Adaptive learning hyper-heuristic optimization method for joint replenishment optimization problem

Abstract

The invention relates to an adaptive learning hyper-heuristic optimization method for a joint replenishment optimization problem, and the method comprises the steps: obtaining a to-be-optimized decision variable in a joint replenishment problem JRP model, determining a hyper-parameter and a sub-algorithm of an adaptive learning hyper-heuristic optimization model, and generating an initial sub-population individual according to the to-be-optimized decision variable; calculating the fitness value of each individual in the sub-populations and the updating times of the sub-populations, and updating the sub-populations; according to the historical information of the sub-populations, whether optimal individual communication among the sub-populations is carried out or not is judged; and generating a random number, judging whether individual migration is carried out among the updated sub-populations according to the random number, calculating the latest fitness value of the sub-populations after individual migration, and obtaining the current optimal individual, thereby generating the current optimal decision variable. The method comprises an adaptive learning strategy and a sub-population communication and migration strategy, has excellent global optimization ability and solving efficiency, and can significantly reduce the total cost when solving the JRP problem.

Description

Technical Field

[0001] The present invention relates to the technical field of optimization algorithms, and in particular to an adaptive learning hyper-heuristic optimization method for joint replenishment optimization problems. Background Art

[0002] The Joint Replenishment Problem (JRP) refers to the process of reducing the total cost by sharing the ordering costs of multiple products through group purchasing strategies (Peng et al., 2022). In the JRP problem, the production of multiple items shares a joint startup cost, and the packaging of each item incurs a secondary startup cost. Due to this special cost structure, significant cost savings can be achieved by coordinating the production cycle of items (Wang Sirui et al., 2024). The JRP problem has been verified to be an NP-hard problem (Wang & Chen, 2020; Wang et al., 2024). There is no polynomial time algorithm. The commonly used solution methods are some heuristic and meta-heuristic methods. However, as the problem scale increases, the solution performance also needs to be improved. It is necessary to design new efficient solution methods.

[0003] Metaheuristic algorithms are algorithms that imitate various operating mechanisms in nature and are widely used in engineering design, operations research, artificial intelligence, and economics (Kaveh et al., 2023). At present, more and more metaheuristic algorithms have been proposed, which can be mainly divided into five categories: population-based algorithms, evolution-based algorithms, physics-based algorithms, human society-based algorithms, and other algorithms. Population-based algorithms mainly simulate the behavior of animals and other organisms. For example, the Snow Geese Algorithm (SGA) imitates the special flight pattern of snow geese during migration (Tian et al., 2024). Evolution-based algorithms are inspired by the natural selection mechanism of "survival of the fittest" and the law of gene transmission in the process of biological evolution, mainly including differential evolution algorithms (DE) and so on (Storn & Price, 1997). Physics-based algorithms mainly establish mathematical models based on various phenomena, concepts, and laws in physics. For example, Deng and Liu (2023) proposed a snow ablation optimizer (SAO) based on the sublimation and melting of snow. Algorithms based on human society mainly simulate human behavior and interaction in social life. For example, Kaveh et al. (2023) simulated horticultural behaviors such as irrigation, fertilization, pruning and grafting, and proposed the Orchard Algorithm (OA). Other algorithms are algorithms inspired by other phenomena. For example, the Arithmetic Optimization Algorithm (AOA) is a metaheuristic optimization algorithm designed based on the idea of ​​mixed operations of four arithmetic operations (Abualigah et al., 2021). Existing metaheuristic algorithms have the following advantages: ① Simple and easy to understand, easy to operate, and easy for non-professionals to understand and modify; ② Highly adaptable and can be applied to various types of practical optimization problems. Although the existing metaheuristic algorithms show the above advantages in solving some optimization problems, according to the free lunch theorem, a single algorithm cannot be applicable to all optimization problems. Joint replenishment optimization is a complex problem. Although the existing metaheuristic algorithms can solve the JRP problem to a certain extent, they still have the following disadvantages:

[0004] First, most single metaheuristic algorithms are only applicable to JRP problems in specific scenarios. A single metaheuristic algorithm works well in solving JRP problems in a certain scenario, but not necessarily in solving JRP problems in another scenario. This is caused by the specific search strategy of the metaheuristic algorithm, which is only applicable to certain scenarios. Therefore, a single metaheuristic algorithm cannot be applied to JRP optimization problems in all scenarios.

[0005] Secondly, some improved metaheuristic algorithms only adopt low-level integration strategies. Low-level integrated metaheuristic algorithms mainly include a simple combination of algorithm components such as multiple hyperparameters of a single metaheuristic algorithm and multiple search strategies. This integration enriches the diversity of the population and enhances the search ability of the algorithm to a certain extent, but the performance is not stable and cannot be adaptively adjusted according to actual conditions. In practical applications, the performance improvement is not significant.

[0006] Finally, the existing metaheuristic algorithm integration framework has poor complementarity. At present, many metaheuristic algorithms have been proposed by domestic and foreign authors, and these algorithm resources can be fully utilized through integration. However, in the current integration framework, the metaheuristic algorithms are almost parallel, and there is almost no complementary communication between these algorithms, lacking versatility. When facing complex or multi-peak optimization problems, the actual effect is not ideal. Summary of the invention

[0007] The purpose of the present invention is to propose an adaptive learning hyperheuristic optimization algorithm (ALHOA) for joint replenishment optimization problem to solve the problems existing in the above-mentioned prior art. The method has excellent global optimization ability and solving efficiency, and can be applied to solving multi-product JRP problems to obtain lower total costs.

[0008] To achieve the above object, the present invention provides the following solutions:

[0009] An adaptive learning hyper-heuristic optimization method for joint replenishment optimization problem, including:

[0010] Step S1, obtaining the decision variables to be optimized in the joint replenishment problem JRP model, determining the hyperparameters and sub-algorithms of the adaptive learning hyper-heuristic optimization model, and generating initial sub-population individuals according to the decision variables to be optimized; wherein each sub-population corresponds to a sub-algorithm, and the sub-algorithm is any meta-heuristic algorithm;

[0011] Step S2, calculating the fitness value of each individual in the sub-population and the number of updates of the sub-population, and updating the sub-population;

[0012] Step S3: judging whether to communicate the best individuals between sub-populations according to the historical information of the sub-populations;

[0013] Step S4, generate a random number and determine whether to migrate individuals between the updated sub-populations based on the random number, calculate the latest fitness value of the sub-population after individual migration, obtain the contemporary optimal individual, and thus generate the current optimal decision variable.

[0014] Optionally, calculating the fitness value of each individual in the subpopulation and the number of updates of the subpopulation includes:

[0015]

[0016] Among them, fitness(·) is the fitness function, U i is the optimal fitness value in subpopulation i, is the best individual in subpopulation i, V i is the update times of subpopulation i, floor(·) is the rounding function, M is the number of subpopulations, np is the number of individuals in the initial subpopulation, is the sum of the optimal fitness values ​​of all sub-populations, U i' is the optimal fitness value in subpopulation i', V M is the number of updates of subpopulation M.

[0017] Optionally, updating the subpopulation includes:

[0018] If count(XA i )≤V i , the first V in the subpopulation are sorted according to the sorting order of the subpopulation i. i Individuals are updated, if count(XA i )>V i , the individuals in the subpopulation are cyclically updated according to the sorting order of the subpopulation i until the calculated number of updates is reached; wherein, count(XA i ) is the population size of subpopulation i, count(·) is the function for calculating the number of individuals in the population, V i is the number of updates of subpopulation i.

[0019] Optionally, communicating the best individuals between subpopulations includes:

[0020] Calculating the fitness value of each individual in the updated sub-population, re-ranking each individual in the sub-population according to the fitness value of each individual in the updated sub-population, and determining whether to communicate the best individuals between the sub-populations;

[0021] Determining whether to communicate the best individuals between sub-populations includes:

[0022] like , then the best individual in subpopulation i is replaced by the best individual in the population; represents the fitness value of the best individual in subpopulation i at the tth iteration, represents the fitness value of the historical best individual in subpopulation i, fitness(·) is the fitness function, is the best individual in subpopulation i at the tth iteration, is the historical best individual in subpopulation i.

[0023] Optionally, generating a random number to determine whether to perform individual migration between subpopulations includes:

[0024] Generating the random number includes:

[0025] like When , subpopulation i is the optimal subpopulation, its mean parameter is calculated, and the corresponding random number is generated;

[0026] According to the mean parameter of the optimal subpopulation, the method for generating the corresponding random number is:

[0027]

[0028] Among them, x best (t) represents the best individual in all subpopulations at the tth iteration, mean best (t) represents the mean parameter of the normal distribution corresponding to the subpopulation with the best individual at the tth iteration, Max_t is the maximum number of iterations, and rand represents a random number between 0 and 1. Indicates the mean best (t) and a normal distribution with variance 1, rand best (t) represents the random number corresponding to the subpopulation with the best individual at the tth iteration;

[0029] like When , subpopulation i is a non-optimal subpopulation, its mean parameter is calculated, and the corresponding random number is generated;

[0030] According to the mean parameter of subpopulation i, the method of generating the corresponding random number is:

[0031]

[0032]

[0033] Among them, mean i (t) is the mean parameter of the normal distribution corresponding to subpopulation i at the tth iteration, rand i (t) is the random number corresponding to subpopulation i at the tth iteration, and Max_t is the maximum number of iterations;

[0034] The basis for judging whether to migrate individuals between sub-populations:

[0035] If rand i (t) <rand best(t), individuals are migrated between subpopulation i and the optimal subpopulation; otherwise, individuals are not migrated between subpopulation i and the optimal subpopulation.

[0036] Optionally, performing individual migration between the subpopulation i and the optimal subpopulation includes:

[0037] If count(XA i )≥2×np×threshold, the last np×threshold individuals in subpopulation i are transferred to the subpopulation with the best individual of the contemporary era, where threshold is the threshold for retaining the uniqueness of the subpopulation; if count(XA i )>np×threshold, then the last count(XA i )-np×threshold individuals are transferred to the subpopulation with the best individual of the current era; if count(XA i )≤np×threshold, in order to preserve the uniqueness of subpopulation i, no individual migration is performed between the updated subpopulations.

[0038] Optionally, generating the optimal decision variable includes:

[0039] Determine whether the preset number of update iterations has been reached. If the preset number of update iterations has not been reached, repeat steps S2-S4 until the preset number of update iterations has been reached. If the preset number of update iterations has been reached, generate the optimal decision variable according to the contemporary optimal individual. The optimal decision variable includes: the corresponding optimal k l and unit time T.

[0040] The beneficial effects of the present invention are:

[0041] Versatility: The present invention is applicable to JRP optimization problems in various scenarios. The idea of ​​integration is a mature and perfect technology that has been applied to many fields and has shown excellent performance. The proposed ALHOA algorithm draws on the idea of ​​integration in machine learning and proposes a collaborative hyper-inspirational integration method. By controlling and manipulating multiple low-level metaheuristic algorithms (sub-algorithms), it makes full use of these algorithm resources to find approximate optimal solutions in the solution space. The proposed algorithm can select multiple arbitrary metaheuristic algorithms as sub-algorithms, and each sub-algorithm corresponds to a subpopulation. Compared with a single metaheuristic algorithm, the ALHOA algorithm has excellent versatility and can achieve excellent performance on JRP optimization problems in various scenarios.

[0042] Ability to perform adaptive adjustment: The adaptive adjustment strategy of the present invention is very important in the integration process of metaheuristic algorithms. It can automatically coordinate the appropriate sub-algorithm to match the problem to be optimized based on historical information, and maximize the search ability of the sub-algorithm. This technical solution designs an adaptive learning strategy, which tilts computing resources toward the dominant sub-algorithm by learning historical information, increases the number of updates of the dominant sub-algorithm, reduces the number of updates of the inferior sub-algorithm, and strengthens the local search ability of the proposed algorithm.

[0043] Possess good complementarity: The present invention proposes a design of information exchange and migration strategy for sub-populations. Through the communication strategy, the best individuals in the population can guide different sub-populations to explore the area where the theoretical optimal solution is most likely to be found; in addition, the population migration strategy can make the search individuals in the population lean towards the dominant sub-algorithm as the number of iterations increases or decreases, while retaining the independence of the sub-population individuals, enriching the diversity of the population, and enhancing the global search ability of the proposed algorithm and the ability to jump out of the local optimum. Based on this, the proposed algorithm can learn from each other's strengths and weaknesses, focus on exploration, and thus achieve good performance in practical applications.

[0044] The present invention draws on the integration idea in machine learning and proposes a collaborative hyper-inspiration integration method for joint replenishment optimization problems, namely the ALHOA algorithm: by controlling and manipulating multiple low-level meta-heuristic algorithms (sub-algorithms), taking advantage of each other's strengths and weaknesses, making full use of these algorithm resources, searching and optimizing in the solution space; at the same time, an adaptive learning strategy is designed to tilt computing resources toward dominant sub-populations by learning historical information, increase the number of updates of dominant sub-populations, reduce the number of updates of disadvantaged sub-populations, and strengthen the local search ability of the proposed algorithm; finally, through information exchange and migration between populations, while retaining the independence between sub-populations, to enrich the diversity of populations, enhance the global search ability of the proposed algorithm and the ability to jump out of local optimality, and have excellent global optimization ability and solution efficiency. When solving the JRP problem, the ALHOA algorithm can significantly reduce the total cost. These advantages and beneficial effects make the technical solution of the present invention have broad application prospects and market promotion value in the field of optimization algorithms. BRIEF DESCRIPTION OF THE DRAWINGS

[0045] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative labor.

[0046] Figure 1 This is a flow chart of an adaptive learning hyper-heuristic optimization method for the joint replenishment optimization problem according to an embodiment of the present invention. DETAILED DESCRIPTION

[0047] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0048] In order to make the above-mentioned objects, features and advantages of the present invention more obvious and easy to understand, the present invention is further described in detail below with reference to the accompanying drawings and specific embodiments.

[0049] Related terms introduction:

[0050] Introduce the joint replenishment optimization problem: In the JRP model, every k l Item l will be replenished at time intervals of T, so the JRP model aims to determine the appropriate replenishment combination to minimize the total supply chain cost.

[0051] In the JRP model, the number of products is L, l represents the product number and 1≤l≤L, D l represents the demand rate of commodity l, S represents the main ordering cost of each replenishment, and s l is the secondary ordering cost when replenishing item l, h l represents the unit inventory maintenance cost of commodity l. The decision variables in the JRP model are: the unit time T for commodity replenishment, the replenishment cycle multiplier k for commodity l l The total cost of the JRP model is shown in formula (1):

[0052]

[0053] Among them, C h represents inventory cost, C o represents the total ordering cost. The optimal T can be further obtained as shown in formula (2):

[0054]

[0055] The technical problem to be solved by this embodiment is to provide an adaptive learning hyper-heuristic optimization method for the joint replenishment optimization problem. This method draws on the integration idea in machine learning and proposes a collaborative hyper-heuristic integration method. The proposed integration method can select multiple arbitrary meta-heuristic algorithms as sub-algorithms, and each sub-algorithm corresponds to a sub-population. By controlling and manipulating multiple low-level meta-heuristic algorithms (sub-algorithms), the strengths of these algorithms are complemented, and the resources of these algorithms are fully utilized to search and optimize in the solution space; at the same time, an adaptive learning strategy is designed to tilt the computing resources toward the dominant sub-population by learning historical information, increase the number of updates of the dominant sub-population, reduce the number of updates of the disadvantaged sub-population, and strengthen the local search ability of the proposed algorithm; finally, through information exchange and migration between populations, while retaining the independence of sub-populations, to enrich the diversity of populations, enhance the global search ability of the proposed algorithm and the ability to jump out of local optimality. The specific process of ALHOA is as follows: Figure 1 As shown:

[0056] Step S1, obtaining the decision variables to be optimized in the joint replenishment problem JRP model, determining the hyperparameters and sub-algorithms of the adaptive learning hyper-heuristic optimization model, and generating initial sub-population individuals according to the decision variables to be optimized; wherein each sub-population corresponds to a sub-algorithm, and the sub-algorithm is any meta-heuristic algorithm;

[0057] Step S2, calculating the fitness value of each individual in the sub-population and the number of updates of the sub-population, and updating the sub-population;

[0058] Step S3: judging whether to communicate the best individuals between sub-populations according to the historical information of the sub-populations;

[0059] Step S4, generating a random number and judging whether to migrate individuals between the updated sub-populations according to the random number, calculating the latest fitness value of the sub-population after the individual migration, obtaining the contemporary optimal individual, and thus generating the current optimal decision variable;

[0060] Among them, the optimal decision variables include: the corresponding optimal k l and unit time T.

[0061] Specifically, step 1: parameter setting and initialization of each sub-population. Set ALHOA related parameters and generate the initial sub-population XA i (a total of M subpopulations), as shown in formula (3):

[0062]

[0063] Where DM is the dimension, np is the number of individuals in the initial subpopulation, and xa i,j,k represents the kth dimension of the jth individual in the i-th subpopulation. And xa i,j=round(rand(1)×(UB-LB)+LB), where round(·) represents the rounding function, and UB and LB represent the maximum and minimum values ​​of the decision variable k, respectively. In ALHOA, the decision variable k l The maximum value UB is usually set to a larger value, and the decision variable k l The minimum value is usually set to 1.

[0064] Furthermore, calculating the fitness value of each individual in the subpopulation and the number of updates of the subpopulation includes:

[0065] Calculating the fitness value of each individual in the sub-population, and sorting the individuals in the sub-population according to the fitness value. Then, calculating the number of updates of the sub-population includes:

[0066]

[0067]

[0068] Among them, fitness(·) is the fitness function, U i is the optimal fitness value in subpopulation i, is the best individual in subpopulation i, V i is the update times of subpopulation i, floor(·) is the rounding function, U i is the optimal fitness value in subpopulation i, M is the number of subpopulations, np is the number of individuals in the initial subpopulation, is the sum of the optimal fitness values ​​of all sub-populations, U i' is the optimal fitness value in subpopulation i', V M is the number of updates of subpopulation M.

[0069] Specifically, step 2: Calculate the update times of each subpopulation according to formulas (5) and (6):

[0070]

[0071]

[0072] Among them, fitness(·) is the fitness function, U i is the optimal fitness value in subpopulation i, is the best individual in subpopulation i, V i is the update times of subpopulation i, floor(·) is the rounding function, U i is the optimal fitness value in subpopulation i, M is the number of subpopulations, np is the number of individuals in the initial subpopulation, is the sum of the optimal fitness values ​​of all sub-populations, U i'is the optimal fitness value in subpopulation i', V M is the update number of subpopulation M. Through the adaptive learning strategy, ALHOA can tilt the computing resources toward the dominant subpopulation, increase the update number of the dominant subpopulation, reduce the update number of the disadvantaged subpopulation, and strengthen the local search ability of the proposed algorithm.

[0073] Further, updating the subpopulation includes: if count(XA i )≤V i , the first V in the subpopulation are sorted according to the sorting order of the subpopulation i. i Individuals are updated, if count(XA i )>V i , the individuals in the subpopulation are cyclically updated according to the sorting order of the subpopulation i until the calculated number of updates is reached; wherein, count(XA i ) is the population size of subpopulation i, count(·) is the function for calculating the number of individuals in the population, V i is the number of updates of subpopulation i.

[0074] Specifically, step 3: If count(XA i )≤V i , sort the first V in the subpopulation according to the sorting order of subpopulation i i Individuals are updated, if count(XA i )>V i , according to the sorting order of the subpopulation i, the individuals in the subpopulation are cyclically updated until the calculated number of updates is reached; wherein, count(XA i ) is the population size of subpopulation i, count(·) is the function for calculating the number of individuals in the population, V i is the number of updates of subpopulation i.

[0075] Furthermore, communicating the best individuals between subpopulations includes: calculating the fitness value of each individual in the updated subpopulation, re-ranking each individual in the subpopulation according to the fitness value of each individual in the updated subpopulation, and determining whether to communicate the best individuals between subpopulations;

[0076] Determining whether to communicate the best individuals between sub-populations includes:

[0077] like , then the best individual in subpopulation i is replaced by the best individual in the population; represents the fitness value of the best individual in subpopulation i at the tth iteration, represents the fitness value of the historical best individual in subpopulation i, fitness(·) is the fitness function, is the best individual in subpopulation i at the tth iteration, is the historical best individual in subpopulation i.

[0078] Specifically, step 4: If Replace the best individual in the current subpopulation i with the best individual in the current population; where, represents the fitness value of the best individual in subpopulation i at the tth iteration, represents the fitness value of the historical best individual in subpopulation i, fitness(·) is the fitness function, is the best individual in subpopulation i at the tth iteration, is the historical best individual in subpopulation i. The communication of the best individuals between subpopulations can guide different subpopulations to explore the area where the theoretical optimal solution is most likely to be found.

[0079] Furthermore, a random number is generated to determine whether to migrate individuals between subpopulations:

[0080] Generating the random number includes:

[0081] like When , subpopulation i is the optimal subpopulation, its mean parameter is calculated, and the corresponding random number is generated;

[0082] According to the mean parameter of the optimal subpopulation, the method for generating the corresponding random number is:

[0083]

[0084] Among them, x best (t) represents the best individual in all subpopulations at the tth iteration, mean best (t) represents the mean parameter of the normal distribution corresponding to the subpopulation with the best individual at the tth iteration, Max_t is the maximum number of iterations, and rand represents a random number between 0 and 1. Indicates the mean best (t) and a normal distribution with variance 1, rnad best (t) represents the random number corresponding to the subpopulation with the best individual at the tth iteration.

[0085] like When , subpopulation i is a non-optimal subpopulation, its mean parameter is calculated, and the corresponding random number is generated;

[0086] According to the mean parameter of subpopulation i, the method of generating the corresponding random number is:

[0087]

[0088]

[0089] Among them, mean i (t) is the mean parameter of the normal distribution corresponding to subpopulation i at the tth iteration, The mean function is mean i (t) and a normal distribution with variance 1, rand i (t) is the random number corresponding to subpopulation i at the tth iteration, and Max_t is the maximum number of iterations.

[0090] The basis for judging whether to migrate individuals between sub-populations:

[0091] If rand i (t) <rand best (t), individuals are migrated between subpopulation i and the optimal subpopulation; otherwise, individuals are not migrated between subpopulation i and the optimal subpopulation.

[0092] Specifically, step 5: If According to formulas (7) and (8), the parameter mean is calculated best (t), and generate a random number rand best (t):

[0093]

[0094] Among them, x best (t) represents the best individual in all subpopulations at the tth iteration, mean best (t) represents the mean parameter of the normal distribution corresponding to the subpopulation with the best individual at the tth iteration, Max_t is the maximum number of iterations, and rand represents a random number between 0 and 1. Indicates the mean best (t) and a normal distribution with variance 1, rand best (t) represents the random number corresponding to the subpopulation with the best individual at the tth iteration.

[0095] if According to formulas (9) and (10), the parameter mean is calculated i (t), and generate a random number rand i (t):

[0096]

[0097] Among them, mean i(t) is the mean parameter of the normal distribution corresponding to subpopulation i at the tth iteration, The mean function is mean i (t) and a normal distribution with variance 1, rand i (t) is the random number corresponding to subpopulation i at the tth iteration, and Max_t is the maximum number of iterations.

[0098] The basis for judging whether to migrate individuals between sub-populations:

[0099] If rand i (t) <rand best (t), individuals are migrated between subpopulation i and the optimal subpopulation; otherwise, individuals are not migrated between subpopulation i and the optimal subpopulation.

[0100] Furthermore, performing individual migration between the subpopulation i and the optimal subpopulation includes: if count(XA i )≥2×np×threshold, the last np×threshold individuals in subpopulation i are transferred to the subpopulation with the best individual of the current era, where threshold is the threshold for retaining the uniqueness of the subpopulation; if count(XA i )>np×threshold, then the last count(XA i )-np×threshold individuals are transferred to the subpopulation with the best individual of the current era; if count(XA i )≤np×threshold, in order to preserve the uniqueness of subpopulation i, no individual migration is performed between the updated subpopulations.

[0101] Specifically, step 6: when count(XA i )≥2×np×threshold, the last np×threshold individuals in subpopulation i are transferred to the subpopulation of the best individual of the current era, where threshold is the threshold for retaining the uniqueness of the subpopulation; when count(XA i )>np×threshold, the latter count (XA i )-np×threshold individuals are transferred to the subpopulation with the best individual of the contemporary era; when count(XA i )≤np×threshold, in order to preserve the uniqueness of subpopulation i, no individual migration is performed between the updated subpopulations.

[0102] According to the above description, we can find that at the initial moment, the number of individuals in each subpopulation is equal, but after each iteration, the inferior subpopulation may lose some individuals, and these individuals are transferred to the dominant subpopulation. As the number of iterations increases, the number of search individuals in the population tilts towards the dominant sub-algorithm. In addition, the migration between populations can enrich the diversity of the population while retaining the independence of the subpopulation individuals, and enhance the global search ability of the proposed algorithm and the ability to jump out of the local optimum.

[0103] Furthermore, generating the optimal decision variable includes: determining whether a preset number of update iterations has been reached; if the preset number of update iterations has not been reached, repeating steps S2-S4 until the preset number of update iterations has been reached; if the preset number of update iterations has been reached, generating the optimal decision variable based on the contemporary optimal individual.

[0104] Step 7: According to the fitness values ​​of individuals in all subpopulations, obtain the contemporary optimal individual.

[0105] Step 8: Update the number of iterations and determine whether the maximum number of iterations has been reached. If so, go to step 9; if not, return to step 2.

[0106] Step 9: Until the maximum number of iterations is reached, the TC cost value and the corresponding optimal k are obtained. l and T value.

[0107] This technical solution proposes an adaptive learning hyper-heuristic optimization method for the joint replenishment optimization problem. By controlling and manipulating the search and optimization of multiple low-level meta-heuristic algorithms (sub-algorithms) in the solution space, it has excellent global optimization ability and solution efficiency. For the joint replenishment optimization problem, many studies have shown that the use of meta-heuristic algorithms such as differential evolution (DE) can achieve good results (Cui et al., 2023).

[0108] Metaheuristic algorithms have the following advantages: ① simple to understand, easy to operate, and easy for non-professionals to understand and modify; ② highly adaptable and can be applied to various types of practical optimization problems. Although existing metaheuristic algorithms have demonstrated the above advantages in solving some optimization problems, according to the free lunch theorem, a single algorithm cannot be applicable to all optimization problems. Joint replenishment optimization is a complex problem that requires targeted design solutions. Therefore, this technical solution draws on the integration idea in machine learning and proposes a collaborative hyper-inspiration integration method, namely the ALHOA algorithm. The algorithm includes adaptive learning strategies, subpopulation communication and migration strategies, and has excellent global optimization capabilities and solution efficiency.

[0109] The main idea is to use the integration idea to control and manipulate multiple low-level meta-heuristic algorithms (sub-algorithms), take advantage of their strengths and overcome their weaknesses, and find the optimal solution in the solution space.

[0110] The second is the adaptive learning strategy. By allocating the update times of each subpopulation through historical information, the computing resources are tilted towards the dominant subpopulation, the update times of the dominant subpopulation are increased, the update times of the disadvantaged subpopulation are reduced, and the local search ability of the proposed algorithm is strengthened. The calculation method of the update times is as follows.

[0111]

[0112]

[0113] Then update the individuals in different subpopulations according to the following rules: i )≤V i , sort the first V in the subpopulation according to the sorting order of subpopulation i i Individuals are updated, if count(XA i )>V i , the individuals in the subpopulation are cyclically updated according to the sorting order of the subpopulation i until the calculated number of updates is reached.

[0114] Finally, there is the subpopulation communication and migration strategy. First, the subpopulation communication strategy. The communication of the best individuals between subpopulations can guide different subpopulations to explore the area where the theoretical optimal solution is most likely to be found. The specific rules are as follows: If Replace the best individual in the current subpopulation i with the best individual in the current population. The second is the migration strategy of the subpopulation, which determines whether to migrate individuals between subpopulations by generating random numbers. The specific rules are as follows: If Calculation parameters And generate random numbers if Calculation parameters And generate random numbers If rand i (t) <rand best (t), then individual migration between populations will be carried out, otherwise individual migration between populations will not be carried out.

[0115] The migration rules between populations are as follows: When count(XA i )≥2×np×threshold, the last np×threshold individuals in subpopulation i are transferred to the subpopulation with the best individual of the current era; when count(XA i)>np×threshold, the latter count (XA i )-np×threshold individuals are transferred to the subpopulation with the best individual of the contemporary era; when count(XA i )≤np×threshold, in order to preserve the uniqueness of each sub-population, no migration between sub-populations is performed.

[0116] In order to verify the effect of the adaptive learning hyper-heuristic optimization algorithm (ALHOA) to be adopted in this embodiment to optimize the joint replenishment problem, two comparative schemes are designed: differential evolution algorithm (DE) and arithmetic optimization algorithm (AOA). Table 1 gives the basic data settings of the JRP model. The number of items is set to 30, 60 and 90 respectively. S is set to 0.5 times and 1.0 times the secondary ordering cost, that is, V=0.5 and V=1.0 respectively. Other data are randomly generated within the data range shown in Table 1. Table 2 shows the solution results of the three schemes for the JRP problem. It can be concluded from Table 2 that the average value, minimum and maximum values ​​of the total cost obtained by the ALHOA algorithm are better than those of the AOA and DE algorithms.

[0117] Table 1

[0118]

[0119] Table 2

[0120]

[0121]

[0122] The above-described embodiments are merely descriptions of the preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Without departing from the design spirit of the present invention, various modifications and improvements made to the technical solutions of the present invention by ordinary technicians in this field should all fall within the protection scope of the claims of the present invention.

Claims

1. An adaptive learning hyper-heuristic optimization method for joint replenishment optimization problem, characterized by: include: Step S1, obtaining the decision variables to be optimized in the joint replenishment problem JRP model, determining the hyperparameters and sub-algorithms of the adaptive learning hyper-heuristic optimization model, and generating initial sub-population individuals according to the decision variables to be optimized; wherein each sub-population corresponds to a sub-algorithm, and the sub-algorithm is any meta-heuristic algorithm; Step S2, calculating the fitness value of each individual in the sub-population and the number of updates of the sub-population, and updating the sub-population; Step S3: judging whether to communicate the best individuals between sub-populations according to the historical information of the sub-populations; Step S4, generate a random number and determine whether to migrate individuals between the updated sub-populations based on the random number, calculate the latest fitness value of the sub-population after individual migration, obtain the contemporary optimal individual, and thus generate the current optimal decision variable.

2. The adaptive learning hyper-heuristic optimization method for joint replenishment optimization problem according to claim 1, characterized in that: Calculating the fitness value of each individual in the subpopulation and the number of updates of the subpopulation includes: Among them, fitness(·) is the fitness function, U i is the optimal fitness value in subpopulation i, is the best individual in subpopulation i, V i is the update times of subpopulation i, floor(·) is the rounding function, M is the number of subpopulations, np is the number of individuals in the initial subpopulation, is the sum of the optimal fitness values ​​of all sub-populations, U i' is the optimal fitness value in subpopulation i', V M is the number of updates of subpopulation M.

3. The adaptive learning hyper-heuristic optimization method for joint replenishment optimization problem according to claim 2, characterized in that: Updating the subpopulation comprises: If count(XA i )≤V i , the first V in the subpopulation are sorted according to the sorting order of the subpopulation i. i Individuals are updated, if count(XA i )>V i , the individuals in the subpopulation are cyclically updated according to the sorting order of the subpopulation i until the calculated number of updates is reached; wherein, count(XA i ) is the population size of subpopulation i, count(·) is the function for calculating the number of individuals in the population, V i is the number of updates of subpopulation i.

4. The adaptive learning hyper-heuristic optimization method for joint replenishment optimization problem according to claim 1, characterized in that: The communication of the best individuals between subpopulations includes: Calculating the fitness value of each individual in the updated sub-population, re-ranking each individual in the sub-population according to the fitness value of each individual in the updated sub-population, and determining whether to communicate the best individuals between the sub-populations; Determining whether to communicate the best individuals between sub-populations includes: like , then the best individual in subpopulation i is replaced by the best individual in the population; represents the fitness value of the best individual in subpopulation i at the tth iteration, represents the fitness value of the historical best individual in subpopulation i, fitness(·) is the fitness function, is the best individual in subpopulation i at the tth iteration, is the historical best individual in subpopulation i.

5. The adaptive learning hyper-heuristic optimization method for joint replenishment optimization problem according to claim 1, characterized in that: Generating random numbers to determine whether to migrate individuals between subpopulations includes: Generating the random number comprises: like When , subpopulation i is the optimal subpopulation, its mean parameter is calculated, and the corresponding random number is generated; According to the mean parameter of the optimal subpopulation, the method for generating the corresponding random number is: Among them, x best (t) represents the best individual in all subpopulations at the tth iteration, mean best (t) represents the mean parameter of the normal distribution corresponding to the subpopulation with the best individual at the tth iteration, Max_t is the maximum number of iterations, and rand represents a random number between 0 and 1. Indicates the mean best (t) and a normal distribution with variance 1, rand best (t) represents the random number corresponding to the subpopulation with the best individual at the tth iteration; like When , subpopulation i is a non-optimal subpopulation, its mean parameter is calculated, and the corresponding random number is generated; According to the mean parameter of subpopulation i, the method of generating the corresponding random number is: Among them, mean i (t) is the mean parameter of the normal distribution corresponding to subpopulation i at the tth iteration, rand i (t) is the random number corresponding to subpopulation i at the tth iteration, and Max_t is the maximum number of iterations; The basis for judging whether to migrate individuals between subpopulations: If rand i (t) <rand best (t), individuals are migrated between subpopulation i and the optimal subpopulation; otherwise, individuals are not migrated between subpopulation i and the optimal subpopulation.

6. The adaptive learning hyper-heuristic optimization method for joint replenishment optimization problem according to claim 5, characterized in that: The individual migration between the subpopulation i and the optimal subpopulation includes: If count(XA i )≥2×np×threshold, the last np×threshold individuals in subpopulation i are transferred to the subpopulation with the best individual of the current era, where threshold is the threshold for retaining the uniqueness of the subpopulation; if count(XA i )>np×threshold, then the last count(XA i )-np×threshold individuals are transferred to the subpopulation with the best individual of the current era; if count(XA i )≤np×threshold, in order to preserve the uniqueness of subpopulation i, no individual migration is performed between the updated subpopulations.

7. The adaptive learning hyper-heuristic optimization method for joint replenishment optimization problem according to claim 1, characterized in that: Generating the optimal decision variables includes: Determine whether the preset number of update iterations has been reached. If the preset number of update iterations has not been reached, repeat steps S2-S4 until the preset number of update iterations has been reached. If the preset number of update iterations has been reached, generate the optimal decision variable according to the contemporary optimal individual. The optimal decision variable includes: the corresponding optimal k l and unit time T.

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