Multi-target artificial protozoa optimization method for solving knapsack problem

Abstract

The invention provides a multi-target artificial protozoa optimization method for solving a knapsack problem, and the method comprises the following steps: S1, setting artificial protozoa population parameters and control parameter values, and initializing the position and adaptive value of each artificial protozoa in an artificial protozoa population; s2, simulating foraging, dormancy and reproduction behaviors of the artificial protozoa; s3, updating the position and the adaptive value of the artificial protozoa according to the step S2; s4, converting the continuous solution space into a discrete solution space through discretization processing; s5, selectively selecting part of the articles according to the characteristics of all the articles; and S6, repeating the steps S2-S5, and iteratively updating the position and the adaptive value of the artificial protozoa until an iteration termination condition is satisfied, the method simulates foraging, dormancy and reproduction stages, and for a multi-objective optimization problem, a search direction can be guided to turn to other areas which are not fully explored, so that a better global solution can be possibly found.

Description

Technical Field

[0001] The invention relates to the field of optimization calculation of knapsack problems, and in particular to a multi-objective artificial protozoan optimization method for solving knapsack problems. Background Art

[0002] With the rapid development of science and technology, optimization problems in production practice, scientific research, economic management and daily life have become more complex than ever before. In the face of these challenges, traditional computing methods face the problems of high computational complexity and long computational time. Especially when dealing with NP-hard problems, classical algorithms, constructive algorithms and local search algorithms often find it difficult to find satisfactory solutions within a reasonable time. In order to overcome this problem, the scientific community has actively explored and innovated a series of heuristic intelligent optimization methods.

[0003] The knapsack problem is an NP-complete problem of combinatorial optimization. Its core lies in how to select a set of items to maximize the total value under a given total weight limit. The multi-objective knapsack problem is a multi-objective version of the knapsack problem. Its core lies in selecting a subset from a set of items to maximize (or minimize) multiple objective functions. This problem has a wide range of applications in production practice, scientific research, economic management and daily life.

[0004] With the increasing number of items in the backpack and the complexity of their properties, the challenges of solving the multi-objective knapsack problem have increased significantly, and its solution space and time complexity have expanded dramatically, showing an exponential growth trend. Faced with large-scale multi-objective knapsack problems, the huge solution space makes traditional computational methods not only time-consuming and costly, but also difficult to effectively search for high-quality optimal solutions. In order to overcome these difficulties, it is particularly important to study and apply efficient search methods to achieve a good balance between solution time and solution accuracy. In this context, random optimization algorithms that simulate natural evolution processes, such as the multi-objective whale optimization algorithm (MOWOA), the multi-objective particle swarm optimization algorithm (MOPSO), the multi-objective grey wolf optimization algorithm (MOGWO), and the multi-objective artificial bee colony optimization algorithm (MOABC), have shown unique advantages.

[0005] Václav The team proposed an artificial protozoan optimization algorithm in 2024, which has attracted widespread attention due to its strong competitiveness. However, the artificial protozoan optimization algorithm can only solve continuous single-objective optimization problems, and there is a large exploration space for discrete multi-objective optimization problems. Therefore, the multi-objective artificial protozoan optimization method for solving the knapsack problem has certain research value. Summary of the invention

[0006] The purpose of the present invention is to solve the technical problems in the above background, and to propose a multi-objective artificial protozoan optimization method for solving the knapsack problem, comprising the following steps:

[0007] S1. Setting the artificial protozoan population parameters and control parameter values, initializing the position and fitness value of each artificial protozoan in the artificial protozoan population;

[0008] S2, simulate the foraging, dormancy and reproduction behaviors of artificial protozoa;

[0009] S3, according to step S2, updating the position and fitness value of the artificial protozoa;

[0010] S4, converting the continuous solution space into a discrete solution space through discretization processing;

[0011] S5. Select some items selectively based on the characteristics of all items;

[0012] S6. Repeat steps S2 to S5 to iteratively update the position and fitness value of the artificial protozoa until the iteration termination condition is met.

[0013] In the preferred solution, in step S1, the problem dimension is first determined according to the actual problem, and after setting the maximum number of population iterations, the optimization calculation is carried out.

[0014] In the preferred solution, during the initialization process, the position of the artificial protozoa is randomly discretized, and the mathematical formula is:

[0015] X j ∈{0,1},j=1,2,3,…,n;

[0016] Where j is the population dimension, X j is the position of the artificial protozoa corresponding to each population dimension.

[0017] In the preferred solution, step S4 further includes the following steps: first, the problem space is converted to the space [0,1) using the transfer function T(X), and then the data is converted to the discrete space using the mapping method, and the mathematical formula is:

[0018]

[0019] In the preferred solution, step S5 also includes the following steps: Sort from large to small, randomly select one item from the top percentage a and put it into the backpack. The mathematical formula is:

[0020] c1=n*a,c2∈{1,2,…c1};

[0021] X k=1, k = B(c2);

[0022] Where n is the dimension of the item, a is 10, c1 is the number of items in the first a percent, c2 is the random item selection index, and k is the mapping of the item index selected in the sorted backpack.

[0023] In a preferred solution, in step S6, when the iteration termination condition is met, the final non-dominated solution set is output.

[0024] In the preferred solution, the position update formula of the artificial protozoa's foraging behavior under the influence of internal factors is:

[0025]

[0026] in, is the updated position, leader i is the leader of the ith iteration selected from the archive, f is the foraging factor between [0,1], ps is the size of the population, and np refers to the number of neighbor pairs in the external factor. max Refers to the maximum number of neighbor pairs in the external factors, X i is the position of the ith artificial protozoa, X j represents the jth artificial protozoan position randomly selected, ⊙ represents the Hadamard product, iter represents the current iteration number, and iter max represents the maximum number of iterations, X k- represents an artificial protozoan with a ranking lower than i that is randomly selected in the kth pairing neighborhood, X k+ represents another randomly selected artificial protozoan with a higher ranking than i in the kth pairing neighborhood, w a is the weight factor, and M f is a mapping vector (1*dim) used for foraging, where each element is either 0 or 1, d i is the dimension index, d i ∈{1,2,3,…,dim}.

[0027] In the preferred solution, the position update formula of the artificial protozoa's foraging behavior under the influence of external factors is:

[0028]

[0029] Rand=[rand1,rand2,rand3,…,rand dim, ];

[0030] X near is a neighboring position, X i-k represents the ikth artificial protozoan ranked ik selected from the kth pairing neighborhood, Xi+k represents the i+kth artificial protozoan ranked i+k selected from the kth pairing neighborhood, w h is a weight factor, ± represents X near The i-th artificial protozoa are in different directions, Rand is a random number in the interval [0,1], leader i is the leader of the i-th iteration selected from the archive, r2 represents a random value uniformly distributed in the interval [0,2π], r3 represents a random value uniformly distributed in the interval [-2,2], r4 represents a random value uniformly distributed in the interval [0,1], and r1 is a random value constructed by the following formula:

[0031]

[0032] Among them, a is a constant, t is the current number of iterations, and T is the maximum number of iterations.

[0033] In the preferred solution, the formula for updating the position of the artificial protozoan during dormancy behavior is:

[0034]

[0035] X min =[lb1,lb2,lb3,…,lb dim ],X max =[ub1,ub2,ub3,…,ub dim ];

[0036] Among them, X min and X max represent the lower and upper limits of the vector respectively.

[0037] In the preferred solution, the formula for updating the position of artificial protozoa during reproduction behavior is:

[0038]

[0039] The “±” indicates that the disturbance can be positive or negative, and M r is a mapping vector in reproductive behavior.

[0040] Compared with the prior art, the present invention has the following beneficial effects:

[0041] (1) The present invention inherits the basic framework of artificial protozoa, simulates the foraging, dormancy, and reproduction stages, and can guide the search direction to other underexplored areas for multi-objective optimization problems, thereby making it possible to find a better global solution.

[0042] (2) During the position update process, the characteristics of the items are fully considered, and the selected items have greater value and smaller weight, which can promote the population to form a better Pareto frontier. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Figure 1 Schematic diagram of the Pareto frontier obtained by different algorithms on the F1-20 test function.

[0044] Figure 2 This is a schematic diagram of the Pareto frontier obtained by different algorithms on the F1-200 test function.

[0045] Figure 3 It is a schematic diagram of the Pareto frontier obtained by different algorithms on the F1-450 test function.

[0046] Figure 4 This is a schematic diagram of the Pareto frontier obtained by different algorithms on the F1-750 test function.

[0047] Figure 5 This is a schematic diagram of the Pareto frontier obtained by different algorithms on the F2-20 test function.

[0048] Figure 6 This is a schematic diagram of the Pareto frontier obtained by different algorithms on the F2-200 test function.

[0049] Figure 7 It is a schematic diagram of the Pareto frontier obtained by different algorithms on the F2-450 test function.

[0050] Figure 8 This is a schematic diagram of the Pareto frontier obtained by different algorithms on the F2-750 test function. DETAILED DESCRIPTION

[0051] The specific implementation modes of the present invention are described in detail below with reference to the accompanying drawings.

[0052] The knapsack problem function used in this example is shown in Table 1, and the parameter settings of each algorithm are shown in Table 2.

[0053] Table 1 Knapsack problem function used in optimization calculation

[0054]

[0055] Table 2 Algorithm parameter settings

[0056] parameter MOWOA MOABC MOPSO MOGWO IMOAPO Population size 100 100 100 100 100 Archive size 100 100 100 100 100 Beta - - 4 4 - Gamma - - 2 2 -

[0057] When using the multi-objective artificial protozoa optimization algorithm for optimal value iterative solution, first determine the problem dimension n according to the actual problem (n = 10, 200, 450 and 750 in the example), set the maximum population iteration number to 400 to carry out the optimization calculation.

[0058] In the second step, a random parent population is generated in the problem space according to the value range and dimension of the problem to be solved, and the fitness value of each parent population is calculated according to the knapsack problem.

[0059] Step 3: Iterative Search

[0060] First, according to the number of iterations, determine whether the iteration termination condition is met. If the iteration termination condition is not met, continue to update the population position, otherwise output the final non-dominated solution set, and the mapping of these non-dominated solution sets in the target space constitutes the Pareto frontier.

[0061] The population position update strategy is as follows:

[0062] The non-dominated sorting (NDS) and crowding distance (CD) are applied to the parent population, different position update formulas are adopted according to the different behaviors of artificial protozoa, and the parent population and the new population are merged to create a new parent population. In addition, the non-dominated solution will be saved in the archive. These steps will be repeated until the loop termination condition is met.

[0063] The position update formula of the artificial protozoan foraging behavior under the influence of internal factors is:

[0064]

[0065] in, is the updated position, leader i is the leader of the ith iteration selected from the archive, f is the foraging factor between [0,1], ps is the size of the population, and np refers to the number of neighbor pairs in the external factor. max Refers to the maximum number of neighbor pairs in the external factors, X i is the position of the ith artificial protozoa, X j represents the jth artificial protozoan position randomly selected, ⊙ represents the Hadamard product, iter represents the current iteration number, and iter max represents the maximum number of iterations, X k- represents an artificial protozoan with a ranking lower than i that is randomly selected in the kth pairing neighborhood, X k+ represents another randomly selected artificial protozoan with a higher ranking than i in the kth pairing neighborhood, w a is the weight factor, and M fis a mapping vector (1*dim) used for foraging, where each element is either 0 or 1, d i is the dimension index, d i ∈{1,2,3,…,dim}.

[0066] The position update formula of the artificial protozoan's foraging behavior under the influence of external factors is:

[0067]

[0068] Rand=[rand1,rand2,rand3,…,rand dim ] (15)

[0069] X near is a neighboring position, Xi-k represents the ikth artificial protozoa ranked ik selected from the kth paired neighborhood, Xi+k represents the i+kth artificial protozoa ranked i+k selected from the kth paired neighborhood, ± represents X near The i-th artificial protozoa can be in different directions, w h is a weight factor, where leader i is the leader of the i-th iteration selected from the archive, r2 represents a random value uniformly distributed in the interval [0,2π], r3 represents a random value uniformly distributed in the interval [-2,2], r4 represents a random value uniformly distributed in the interval [0,1], and r1 is a random value constructed by formula (16):

[0070]

[0071] Among them, represents a, a constant which is usually assumed to be 2, t represents the current iteration number, and T represents the maximum iteration number.

[0072] The position update formula of artificial protozoa in dormancy behavior is:

[0073]

[0074] X min =[lb1,lb2,lb3,…,lb dim ],X max =[ub1,ub2,ub3,…,ub dim ] (18)

[0075] Among them, X min and X max represent the lower and upper limits of the vector respectively.

[0076] The formula for updating the position of artificial protozoa during reproduction behavior is:

[0077]

[0078] The “±” indicates that the disturbance can be positive or negative, and M r is a mapping vector in reproductive behavior.

[0079] The above process is a Monte Carlo numerical simulation calculation process. In order to reduce the influence of random effects, N = 10 Monte Carlo calculations are performed in this example.

[0080] In order to evaluate the performance of each algorithm, the evaluation index Inverse Generation Distance (IGD) is used to evaluate the advantages and disadvantages of each algorithm. IGD mainly measures the distance between a set of non-dominated solutions (solutions found by the optimization algorithm) and the reference Pareto optimal front. The smaller the IGD value, the closer the solution obtained by the algorithm is to the ideal Pareto optimal solution. Table 3 shows the simulation results of different algorithms on different test functions for the IGD evaluation index. Figures 1 to 8 Shows the Pareto front formed by different algorithms on different knapsack problems.

[0081] Table 3 Simulation results of different algorithms for IGD evaluation indicators on different test functions

[0082]

[0083]

[0084] It can be seen that the multi-objective artificial protozoa optimization algorithm (IMOAPO) proposed in the present invention is significantly superior to other multi-objective optimization algorithms, indicating that IMPAPO can effectively balance global search and local search, avoid falling into local optimal solutions, and thus provide a high-quality solution set for multi-objective optimization problems.

[0085] The above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit the same. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that the technical solutions described in the aforementioned embodiments may still be modified, or some of the technical features may be replaced by equivalents. However, these modifications or replacements do not deviate the essence of the corresponding technical solutions from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A multi-objective artificial protozoan optimization method for solving the knapsack problem, characterized by: The following steps are involved: S1. Setting the artificial protozoan population parameters and control parameter values, initializing the position and fitness value of each artificial protozoan in the artificial protozoan population; S2, simulate the foraging, dormancy and reproduction behaviors of artificial protozoa; S3, according to step S2, updating the position and fitness value of the artificial protozoa; S4, converting the continuous solution space into a discrete solution space through discretization processing; S5. Select some items selectively based on the characteristics of all items; S6. Repeat steps S2 to S5 to iteratively update the position and fitness value of the artificial protozoa until the iteration termination condition is met.

2. A multi-objective artificial protozoan optimization method for solving the knapsack problem according to claim 1, characterized in that: In step S1, the problem dimension is first determined according to the actual problem, and after setting the maximum number of population iterations, the optimization calculation is carried out.

3. A multi-objective artificial protozoan optimization method for solving the knapsack problem according to claim 1, characterized in that: During the initialization process, the position of the artificial protozoa is randomly discretized, and its mathematical formula is: X j ∈{0,1},j=1,2,3,…,n): Where j is the population dimension, X j is the position of the artificial protozoa corresponding to each population dimension.

4. A multi-objective artificial protozoan optimization method for solving the knapsack problem according to claim 1, characterized in that: Step S4 The following steps are also included: first, the space of the problem is transformed into the space [0,1) using the transfer function T(X), and then the data is transformed into a discrete space using the mapping method. The mathematical formula is:

5. A multi-objective artificial protozoan optimization method for solving the knapsack problem according to claim 1, characterized in that: Step S5 also includes the following steps: Sort from large to small, randomly select one item from the top percentage a and put it into the backpack. The mathematical formula is: c1=n*a,c2∈{1,2,…c1}; X k =1,k=B(c2); Where n is the dimension of the item, a is 10, c1 is the number of items in the first a percent, c2 is the random item selection index, and k is the mapping of the item index selected in the sorted backpack.

6. A multi-objective artificial protozoan optimization method for solving the knapsack problem according to claim 1, characterized in that: In step S6, when the iteration termination condition is met, the final non-dominated solution set is output.

7. A multi-objective artificial protozoan optimization method for solving the knapsack problem according to claim 1, characterized in that: The position update formula of the artificial protozoan foraging behavior under the influence of internal factors is: X i =sort(X i ); in, is the updated position, leader i is the leader of the ith iteration selected from the archive, f is the foraging factor between [0,1], ps is the size of the population, and np refers to the number of neighbor pairs in the external factor. max Refers to the maximum number of neighbor pairs in the external factors, X i is the position of the ith artificial protozoa, X j represents the jth artificial protozoan position randomly selected, ⊙ represents the Hadamard product, iter represents the current iteration number, and iter max represents the maximum number of iterations, X k- represents an artificial protozoan with a ranking lower than i that is randomly selected in the kth pairing neighborhood, X k+ represents another randomly selected artificial protozoan with a higher ranking than i in the kth pairing neighborhood, w a is the weight factor, and M f is a mapping vector (1*dim) used for foraging, where each element is either 0 or 1, d i is the dimension index, d i ∈{1,2,3,…,dim}.

8. A multi-objective artificial protozoan optimization method for solving the knapsack problem according to claim 1, characterized in that: The position update formula of the artificial protozoan's foraging behavior under the influence of external factors is: Row=[row1,row2,row3,…,row dim ]; X near is a neighboring position, X i-k represents the ikth artificial protozoan ranked ik selected from the kth pairing neighborhood, X i+k represents the i+kth artificial protozoan ranked i+k selected from the kth pairing neighborhood, w h is a weight factor, ± represents X near The i-th artificial protozoa are in different directions, Rand is a random number in the interval [0,1], leader i is the leader of the i-th iteration selected from the archive, r2 represents a random value uniformly distributed in the interval [0,2π], r3 represents a random value uniformly distributed in the interval [-2,2], r4 represents a random value uniformly distributed in the interval [0,1], and r1 is a random value constructed by the following formula: Among them, a is a constant, t is the current number of iterations, and T is the maximum number of iterations.

9. A multi-objective artificial protozoan optimization method for solving the knapsack problem according to claim 1, characterized in that: The position update formula of artificial protozoa in dormancy behavior is: X min =[lb1,lb2,lb3,…,lb dim ],X max =[ub1,ub2,ub3,…,ub dim ]; Among them, X min and X max represent the lower and upper limits of the vector respectively.

10. A multi-objective artificial protozoan optimization method for solving the knapsack problem according to claim 1, characterized in that: The formula for updating the position of artificial protozoa during reproduction behavior is: "±" indicates that the disturbance can be positive or negative, and M r is a mapping vector in reproductive behavior.

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