined to include all aspects of the invention. It should be appreciated that the various concepts and embodiments described above, as well as those described in greater detail below, may be implemented in any of numerous ways, as the disclosed concepts and embodiments are not limited to any one implementation. In addition, some aspects of the present disclosure may be used alone or in any suitable combination with other aspects of the present disclosure.
The above is only a preferred embodiment of the present invention, and the protection scope of the present invention is not limited to the above-mentioned embodiments, and all technical solutions belonging to the idea of the present invention belong to the protection scope of the present invention. It should be noted that modifications and embellishments within the scope of the invention may be made by those skilled in the art without departing from the principle of the invention.
Claims (10)
1. A mixed frog leap solving method for a multi-target backpack problem is characterized by comprising the following steps:
s1, reading information input by the problem, defining an optimization target and setting constraint conditions;
s2, initializing a novel multi-target mixed frog leaping algorithm parameter;
s3, generating an initial population, calculating an optimized target value f after processing by using a relaxation constraint repair strategyj；
S4, selecting all non-dominated liberations in the current population Pop according to the target value and putting the non-dominated liberations in an external memory A;
s5, entering a rapid convergence stage: non-dominance to population Pop according to speedSorting results Using an "S" -type grouping approach to divide the subgroups MPopsi；
S6, searching each subgroup locally;
s7, shuffling all subgroups, updating the external memory A by using a domination relation, judging whether a termination condition of a rapid convergence stage is met, if so, terminating iteration of the rapid convergence stage, and entering the next stage, otherwise, turning to S4;
s8, entering the exploration expansion phase: according to the target f in turnjExtracting a bootstrap set A for bootstrap of each subgroup exploration from an external memory A using a leading edge partition strategyjIn turn according to the target fjDividing population Pop into m sub-populations CPopjFor each subgroup, the subgroup MPop is divided in an "S" type grouping manner according to the fast non-dominated sorting resulti；
S9, for each subgroup CPopjEach subgroup MPop ofiLocal searching;
s10, shuffling all subgroups, carrying out de-duplication operation on repeated solutions of the target vector, updating the external memory A by using a domination relation, judging whether a search expansion stage termination condition is met, if so, terminating search expansion stage iteration, entering the next stage, otherwise, turning to the step S8;
s11, entering an extremum mining stage: according to the target f in turnjDividing population Pop into m sub-populations CPopjFor each subgroup, the subgroup MPop is divided in an S-type grouping manner according to the descending sorting resulti；
S12, for each subgroup CPopjEach subgroup MPop ofiLocal searching;
s13, shuffling all subgroups, carrying out weight reduction operation on repeated solutions, updating the external memory A by using a domination relation, judging whether an extreme value mining stage termination condition is met, if so, terminating iteration of the extreme value mining stage, and outputting the external memory A; otherwise go to S11.
2. The method for solving mixed frog leaps of multi-target knapsack problem according to claim 1, wherein in S1, the input information of the problem includes value and weight information of each cargo of each knapsack and problem scale n; the optimization objective is to maximize the total profit for the items in each backpack; the constraint is that each backpack has an upper weight limit, and the total weight of the items placed in each backpack cannot exceed the upper weight limit of the backpack.
3. The method for solving mixed frog leap of multi-objective knapsack problem according to claim 2, wherein in S1, the process of reading the information inputted by the problem and defining the optimization objective and setting the constraint condition comprises the following steps:
setting the scale of the problem to represent the number n of articles which can be put into each backpack;
defining the optimization objective body as the size of the total profit for each knapsack, which is defined as:
wherein f isjRepresents the total profit for the jth backpack; x denotes a decision variable XiA set of (a); x is the number ofiIndicating whether the ith item in the backpack is put into each backpack; p is a radical ofijRepresents the profit for the ith item in the jth backpack;
defining constraints includes the following two:
(1) weight limit of jth backpack cjNamely:
wherein n is the number of items that can be placed in the jth backpack; w is aijIs the weight of the ith item that can be placed in the jth backpack;
(2) the total weight of the items placed in the jth backpack must not exceed the weight limit of the backpack, i.e.:
4. the method for solving the mixed frog leap of the multi-target knapsack problem according to claim 1, wherein in S2, after each generation of a new individual, if the new individual exceeds the constraint condition limit, a relaxed constraint repairing strategy is used for processing to repair the infeasible solution into a feasible solution, and the specific steps are as follows:
s21, calculating profit p of each item in each knapsackijW/weightij；
S22, calculating profit p of ith item in all backpacks in sequenceijW/weightijAverage value AK ofiRepeating S22 until the average value of the profit-to-weight ratios of each article in all the backpacks is calculated;
s23, for the articles in the backpack, AKiThe smallest one removed from the backpack;
s24, for the articles outside the backpack, in AKiRandomly selecting one of k articles with the largest value to be placed in a backpack; if the number of the articles outside the backpack is more than 5, making k equal to 5, otherwise, making k equal to the number of the articles outside the backpack;
and S25, calculating the total weight of the articles in each backpack, outputting a feasible solution if the weight limit of each backpack is met, and turning to S23 if the weight limit of each backpack is not met.
5. The method for solving the mixed frog leap of the multi-target backpack problem according to claim 1, wherein in S5, the grouping manner of the "S" type is specifically as follows:
assuming that the sorted population Pop with the size of m × N is divided into e subgroups, each group of m × b individuals, satisfying m × N ═ e × m × b; the individuals from 1 to e are sequentially placed into the groups from 1 to e, the individuals from w +1 to 2 × e are placed into the groups from w to 1, the individuals from 2 × e +1 to 3 × e are sequentially placed into the groups from 1 to e, and so on.
6. The method for solving mixed frog leap of multi-target knapsack problem of claim 1, wherein S6 comprises randomly selecting w individuals from external memory A to be allocated to each subgroup MPopiIs given by the exclusive global optimal solution XigAdopting discrete jump rule and greedy generation strategy as individual generation mode, and updating local worst X according to the rule of high or lowiwEach subgroup is searched locally L times, t is t + m N;
the discrete jump rule is implemented by the following steps:
s61, determining inferior solution X needing jumpingwAnd the learned optimal solution Xb；
S62, calculating optimal solution XbAnd poor solution XwThe hamming distance dis between the two sets, the gene bit indexes with different values of the superior solution and the inferior solution are stored in the set dif, and the gene bit indexes with the same value are stored in the set same;
s63, randomly selecting rounded up beta%. multidot.dis gene sites from the set dif, and solving the superior X on the corresponding gene sitesbImpartation of inferior solution XwAt the corresponding gene position; beta% is the information reception percentage;
s64, selecting a gene locus in the set same, and generating a [0, 1 ]]Random number rand in between, if rand<pm, then inferior solution XwTurning over the value of the corresponding gene position, and repeating the step S64 until the gene position in the set same is selected;
s65, outputting inferior solution XwNew solution X improved by discrete skip rulewn。
7. The method for solving the mixed frog leap of the multi-target knapsack problem according to claim 6, wherein in S6, the greedy generation strategy is implemented by the following steps:
s66, determining a new solution X needing to be generatedwnWeight and profit information for all items in the backpack;
s67, selecting a Knapsack Knapack using the strategy of roulettejSimultaneously select the backpackWeight and profit information for the item;
s68, selecting the profit p of the ith itemijAnd weight wijInformation, will pij/wijThis ratio is mapped to a probability P using the sigmod functionijWhile generating one [0, 1 ]]Random number rand betweeniIf randi<PijIf yes, putting the ith item into the backpack, otherwise, not putting the ith item into the backpack, and repeating the step S608 until all the n items are judged to be finished;
s69, outputting a new solution X generated by the greedy generation strategywn。
8. The method for solving the mixed frog leap of the multi-target backpack problem according to claim 1, wherein in S8, the front edge division strategy comprises the following specific steps:
s81, calculating the size of the external memory A, and obtaining the number num of the solutions to be selected through the rounded up alpha% size; alpha% is the segmentation percentage;
s82, sorting according to the jth target in descending order, selecting the first num individuals to be put into the guide set AjAnd repeating the step S82 until all the boot sets are generated and output.
9. The method for solving mixed frog leap of multi-target knapsack problem according to claim 1, wherein in S8, the method is performed according to the target fjDividing population Pop into m sub-populations CPopjThe method comprises the following specific steps:
s83, determining the Pop of the population to be divided;
s84, sorting according to the jth target in descending order, selecting the first N individuals to be put into a subgroup CPopjAnd repeating S84 until m sub-groups CPopjAnd finishing and outputting.
10. The method for solving the mixed frog leap of the multi-target backpack problem according to claim 1, wherein in S10, the specific steps of the weight reduction operation are as follows:
s101, calculating the profit p of the existing articles in each backpackijW/weightij；
S102, calculating profits p of ith item in all backpacks in sequenceijW/weightijAverage value AK ofiRepeating the step S102 until the average value of the profit-to-weight ratios of all the backpacks of each article is calculated;
s103, average value AKiAnd (5) performing ascending sorting, taking the first sorted article out of the backpack, and constructing a new solution.
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