A multi-objective optimization method for nurse assignment phase in HHCSRP

Abstract

The application belongs to the field of home health care service route planning, and discloses a multi-objective optimization method for a nurse allocation stage in HHCSRP, which comprises the following steps: establishing a four-objective optimization model for the nurse allocation stage; proposing a multi-objective optimization algorithm D-TA2 based on diversity improvement; and proposing a TOPSIS decision method based on subjective and objective weights. The method has the following advantages: a four-objective model is established, which respectively minimizes the cost of a nursing institution, minimizes the income difference of nurses of different grades, minimizes the workload difference between nurses of the same grade, and maximizes the satisfaction of patients; an improved algorithm D-TA2 is proposed to solve the proposed model, which promotes the convergence and diversity of the population; redundancy analysis is added in the diversity archive, which promotes the diversity of the population; a TOPSIS decision method based on subjective and objective weights is proposed, which combines the practical significance of HHCSRP and provides an optimal allocation scheme in the nurse allocation stage.

Description

Technical Field

[0001] This invention belongs to the field of home care pathway planning and relates to a multi-objective optimization method for the nurse allocation stage in HHCSRP. Specifically, it relates to providing a solution for the home care pathway planning problem based on a multi-objective evolutionary algorithm. Background Technology

[0002] The purpose of Home Health Care Scheduling and Routing (HHCSRP) is to establish specific work arrangements for several caregivers to provide care services to families in need. Many elderly people with chronic illnesses or other physical limitations prefer to receive care and treatment at home, as this care and treatment often requires long-term commitment. Providing care and treatment at home respects the patient's wishes, offers convenience, and significantly reduces the burden on hospitals. With the aging population, home health care (HHC) has developed rapidly in developed countries. In developed countries such as the UK and France, an increasing number of care facilities offer such services, and the development of smartphone data has greatly facilitated home healthcare.

[0003] The primary task of HHC (Hospital Care Unit) is to assemble nursing staff within the institution, receive patient needs, allocate qualified nursing staff according to those needs, and plan the nursing staff's care sequence and travel arrangements. Research on HHCSRP (Hospital Care System Responsibility Planning) began relatively late both domestically and internationally, and most studies involve small sample sizes; however, it has gained considerable attention in recent years. The key focus of HHCSRP is how to rationally allocate healthcare personnel to care for patients according to specific requirements.

[0004] In traditional nursing pathway planning, the patient-nurse assignment stage is mostly done manually by experienced nurses. The matching principle is based on the skill and time availability of both parties, with less attention paid to the profitability of the nursing facility and the satisfaction of both parties. In large-sample situations, patients' nursing needs are divided into different levels, and nurses' nursing capabilities are also divided into different levels. Therefore, it is of great significance to be able to consider multiple aspects of satisfaction and use algorithms to quickly and accurately assign nurses to different patients.

[0005] When addressing the issue of nurses' nursing pathway allocation, the following four issues should be considered: (1) reducing the cost of nursing institutions when allocating personnel; (2) ensuring that the income difference between nurses of different levels is low; (3) balancing the workload among nurses of the same level to keep them at the same level; and (4) ensuring patient satisfaction when receiving nursing services.

[0006] In nursing pathway planning, the interests of nurses, patients, and nursing institutions are involved. Traditional allocation schemes often only consider a single demand and ignore other aspects. In the case of small samples, the drawbacks may not be apparent in the short term, but as the sample size increases, it will inevitably lead to more intense conflicts in the long run. Summary of the Invention

[0007] The purpose of this invention is to propose a novel optimization method for nurse allocation in HHCSRP (Hospital Health Care System Response Plan) from multiple perspectives, including nursing facility costs, nurse income disparities and workload balance, and patient satisfaction. This method aims to reduce nursing facility costs, balance income disparities among nurses of different skill levels and workload differences among nurses of the same skill level, and improve patient satisfaction by comprehensively considering various needs.

[0008] The technical solution adopted in this invention is as follows:

[0009] A multi-objective optimization method for the nurse allocation phase in HHCSRP, comprising the following steps:

[0010] (1) A four-objective optimization model for nurse allocation is established, including: after comprehensively considering the needs and interests of nurses, patients, and nursing institutions, constructing corresponding objective functions with the optimization objectives of minimizing the cost of nursing institutions, minimizing the income difference between nurses of different levels, minimizing the workload difference between nurses of the same level, and maximizing patient satisfaction, thus establishing a four-objective optimization model for nurse allocation; where the sets of nurses and patients of different levels are N respectively. i With P i The nursing time of nurse i for patient j is t. ij The income standards and number of nurses at different levels are respectively a i and n i 'g' represents different sets of levels, and 'b' represents the set of patients cared for by nurse i. ip ;

[0011] (1.1) Minimize the cost of nursing facilities

[0012] HHC nursing facilities are for-profit organizations, so cost is the primary consideration when allocating staff. The total cost is calculated using simple and direct indicators during staff allocation; the cost required for a level K nurse is [Cost]. k ,

[0013]

[0014] Where, N k This represents the set of nurses at grade k.

[0015] Total_Cost is the sum of the costs required for nurses of all levels, expressed as follows:

[0016]

[0017] (1.2) Minimize income disparities among nurses of different skill levels

[0018] Different levels of patients require different levels of nursing staff. Higher-level nurses can care for patients of the same or lower level, but lower-level nurses cannot care for patients with higher needs. Higher-level nurses generally receive higher salaries than lower-level nurses, which can easily lead to a large income gap between nurses of different levels. To avoid conflict between nurses of different levels, this objective function aims to minimize this income gap. The average income of all nurses is Ave_Cost.

[0019]

[0020] Where, n i This indicates the number of nurses at level i.

[0021] The income difference among all nurses is Dif_Cost.

[0022]

[0023] (1.3) Minimize the workload difference among nurses of the same level

[0024] Nurses of the same level care for patients of the same level and receive the same salary. Considering nurse satisfaction, it's desirable that nurses of the same level also have similar workloads; ensuring comparable income and preventing situations where some nurses are very busy while others are very idle is relatively fair for nurses of the same level. The total workload of a nurse of level k is Sum_Load. k ,

[0025]

[0026] The average workload of a nurse at level k is Ave_Load k ,

[0027]

[0028] The workload difference for each nurse i at level k is Dif_Load i ,

[0029]

[0030] (1.4) Maximize patient satisfaction

[0031] Patient satisfaction is a concern for both HHC nursing facilities and patients. Improving patient satisfaction leads to greater revenue for HHC nursing facilities, and patients enjoy higher quality services. While higher-level nurses caring for lower-level patients increases patient satisfaction, solely assigning higher-level nurses to these patients can create significant income and workload disparities among nurses. Therefore, this objective function contradicts the previous three. The significance of multi-objective optimization lies in achieving equilibrium by balancing multiple conflicting objectives. Patient satisfaction is defined as Fit. To facilitate calculation and minimize the values ​​of all four objective functions, the reciprocal of satisfaction is deliberately used as the final function value, as defined below:

[0032]

[0033] (2) A multi-objective optimization algorithm D-TA2 based on diversity improvement is proposed to solve the four-objective optimization model and obtain multiple different nurse allocation sets;

[0034] A multi-objective optimization algorithm, D-TA2, based on diversity improvement, is proposed. D-TA2 adds an index I based on translation density estimation to the convergence archive CA. D Compared with the original indicator I ε+ By introducing the Stochastic Bubble Sort (SRA) algorithm to dynamically balance the impact on individuals in the population, redundancy analysis is introduced into the diversity archive DA, and duplicate operations are performed on individuals with high overlap. Through improvements to CA and DA, the diversity and convergence of the population are improved.

[0035] (2.1) Update CA based on random bubble sort with two indicators

[0036] In the original I ε+ The indicator I, which promotes diversity, was introduced based on the CA indicator update. D Premature convergence leads to poor algorithm diversity. However, by introducing a metric to promote diversity, random bubble sort was used to dynamically balance the two metrics. The weight of the two metrics during sorting was adjusted according to the specific problem. Then, the entire population was sorted based on the two metrics, and the top N best individuals were selected for updating. This improved convergence while maintaining the diversity of CA, effectively enhancing the individual performance in CA. The specific process is as follows:

[0037] After the crossover and mutation breeding phase, the parent population of CA is N in size, and the offspring population is 2N in size. During the CA update process, the offspring and parent populations need to be merged to form a population of size 2N. The top N individuals are then selected from this population to form the output. At this point, individuals with good performance of size N from the offspring population need to be selected to merge with the parent population. Therefore, before merging, the fitness values ​​of the individuals in the offspring population of CA need to be calculated, and the top N individuals with smaller fitness values ​​are selected as the offspring population to merge with the parent population, forming a population POP of size 2N, which then enters the update phase.

[0038] First, the objective function value of the POP population is extracted to obtain obj. Then, the fitness value of the population is calculated to obtain the value of the first index, I1. ε+ This describes the minimum distance by which one solution dominates another. F(x) is the formula for calculating the fitness value, and I1 is the set of functions of F(x) that include all individuals and all objective functions. The specific calculation formula is as follows:

[0039] I ε+ (x1,x2)=min(f i (x1)-ε≤f i (x2), 1 ≤ i ≤ m)

[0040]

[0041] I1={F i (x1),F i (x2),...,F i (x N )},(i=1,2,...,m)

[0042] Where x1 and x2 represent two different individuals in the population POP, f(x) represents the calculation of the objective function, and m represents the number of objective functions.

[0043] Then select two individuals y and x whose index values ​​in obj are one before and one after the other in the same target, and calculate I based on the translation density estimate. D The index is obtained by taking the minimum value of each row of the distance matrix to form a column vector, resulting in the second index value I2. D The index can measure not only the convergence of a population but also its diversity. Its calculation formula is as follows:

[0044]

[0045] I2(x)=min(I D (x,y)),y∈obj

[0046] wherein, f i (x) and f i (y) respectively represent calculation results of individual x and individual y in the i-th objective function f(x).

[0047] Finally, after the two indicators are calculated, the process enters a random ranking stage, and the method adopted is the random bubble sort algorithm. First, the indicator value p c , which is the weight between the two indicators, is selected, and then the entire POP population is sorted: a random number p between (0,1) is generated, if p c < p, compare the I1 indicator of individuals in the population, place the individual with a larger indicator value at a front position, otherwise compare the I2 indicator of individuals in the population, and place the individual with a larger indicator value at a front position, so as to finally select the top N individuals in the ranking to complete the update;

[0048] (2.2) Update DA based on redundancy analysis

[0049] Using Pareto dominance to promote diversity may fail in high-dimensional space, and as the number of decision variables increases, the higher the probability that individuals in the population repeat. Based on this problem, a repeated analysis method is added to filter redundant solutions, and promote population diversity by deleting repeated solutions;

[0050] The main idea of redundancy analysis is to use the Manhattan distance between solutions in the decision space to estimate their degree of difference, so as to decide which repeated solutions in the objective space should be deleted; the dissimilarity of a solution is calculated by its minimum Manhattan distance to all other solutions in the decision space, and its value should be between 0 and 1; the Manhattan distance between two decision vectors actually reveals the degree of difference between them; first, a removal threshold δ is set according to the number of objective functions and the number of decision variables, and the dissimilarity of a solution is determined by calculating the minimum Manhattan distance between the solution and other solutions in the decision space; then these repeated solutions are divided into two groups according to their dissimilarity, one group is remote solutions whose dissimilarity is higher than or equal to the removal threshold δ, and the other group is clustered solutions whose dissimilarity is lower than the removal threshold δ; all remote solutions are retained, while only one clustered solution is retained, which requires randomly selecting one solution from all clustered solutions and removing all other solutions; therefore, the larger the removal threshold δ, the more likely repeated solutions are to be removed;

[0051] (3) A TOPSIS decision-making method based on subjective and objective weights is proposed, which selects an optimal allocation scheme from multiple allocation sets;

[0052] The main idea of ​​the TOPSIS decision-making method is as follows: First, determine the weights of each objective to obtain a weighted normalized decision set; then, obtain the ideal solution by analyzing the weighted normalized decision set; next, calculate the distance of all individuals in the weighted normalized decision set from the ideal solution; finally, rank the individuals in the decision set according to their preferences based on the calculated distances. The detailed steps are as follows:

[0053] (3.1) Determine subjective weights based on complementary judgment matrices

[0054] First, the decision-maker provides a supplementary judgment matrix based on subjective experience and wishes: x needs to be guaranteed. ij ∈[0.1,0.9] and i∈m,j∈m,x ij The meaning is the importance of the i-th objective relative to the j-th objective; the subjective weight of each objective is a vector w. s =(w s1 ,w s2 ,...,w sm The specific calculation formula is as follows:

[0055]

[0056]

[0057] (3.2) Determining objective weights based on the entropy weight method

[0058] The entropy weight method derives weights by analyzing sample information in the decision set, exhibiting strong objectivity. The smaller the decision weight of the objective, the smaller its impact on the final decision. The specific steps are as follows: First, use the objective value provided by the evolutionary algorithm as the decision set. If there are n individuals in the solution set, the decision set can be represented as:

[0059]

[0060] Use the following formula for normalization:

[0061]

[0062] The standardized decision set is V:

[0063]

[0064] The proportion of individual target values ​​in the decision set for each population is calculated using the following formula:

[0065]

[0066] According to the definition of entropy, the entropy value of the j-th target can be calculated by the following formula:

[0067]

[0068] Finally, the j-th target w o =(w o1 ,w o2 ,...,w om The target weight is calculated using the following formula:

[0069]

[0070] (3.3) Comprehensive calculation of subjective and objective weights

[0071] The obtained subjective weight w s =(w s1 ,w s2 ,...,w sm ) and objective weight w o =(w o1 ,w o2 ,...,w om The linear combination of objective and subjective weights for the j-th objective is expressed as follows:

[0072] w j =λw sj +(1-λ)w oj

[0073] Wherein, λ is used to adjust the proportion of subjective weight and objective weight respectively;

[0074] (3.4) Constructing a weighted decision matrix

[0075] Construct a weighted normalized decision matrix R = (r ij ) n×m The formula is as follows:

[0076] r ij =w j v ij

[0077] (3.5) Calculate the positive and negative ideal solutions

[0078] In a multi-objective optimization model, if the objective value is always as small as possible, then the positive and negative ideal solutions are represented as follows:

[0079] S + =[min(r1),min(r2),...,min(r)] m )]

[0080] S - =[max(r1),max(r2),...,max(r)] m )]

[0081] (3.6) Calculate the distance between the individual and the ideal solution.

[0082] Calculate R = (r) sequentially ij ) n×m The distance between an individual and the positive and negative ideal solutions:

[0083]

[0084]

[0085] (3.7) Calculate individual satisfaction

[0086] Calculate the satisfaction level of all individuals in the decision set sequentially:

[0087]

[0088] Individuals with higher satisfaction are better and rank higher. The Ideal Solution Similarity Ranking Decision Method (TOPSIS) based on subjective and objective weights mainly consists of three processes: subjective weight determination based on complementary judgment matrix, objective weight determination based on entropy weight method, and decision based on TOPSIS.

[0089] In this invention, determining subjective weights first requires the decision-maker to provide a complementary judgment matrix reflecting the importance of multiple objectives. Then, a novel method for ranking these complementary judgment matrices is used to determine the subjective weight of each objective. On the other hand, in multi-objective decision-making scenarios, some objectives exhibit less fluctuation in the decision set, and choosing any solution in the solution set may not significantly affect the performance of that objective. To fully utilize this objective information in the decision set, an entropy method is used to determine the objective weights. Finally, a combined weight is determined using both subjective and objective weights, and a decision is made based on TOPSIS.

[0090] This invention first initializes the levels and numbers of patients and nurses, and then assigns nurses to patients according to requirements. Therefore, each patient is a decision variable, and the set of patients constitutes a population. The assigned scheme is fed into the objective function, the objective function value is solved, and a descendant population is generated according to the requirements of the D-TA2 algorithm, and the process is iterated until the optimal population is selected in the final generation. Finally, using the TOPSIS decision method based on subjective and objective weights, an optimal assignment scheme is selected from the optimal population.

[0091] The beneficial effects of this invention are:

[0092] 1. This invention comprehensively considers the actual needs and interests of nurses, patients and nursing institutions, and establishes a four-objective model, namely, minimizing the cost of nursing institutions, minimizing the income difference between nurses of different levels, minimizing the workload difference between nurses of the same level, and maximizing patient satisfaction.

[0093] 2. This invention proposes an improved algorithm, D-TA2, to solve the proposed model. In order to balance the diversity and convergence of the algorithm, a translation density estimation index is introduced into the convergence archive (CA), which is combined with the original index to jointly promote the convergence and diversity of the population.

[0094] 3. This invention adds redundancy analysis to the diversity profile (DA) and performs deduplication during population iteration, which effectively improves the efficiency of the algorithm and promotes population diversity.

[0095] 4. This invention proposes a TOPSIS decision-making method based on subjective and objective weights. Combined with the practical significance of HHCSRP, the nurse allocation stage needs to provide an optimal allocation scheme at the end. The algorithm iteration will eventually form a set of allocation schemes, which contains multiple feasible solutions. Using this decision-making method, an optimal solution can be found in the solution set. Attached Figure Description

[0096] Figure 1 This is a flowchart of the present invention;

[0097] Figure 2 This is a flowchart of the multi-objective optimization algorithm D-TA2 based on diversity improvement of the present invention;

[0098] Figure 3 This is a flowchart of the TOPSIS decision-making method based on subjective and objective weights according to the present invention.

[0099] Figure 4 This is a bar chart showing the values ​​of the first objective function calculated for the optimal individuals obtained by the five multi-objective evolution algorithms Two_arch2, MOEA-D, PeEA, DWU, and PREA in this invention.

[0100] Figure 5 This is a bar chart showing the values ​​of the second objective function calculated for the optimal individuals obtained by the five multi-objective evolution algorithms Two_arch2, MOEA-D, PeEA, DWU, and PREA in this invention.

[0101] Figure 6 This is a bar chart showing the values ​​of the third objective function calculated for the optimal individuals obtained by the five multi-objective evolution algorithms Two_arch2, MOEA-D, PeEA, DWU, and PREA in this invention.

[0102] Figure 7 This is a bar chart showing the value of the fourth objective function calculated for the optimal individuals obtained by the five multi-objective evolution algorithms Two_arch2, MOEA-D, PeEA, DWU, and PREA in this invention. Detailed Implementation

[0103] The specific embodiments of the present invention will be further described below with reference to the accompanying drawings and technical solutions.

[0104] The method of this invention first constructs a four-objective optimization model, initializing a nurse set containing 80 nurses' IDs, maximum working hours, and levels, and a patient set containing 200 patients' IDs, required nursing hours, and levels. It adheres to the constraints that the total time required for a nurse to care for all patients does not exceed their maximum working hours and level restrictions. A nurse is assigned to each patient, and all patient assignment schemes constitute one solution in the population, generating 100 distinct solutions. This completes the initialization, producing a parent population of 100 individuals. The objective function value of each individual in the population is calculated, and offspring are generated through crossover and mutation. The offspring population and the parent population are merged into a new population, which then enters the subsequent iteration process of the D-TA2 algorithm. The CA is updated using a random bubble sort method based on dual indices, and the DA is updated using non-dominated sorting and redundancy analysis. The populations with updated CA and DA are merged into a new parent population, and the next iteration begins, until the maximum number of iterations is met, outputting the optimal population. Then, the TOPSIS decision method based on subjective and objective weights is used to select an optimal solution from the 100 solutions of the optimal population that satisfies all objective functions and constraints, thus completing the assignment task.

[0105] To verify the effectiveness of this invention, a comparative experiment was conducted with five multi-objective evolutionary algorithms—Two_arch2, MOEA-D, PeEA, DWU, and PREA—while maintaining the same initialization. The optimal population was obtained through the iterative process of each algorithm, and the TOPSIS decision-making method based on subjective and objective weights was uniformly applied to select the best individual from the optimal population. For ease of understanding and visualization, the values ​​of four objective functions were calculated for the best individual obtained by each algorithm, and the results are displayed using bar charts, such as... Figures 4 to 7 As shown, the optimal solution obtained by this invention is minimized in all three objective functions, except for one objective function where it is not dominant. Overall, the result obtained by this invention is the most ideal.

[0106] Figure 1 , Figure 2 , Figure 3 These are flowcharts for each step, among which... Figure 2 This is a flowchart of the D-TA2 multi-objective optimization algorithm based on diversity improvement. The specific steps of the process are as follows:

[0107] 1) Initialize the population;

[0108] 2) Calculate the objective function;

[0109] 3) Generate offspring using simulated binary crossover and polynomial mutation;

[0110] 4) Update the convergence profile (CA) based on the introduced translation density estimation index and the original Iε+ index;

[0111] 5) Update the diversity profile (DA) based on the improved redundancy analysis strategy;

[0112] 6) Repeat steps 3)-5) above until the termination condition is met;

[0113] 7) Output DA.

[0114] DA represents the optimal population, which includes multiple allocation schemes. After calculation using the TOPSIS decision-making method based on subjective and objective weights, an optimal individual is selected as the optimal allocation scheme under the current initial population conditions.

Claims

1. A multi-objective optimization method for the nurse allocation phase in HHCSRP, characterized in that, The method includes the following steps: (1) A four-objective optimization model for nurse allocation is established, including: after comprehensively considering the needs and interests of nurses, patients, and nursing institutions, constructing corresponding objective functions with the optimization objectives of minimizing the cost of nursing institutions, minimizing the income difference between nurses of different levels, minimizing the workload difference between nurses of the same level, and maximizing patient satisfaction, thus establishing a four-objective optimization model for nurse allocation; where the sets of nurses and patients of different levels are N respectively. i With P i The nursing time of nurse i for patient j is t. ij The income standards and number of nurses at different levels are respectively a i and n i 'g' represents different sets of levels, and 'b' represents the set of patients cared for by nurse i. ip ; (1.1) Minimize the cost of nursing facilities HHC nursing facilities are for-profit organizations, so cost is the primary consideration when allocating staff. The total cost is calculated using simple and direct indicators during staff allocation; the cost required for a level K nurse is [Cost]. k , Total_Cost is the sum of the costs required for nurses of all levels, expressed as follows: (1.2) Minimize income disparities among nurses of different skill levels Different patient needs require different levels of nursing staff. Higher-level nurses care for patients of the same or lower level, but lower-level nurses cannot care for patients with higher needs. Higher-level nurses generally receive higher salaries than lower-level nurses, which can easily lead to a large income gap between nurses of different levels. To avoid conflict between nurses of different levels, this objective function aims to minimize this income gap. The average income of all nurses is Ave_Cost. Where, n i Indicates the number of nurses at level i; The income difference among all nurses is Dif_Cost. (1.3) Minimize the workload difference among nurses of the same level Nurses of the same level care for patients of the same level and receive the same salary. Considering nurse satisfaction, it's desirable that nurses of the same level also have similar workloads; ensuring comparable income and preventing situations where some nurses are very busy while others are very idle is relatively fair for nurses of the same level. The total workload of a nurse of level k is Sum_Load. k , The average workload of a nurse at level k is Ave_Load k , The workload difference for each nurse i at level k is Dif_Load i , (1.4) Maximize patient satisfaction Patient satisfaction is a concern for both HHC nursing facilities and patients. Improving patient satisfaction leads to greater revenue for HHC nursing facilities, and patients enjoy higher quality services. While higher-level nurses caring for lower-level patients increases patient satisfaction, solely assigning higher-level nurses to these patients can create significant income and workload disparities among nurses. Therefore, this objective function contradicts the previous three. The significance of multi-objective optimization lies in achieving equilibrium by balancing multiple conflicting objectives. Patient satisfaction is defined as Fit. To facilitate calculation and minimize the values ​​of all four objective functions, the reciprocal of satisfaction is deliberately used as the final function value, as defined below: (2) A multi-objective optimization algorithm D-TA2 based on diversity improvement is proposed to solve the four-objective optimization model and obtain multiple different nurse allocation sets; A multi-objective optimization algorithm, D-TA2, based on diversity improvement, is proposed. D-TA2 adds an index I based on translation density estimation to the convergence archive CA. D Compared with the original indicator I ε+ By introducing a random bubble sort algorithm to dynamically balance the impact on individuals in the population, redundancy analysis is introduced into the diversity archive DA, and duplicate operations are performed on individuals with high overlap. Through improvements to CA and DA, the diversity and convergence of the population are improved. (2.1) Update CA based on random bubble sort with two indicators In the original I ε+ The indicator I, which promotes diversity, was introduced based on the CA indicator update. D Premature convergence leads to poor algorithm diversity. However, by introducing a metric to promote diversity, random bubble sort was used to dynamically balance the two metrics. The weight of the two metrics during sorting was adjusted according to the specific problem. Then, the entire population was sorted based on the two metrics, and the top N best individuals were selected for updating. This improved convergence while maintaining the diversity of CA, effectively enhancing the individual performance in CA. The specific process is as follows: After the crossover and mutation breeding phase, the parent population of CA is N in size, and the offspring population is 2N in size. During the CA update process, the offspring and parent populations need to be merged to form a population of size 2N. The top N individuals are then selected from this population to form the output. At this point, individuals with good performance of size N from the offspring population need to be selected to merge with the parent population. Therefore, before merging, the fitness values ​​of the individuals in the offspring population of CA need to be calculated, and the top N individuals with smaller fitness values ​​are selected as the offspring population to merge with the parent population, forming a population POP of size 2N, which then enters the update phase. First, the objective function value of the POP population is extracted to obtain obj. Then, the fitness value of the population is calculated to obtain the value of the first index, I1. ε+ This describes the minimum distance by which one solution dominates another. F(x) is the formula for calculating the fitness value, and I1 is the set of functions of F(x) that include all individuals and all objective functions. The specific calculation formula is as follows: I ε+ (x1,x2)=min(f i (x1)-ε≤f i (x2),1≤i≤m) I1={F i (x1),F i (x2),...,F i (x N )},(i=1,2,...,m) Where x1 and x2 represent two different individuals in the population POP, f(x) represents the calculation of the objective function, and m represents the number of objective functions; Then select two individuals y and x whose index values ​​in obj are one before and one after the other in the same target, and calculate I based on the translation density estimate. D The index is obtained by taking the minimum value of each row of the distance matrix to form a column vector, resulting in the second index value I2. D The index can measure not only the convergence of a population but also its diversity. Its calculation formula is as follows: I2(x)=min(I D (x,y)),y∈obj Among them, f i (x) and f i (y) represent the calculation results of individuals x and y in the i-th objective function f(x), respectively; Finally, after the calculation of the two indicators is completed, it enters the random sorting stage, and the method adopted is the random bubble sort algorithm. First, the indicator value p c , which is the weight between the two indicators, and then the entire POP population is sorted: generate a random number p between (0,1), if p c <p, compare the I1 indicator of individuals in the population, place the individual with a larger indicator value in the front, otherwise compare the I2 indicator of individuals in the population, and place the individual with a larger indicator value in the front, that is, finally select the top N individuals to complete the update; (2.2) Update DA based on redundancy analysis Using Pareto advantage to promote diversity may fail in high-dimensional spaces, and as the number of decision variables increases, the likelihood of individuals in the population being duplicated increases. Based on this problem, a duplication analysis method is added to filter redundant solutions and promote population diversity by removing duplicate solutions. The main idea of ​​redundancy analysis is to use the Manhattan distance between solutions in the decision space to estimate their degree of dissimilarity in order to determine which duplicate solutions in the objective space should be removed. The dissimilarity of a solution is calculated by the minimum Manhattan distance between it and all other solutions in the decision space, and its value should be between 0 and 1. The Manhattan distance between two decision vectors actually reveals the degree of dissimilarity between them. First, a removal threshold δ is set according to the number of objective functions and decision variables. The dissimilarity of a solution is determined by calculating the minimum Manhattan distance between the solution and other solutions in the decision space. Then, these duplicate solutions are divided into two groups according to their dissimilarity: one group consists of long-range solutions with dissimilarity greater than or equal to the removal threshold δ, and the other group consists of clustered solutions with dissimilarity less than the removal threshold δ. All long-range solutions are retained, while only one clustered solution is retained. A solution is randomly selected from all clustered solutions, and all other solutions are removed. Therefore, the larger the removal threshold δ, the more likely duplicate solutions are to be removed. (3) A TOPSIS decision-making method based on subjective and objective weights is proposed to select an optimal allocation scheme from multiple allocation sets; The main idea of ​​the TOPSIS decision-making method is as follows: First, determine the weights of each objective to obtain a weighted normalized decision set; then, obtain the ideal solution by analyzing the weighted normalized decision set; next, calculate the distance of all individuals in the weighted normalized decision set from the ideal solution; finally, rank the individuals in the decision set according to their preferences based on the calculated distances. The detailed steps are as follows: (3.1) Determine subjective weights based on complementary judgment matrices First, the decision-maker provides a supplementary judgment matrix based on subjective experience and wishes: x needs to be guaranteed. ij ∈[0.1,0.9] and i∈m,j∈m,x ij The meaning is the importance of the i-th objective relative to the j-th objective; the subjective weight of each objective is a vector w. s =(w s1 ,w s2 ,...,w sm The specific calculation formula is as follows: (3.2) Determining objective weights based on the entropy weight method The entropy weight method derives weights by analyzing sample information in the decision set, exhibiting strong objectivity. The smaller the decision weight of the objective, the smaller its impact on the final decision. The specific steps are as follows: First, use the objective value provided by the evolutionary algorithm as the decision set. If there are n individuals in the solution set, the decision set can be represented as: Use the following formula for normalization: The standardized decision set is V: The proportion of individual target values ​​in the decision set for each population is calculated using the following formula: According to the definition of entropy, the entropy value of the j-th target can be calculated by the following formula: Finally, the j-th target w o =(w o1 ,w o2 ,...,w om The target weight is calculated using the following formula: (3.3) Comprehensive calculation of subjective and objective weights The obtained subjective weight w s =(w s1 ,w s2 ,...,w sm ) and objective weight w o =(w o1 ,w o2 ,...,w om The linear combination of objective and subjective weights for the j-th objective is expressed as follows: In j =λw sj +(1-λ)in oj Wherein, λ is used to adjust the proportion of subjective weight and objective weight respectively; (3.4) Constructing a weighted decision matrix Construct a weighted normalized decision matrix R = (r ij ) n×m The formula is as follows: r ij =w j v ij (3.5) Calculate the positive and negative ideal solutions In a multi-objective optimization model, if the objective value is always as small as possible, then the positive and negative ideal solutions are represented as follows: S + =[min(r1),min(r2),...,min(r m )] S - =[max(r1),max(r2),...,max(r m )] (3.6) Calculate the distance between the individual and the ideal solution. Calculate R = (r) sequentially ij ) n×m The distance between an individual and the positive and negative ideal solutions: (3.7) Calculate individual satisfaction Calculate the satisfaction level of all individuals in the decision set sequentially: Individuals with higher satisfaction are better and rank higher. The ideal solution similarity ranking decision method based on subjective and objective weights mainly consists of three processes: subjective weight determination based on complementary judgment matrix, objective weight determination based on entropy weight method, and decision based on TOPSIS.

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