Improved performance index-based preference evolution multi-objective optimization method and application

Abstract

The invention discloses a preference evolution multi-objective optimization method based on an improved performance index and application, and belongs to the technical field of multi-objective optimization. S2, preference construction is carried out; s3, generation of offspring; and S4, PIGA selection is carried out. The invention provides a new preference construction strategy based on coordinate transformation, which comprises the following steps of: transforming uniformly distributed reference points from a target space to a preference space by using a transition matrix to obtain preference reference points; an anti-generation distance index (PIGD +) based on preference improvement is designed, the distance between a candidate solution and a preference reference point is defined as a preference distance, the distance between a non-candidate solution and a non-preference reference point is defined as a penalty distance, and a preferred optimal solution is selected according to contribution of the solutions to the PIGD +. And moreover, complex practical engineering problems (RE22 and RE37) can be solved, and compared with some related algorithms, the method has superior practical performance.

Description

Technical Field

[0001] The present invention relates to the technical field of multi-objective optimization, and particularly relates to a preference evolution multi-objective optimization method and application based on improved performance indicators. Background Art

[0002] Multi-objective optimization is to help decision-makers determine an optimal solution that best meets the needs of the decision-makers while considering multiple objective criteria. The process of combining the preferences of decision-makers with multi-objective optimization problems and directly searching for solutions that meet the preference requirements is called preference multi-objective optimization. In multi-objective optimization problems, solving the complete optimal front of multi-objective optimization problems requires a high computational cost. In fact, decision-makers are only interested in some parts of the front, and the search process in other regions wastes computational resources. Moreover, the scale of the solution set describing the entire optimal front is extremely large, so it is quite difficult to perform decision analysis on such a solution set. To facilitate decision-making, considering adding decision-maker preference information to the search process in the form of prior or interactive, directly searching for the regions that decision-makers are interested in will make the search effect more ideal.

[0003] Since the entire Pareto front is not always consistent with the preferences of decision-makers, in order to help decision-makers find the optimal solution, a simple and intuitive idea is to add the preference information of decision-makers to the search process and use the preference information to guide the search for the region of interest (ROI) of decision-makers. The commonly used preference information includes expected level, weight information, trade-off information, etc. According to the position where decision-maker preference information is added, preference-based search methods can be divided into three types: prior decision-making method, posterior decision-making method, and interactive decision-making method. Currently, most preference algorithms perform posterior decision-making. This method first needs to find a Pareto approximate front with good diversity and convergence, and decision-makers need to carefully observe the solutions before making decisions. However, it is difficult to search the complete front in the objective space, and the scale of the solutions will increase exponentially with the increase of the objective space dimension, making posterior decision-making difficult. Therefore, prior decision-making and interactive decision-making are often used to directly or dynamically search for the regions that decision-makers are interested in, reducing the search difficulty and improving the search efficiency.

[0004] Utilizing the preference information of decision-makers is a key issue in preference-based evolutionary multi-objective optimization algorithms (PBEMOAs). Most traditional research in this field can be roughly divided into four main categories: The first category involves improving Pareto dominance or using fuzzy linguistic terms to express the preferences of decision-makers at different levels; the second category focuses on designing diversity management strategies, such as using preference-based reference vectors to guide the distribution of the Pareto front in the region of decision-makers' preferences; the third category combines traditional reference point-based methods with multi-objective optimization algorithms; the fourth category combines the preference information of decision-makers with performance metrics. Among the large number of performance metrics that have been proposed, they evaluate the approximation of the entire Pareto front in terms of convergence and diversity respectively, as well as metrics that evaluate both aspects simultaneously. For example, the Inverted Generational Distance (IGD) metric: The comprehensive performance of the algorithm is evaluated by calculating the distance from the true Pareto optimal front PF to the obtained approximate Pareto optimal solution set PF. The smaller the value of the Inverted Generational Distance, the better the comprehensive performance of the method, including convergence and distribution. However, only a small amount of research has developed performance metrics that incorporate the preferences of decision-makers and applied them to PBEMOAs.

[0005] So first, define IGD + The Inverted Generational Distance Plus (IGD+) metric is one of the performance metrics in multi-objective optimization. Similar to the IGD metric, it also measures the distance from the Pareto front to the obtained optimal solution set. However, the IGD + metric has been modified and optimized in the calculation process, focusing on its application in measuring the distance between non-preferred solutions and reference points. And it emphasizes integrating the preferences of the DM (decision-maker) into the IGD + metric. The purpose of the integration is to improve the selection of optimal solutions that are consistent with the specific requirements and preferences of the DM.

[0006]

[0007] where \(P\) represents the approximate non-dominated solutions, i.e., candidate solutions, and \(Z\) represents a set of reference points.

[0008]

[0009] where \(n\) is the dimension of the decision variables, \(z\) j represents the reference point, and \(s\) j represents the target point. The modified distance metric is different from the Euclidean distance. The smaller the value of the distance between the reference point \(z\) j and the region occupied by the target point \(s\) j is, the better the convergence and diversity of the approximate solutions.

[0010] In addition, current research has also proposed nine general attributes necessary for evaluating preference indicators, highlighting their importance in PBEMOAs. There are various methods to evaluate the performance of preference-based evolutionary algorithms. The nine general attributes (GPs) generally exhibited by preference indicators. Ideally, new performance indicators should possess most of the following desirable characteristics:

[0011] GP1: Evaluate the convergence of solutions within the approximate PF region consistent with the DM's preferences (local convergence).

[0012] GP2: Evaluate the diversity of solutions within the approximate PF region consistent with the DM's preferences (local diversity).

[0013] GP3: Evaluate the performance independent of the number of objective functions (scalability).

[0014] GP4: Evaluate the performance without knowing the PF.

[0015] GP5: Evaluate the performance by integrating preferences expressed through different methods.

[0016] GP6: Evaluate the performance economically and effectively.

[0017] GP7: Evaluate the performance independent of other interaction methods.

[0018] GP8: Evaluate the performance without introducing ambiguous parameters.

[0019] GP9: Evaluate the performance as a complete process rather than as independent steps.

[0020] Performance indicators have been proven effective in solving multi-objective and multi-objective optimization problems. In addition, several advanced PBEMOAs were introduced earlier, but there is limited research tailored for the performance indicators of PBEMOAs. The main challenge lies in how to construct the DM's preferences and integrate them into the performance indicator-based evolutionary multi-objective optimization algorithm to identify the optimal solution and thus solve practical engineering problems.

[0021] To this end, a preference evolutionary multi-objective optimization method based on improved performance indicators is proposed. Summary of the Invention

[0022] The technical problem to be solved by the present invention is: how to construct the DM's preferences and integrate them into the performance indicator-based evolutionary multi-objective optimization algorithm to identify the optimal solution and then apply the algorithm to practical engineering problems, and a preference evolutionary multi-objective optimization method based on improved performance indicators is provided.

[0023] The present invention solves the above technical problem through the following technical solutions. The present invention includes the following steps:

[0024] S1: Initialization

[0025] Randomly generate an initial population P of size N according to predefined parameters 0 , and generate a set of uniformly distributed reference points Z according to the direction angle;

[0026] S2: Preference construction

[0027] When the DM provides a preference point p and a preference angle γ, use coordinate transformation to construct the preference, and obtain the preference reference point Z using the transition matrix T + and the non - preference reference point Z - ;

[0028] S3: Offspring generation

[0029] Apply simulated binary crossover and polynomial mutation to create a new offspring population Q t , and combine the offspring population Q t and the parent population P t to form a new population R t ;

[0030] S4: PIGA selection

[0031] Divide the population into a preference sub - population and a non - preference sub - population, and perform hierarchical sorting on the combined population R based on the PIGD + indicator to find a set of preferred candidate solutions P that meet the decision - maker's preferences t . t+1 .

[0032] Furthermore, in the step S2, the specific processing process is as follows:

[0033] S21: Analyze the relationship between the preference point p and the ideal point Z * ;

[0034] S22: When the preference point p does not coincide with the ideal point Z * , that is, p≠Z * , then normalize the vector from the ideal point Z * to the preference point p to form a unique preference vector r, calculate the corresponding moderate preference basis vector e based on the preference vector r and the preference angle γ pi , and construct the transition matrix T using the moderate preference basis vector e pi ;

[0035] S23: When the preference point p coincides with the ideal point Z * , that is, p = Z * , then define the preference vector r as the moderate reference vector. At this time, the preference vector r is not unique, and calculate the moderate preference basis vector e piWith the extreme reference vector e ci The basis vector e of the corresponding preference region pij , and using the moderate preference basis vector e pi and the extreme preference basis vector Construct the transition matrix T;

[0036] S24: Apply the transition matrix T to the reference point Z for coordinate transformation to obtain the preference reference point Z + , the non-preference reference point Z - The reference point that remains outside the preference region before coordinate transformation.

[0037] Furthermore, in the step S22, the calculation formula of the moderate preference basis vector e pi is as follows:

[0038] e pi = e pi + k i (r - e ci ) = (1 - k i )e ci + k i r

[0039]

[0040] where i = 1, 2,..., m, m represents the dimension of the target space, k i is the proportionality factor between the vector (e pi - e ci ) and the vector (r - e ci ), and e ci is the extreme reference vector in the target space;

[0041] Using the moderate preference basis vector e pi Construct the transition matrix T as follows:

[0042] T = [e p1 e p2 …e pm .

[0043] Furthermore, in the step S23, the calculation formula of the moderate preference basis vector e pi is as follows:

[0044] e pi = e pi + k i (r - e ci ) = (1 - k i )e ci + k i r

[0045]

[0046] where \(i = 1, 2, \ldots, m\), and \(m\) represents the dimension of the target space, \(k\) i is the scaling factor between the vector \((e\) pi - e\) ci ) and the vector \((r - e\) ci ), and \(e\) ci is the extreme reference vector in the target space;

[0047] Extreme preference basis vector is calculated as follows:

[0048]

[0049] Using the moderate preference basis vector \(e\) pi and the extreme preference basis vector to construct the transition matrix \(T\) as follows:

[0050]

[0051] Furthermore, in the step S24, the coordinate transformation formula is as follows:

[0052] Z′ = T·Z

[0053] where \(Z′\) represents the preference reference point after coordinate transformation by applying the transition matrix \(T\), denoted as \(Z\) + .

[0054] Furthermore, in the step S4, PIGD + is an improved inverse generational distance - based metric, and its specific design process is as follows:

[0055] S401: Define the preference distance \(d\) + and the penalty distance \(d\) - , where the preference distance \(d\) + is used to quantify the IGD + metric between the preference sub - population and the preference reference point, and the penalty distance \(d\) - is used to quantify the IGD + metric between the non - preference sub - population and the non - preference reference point. The calculation formulas for the preference distance \(d\) + and the penalty distance \(d\) - are as follows respectively:

[0056] d + = IGD + (P + , Z + )

[0057] d - = IGD + (P -∪{H}, Z - )

[0058] Among them, H represents the lowest point of the population;

[0059] S402: Use the defined preference distance d + and the penalty distance d - to design the PIGD + The indicators are as follows:

[0060]

[0061] Among them, P represents the overall population, that is, the population R t , including multiple candidate solutions.

[0062] Furthermore, in the step S4, the specific process of population classification is as follows:

[0063] S411: Calculate the objective value FR of the population R t ; t ;

[0064] S412: For each candidate solution, determine the angle δ between its corresponding objective vector FR t,i and the preference point p i , where the angle δ i is used as a measure to evaluate the consistency between the candidate solution and the decision-maker's preference;

[0065] S413: The candidate solutions with δ i ≤γ are assigned to the preferred overall population R p , while the remaining candidate solutions are in the non-preferred region, denoted as the population R f , where R p is the preferred sub-population and R f is the non-preferred sub-population.

[0066] Furthermore, in the step S4, the process of obtaining the preferred candidate solution P t+1 is as follows:

[0067] S421: For each candidate solution in the population R t , calculate its normalized value, and its normalized value is equal to the PIGD t value of the population R + minus the PIGD + value when this solution is deleted;

[0068] S422: Each iteration process of calculating the normalized value will generate individuals with negative normalized values, denoted as P 1 , and a group of individuals with positive normalized values, denoted as P 2 ;

[0069] S423: Delete P generated in this iteration 1 from R p , update population R p , delete P generated in this iteration 2 from R f , update population R f , delete P generated in this iteration 1 and P 2 from R t , update population R t ;

[0070] S424: Divide R t into two populations Rank p and Rank f , where Rank p represents candidate solutions with negative contributions, and Rank f represents candidate solutions with positive contributions; then incorporate P 1 into population Rank p , incorporate P 2 into population Rank f , and update populations Rank p and Rank f ;

[0071] S425: When |Rank p | < N, retain the first N - |Rank p | candidate solutions in population Rank p as the preferred candidate solutions P t+1 for output. When |Rank p | > N, select the first N candidate solutions from Rank p as the preferred candidate solutions P t+1 for output. When |Rank p | = N, retain all candidate solutions in Rank p as the preferred candidate solutions P t+1 for output.

[0072] The present invention also provides an application of the above - mentioned preference - evolution multi - objective optimization method based on improved performance metrics. Applying the method to the four - bar truss design problem, minimize the mass of the truss and reduce the vertical displacement at the nodes.

[0073] The present invention also provides an application of the above - mentioned preference - evolution multi - objective optimization method based on improved performance metrics. Applying the method to the rocket injector design problem, minimize the fuel consumption and the wall heat flux.

[0074] The present invention has the following advantages compared with the prior art: The preference evolution multi-objective optimization method based on improved performance indicators proposes a new strategy for constructing preferences based on coordinate transformation. By using a transition matrix, uniformly distributed reference points are transformed from the objective space to the preference space to obtain preference reference points. A preference improvement-based inverted generational distance metric (PIGD + ) is designed. The distance between the candidate solution and the preference reference point is defined as the preference distance, and the distance from the non-candidate solution to the non-preference reference point is defined as the penalty distance. The preferred optimal solution is selected according to the contribution of the solution to PIGD + . This method not only obtains preference solutions theoretically, but also can solve complex practical engineering problems (RE22, RE37). Compared with some related algorithms, it has superior practical performance. BRIEF DESCRIPTION OF THE DRAWINGS

[0075] Figure 1 is a schematic flow chart of generating preference reference points by coordinate transformation in an embodiment of the present invention, where (a) is a schematic diagram of generating uniformly distributed reference points, (b) is a schematic diagram of determining the preference region, and (c) is a schematic diagram of coordinate transformation;

[0076] Figure 2 is a schematic diagram of the comprehensive transformation relationship between basis vectors in an embodiment of the present invention;

[0077] Figure 3 is a schematic diagram of the division of the preference region in an embodiment of the present invention;

[0078] Figure 4 is an example diagram of the preference candidate solution selection process based on the PIGD + metric in an embodiment of the present invention;

[0079] Figure 5 is an example diagram of the PIGA selection program in an embodiment of the present invention;

[0080] Figure 6 is a preference PF diagram obtained by various algorithms for the ZDT2 problem in an embodiment of the present invention, where (a) is the g-NSGA-II algorithm, (b) is the r-NSGA-II algorithm, (c) is the WV-MOEA-P algorithm, and (d) is the PIGA algorithm (the method of the present invention);

[0081] Figure 7 is a preference PF diagram obtained by various algorithms for the ZDT4 problem in an embodiment of the present invention, where (a) is the g-NSGA-II algorithm, (b) is the r-NSGA-II algorithm, (c) is the WV-MOEA-P algorithm, and (d) is the PIGA algorithm (the method of the present invention);

[0082] Figure 8It is the result graph of multiple algorithms in the DTLZ1 problem in the embodiments of the present invention, where (a) is the g-NSGA-II algorithm, (b) is the r-NSGA-II algorithm, (c) is the WV-MOEA-P algorithm, and (d) is the PIGA algorithm (the method of the present invention);

[0083] Figure 9 It is the result graph of multiple algorithms in the DTLZ4 problem in the embodiments of the present invention, where (a) is the g-NSGA-II algorithm, (b) is the r-NSGA-II algorithm, (c) is the WV-MOEA-P algorithm, and (d) is the PIGA algorithm (the method of the present invention);

[0084] Figure 10 It is the distribution graph of the solution sets of multiple algorithms for the RE22 problem in the embodiments of the present invention, where (a) is the g-NSGA-II algorithm, (b) is the r-NSGA-II algorithm, (c) is the WV-MOEA-P algorithm, and (d) is the PIGA algorithm (the method of the present invention);

[0085] Figure 11 It is the distribution graph of the solution sets of multiple algorithms for the RE37 problem in the embodiments of the present invention, where (a) is the g-NSGA-II algorithm, (b) is the r-NSGA-II algorithm, (c) is the WV-MOEA-P algorithm, and (d) is the PIGA algorithm (the method of the present invention);

[0086] Figure 12 It is the schematic flow diagram of the present invention. Detailed implementation manners

[0087] The embodiments of the present invention will be described in detail below. These embodiments are implemented on the premise of the technical solution of the present invention, and detailed implementation manners and specific operation processes are given. However, the protection scope of the present invention is not limited to the following embodiments.

[0088] Embodiment 1

[0089] In the prior art, the evolutionary algorithm based on the IGD + index effectively balances convergence and diversity and approximates the entire PF. However, we cannot determine whether the surviving candidate solutions satisfy the preferences of the DM. To solve this problem, first, the preference region is defined using the preference point p and preference angle γ specified by the DM. If the angle θ(p,z) between the reference point z and the preference point p is less than γ, then z is considered to be within the preference region. Otherwise, z is considered to be outside the preferred region. The definition of θ(p,z) is as follows:

[0090]

[0091] Among them, f j (z) and p jRespectively represent the j-th dimensional values of the reference point z and the preference point p, where j = 1, 2, …, m.

[0092] Therefore, the present invention proposes a preference evolution multi-objective optimization method (PIGA) based on an improved inverse generational distance metric: First, obtain the transition matrix from the preference information provided by the decision maker, and use the transition matrix to transform the uniformly distributed reference points from the objective space to the preference space to obtain the preference reference points; Second, rank the contributions of the solutions to PIGD + and perform PIGA selection; this process continues until a set of solutions that satisfy the decision maker's preferences is determined.

[0093] The specific steps of the method of the present invention are as follows:

[0094] S1: Initialization

[0095] According to the predefined parameters, randomly generate an initial population P of size N 0 ; In addition, generate a set of uniformly distributed reference points Z according to the direction angle.

[0096] S2: Preference construction

[0097] When the DM provides the preference point p and the preference angle γ, use coordinate transformation to construct the preference and obtain the preference reference point Z + and the non-preference reference point Z - .

[0098] S3: Generate offspring

[0099] Apply simulated binary crossover (SBX) and polynomial mutation (PM) to create a new offspring population Q t ; Combine the offspring population Q t and the parent population P t into a new population R t .

[0100] S4: PIGA selection

[0101] First, propose a population classification method to divide the population into a preference subpopulation and a non-preference subpopulation. Then, propose a selection procedure based on the PIGD + metric to rank the combined population R t and find a set of preferred candidate solutions P t+1 that satisfy the decision maker's preferences.

[0102] Table 1 Overall framework of the PIGA method

[0103]

[0104]

[0105] Example 2

[0106] In this embodiment, a further description is given of the coordinate transformation and the process of constructing preferences in step S2 of Embodiment 1:

[0107] 1. Coordinate transformation for preference construction

[0108] Based on the preference point p and the angle γ provided by the DM, we can construct preferences based on coordinate transformation. The reference points are uniformly distributed in the target space, and the preference angle γ can effectively map the reference points to the preference region, guiding the population to move in the PF direction preferred by the decision maker. Figure 1 is a schematic diagram of generating preference reference points through coordinate transformation. First, a set of uniformly distributed reference points {Z 1 , …, Z 10} is generated using the method based on the direction angle, as shown in (a) of Figure 1 . This method ensures uniform distribution by keeping the angle between adjacent points in the target space consistent. Then, as shown in (b) of Figure 1 , based on the preference point p and the preference angle γ, the transition matrix T is calculated and the basis vectors for determining the preference space are involved according to the preferences of the DM. Finally, as shown in (c) of Figure 1 , we project the reference points onto the preference region. {Z′ 1 , …, Z′ 10} is a set of preference points, while {Z 1 , Z 2 , Z 3 , Z 8 , Z 9 , Z 10} is a set of non-preference points. By utilizing the transition matrix T and the reference points and the preferences of the DM, we can generate preference reference points and identify non-preference reference points.

[0109] According to the relationship between the preference point p and the ideal point Z * , the transition matrix T is divided into two different cases:

[0110] When the preference point p does not coincide with the ideal point Z * , there is a unique set of preference basis vectors. Figure 2 gives the comprehensive transformation relationship between the basis vectors. The moderate preference basis vector e pi can be calculated by formulas (1) and (2) according to the preference vector r and the preference angle γ.

[0111] e pi = e pi + k i (r - e ci ) = (1 - k i )e ci + ki r (1)

[0112]

[0113] In formulas (1) and (2), i = 1, 2, …, m, where m represents the dimension of the target space. The vector (e pi -e ci ) and the vector (r - e ci ) are represented by the scale factor k i . Based on the above analysis, we can obtain the moderate preference basis vector e pi . At this time, the transition matrix T is shown in formula (3):

[0114] T = [e p1 e p2 … e pm (3)

[0115] When the preference point p coincides with the ideal point Z * , it indicates that the DM has certain prior knowledge. Two sets of preference reference vectors may be of interest: one is the moderate reference vector r with the same preference degree for each target (different from the preference vector r defined in the previous case), and the other is the extreme reference vector e ci that coincides with the coordinate axes. They can generate a total of m + 1 preference sub-regions. The division of the preference regions is as shown in Figure 3 .

[0116] The method for solving the moderate preference basis vector e pi is the same as in the previous case. The coordinates of the moderate reference vector r are shown in formula (4).

[0117] The basis vector e ci corresponding to the extreme reference vector e pij has coordinates as shown in formula (5), where i, j = 1, 2, …, m.

[0118]

[0119] The transition matrix T in this case is shown in formula (6):

[0120]

[0121] Finally, the reference point is projected onto the preference region using formula (7):

[0122] Z′ = T · Z (7)

[0123] where Z = [Z 1 , Z 2 , …, Z nrepresents the initial reference points of a uniform distribution, and Z′ represents the preference reference points after applying the coordinate transformation of the transition matrix T.

[0124] For clarity, the preference reference point Z + is defined based on the reference points generated after coordinate transformation, while the remaining initial reference points outside the preference region are defined as non-preference reference points Z - . Z + and Z - are crucial for formulating the PIGD + index.

[0125] 2. Steps to construct the DM preference

[0126] S1: Generate a set of uniformly distributed reference points Z based on the direction angle, and then analyze the relationship between the preference point p and the ideal point Z * therebetween.

[0127] S2: If p ≠ Z * , then normalize the vector from the ideal point Z * to the preference point p to form a unique preference vector r. Calculate the moderate preference basis vector e pi according to the preference vector r and the preference angle γ. Use this basis vector to construct the transition matrix T.

[0128] S3: If p = Z * , then define the preference vector r (at this time the preference vector r is not unique) as the moderate reference vector r, and the solution method of the moderate preference basis vector e pi is the same as the previous case, and then find the basis vector e ci corresponding to the preference region of the extreme reference vector e pij . In this case, the transition matrix T is obtained by combining these two sets of basis vectors.

[0129] S4: Apply the coordinate transformation relationship to the set of initial reference points Z to obtain the set of preference reference points Z + . The set of non-preference reference points Z - remains the set of reference points outside the preference region before transformation.

[0130] Table 2 Steps to construct the DM preference using coordinate transformation

[0131]

[0132]

[0133] Example 3

[0134] In this example, the definition and properties of the PIGD + index in step S4 of Example 1 are further illustrated:

[0135] Generate a set of preferred reference points Z using a coordinate transformation strategy + . At the same time, reference points located outside the preference region before this transformation are designated as the non-preferred reference point set Z - . Therefore, the overall population P can be divided into two categories: the sub-population P located within the preferred sub-region + and the sub-population P existing outside the preferred sub-region - . And formulate the PIGD + metric using the preferred reference points and non-preferred reference points

[0136] First, we define the preference distance d + , which quantifies the IGD + metric between the preferred sub-population and the preferred reference points. Similarly, we introduce the penalty distance d - , which measures the IGD + metric between the non-preferred sub-population and the non-preferred reference points. The calculation methods of d + and d - are as follows

[0137] d + = IGD + (P + , Z + ) (8)

[0138] d - = IGD + (P - ∪{H}, Z - ) (9)

[0139] In formula (9), H represents the lowest point of the population. The core concept of PIGD + is to decompose the IGD + metric into two different distance metrics according to the preferences of the DM. The preference distance d + reflects the relationship between the solutions that the DM is directly interested in and the preferred reference points. On the contrary, the penalty distance d - captures the relationship between the solutions outside the DM's interest and the non-preferred reference points. Using these two metrics, the PIGD + metric is designed as follows

[0140]

[0141] As shown in formula (10), the smaller d + , the closer the solutions within the preference range are to the PF. Use the component d - to evaluate whether all solutions belong to the preference region. If any solution is outside the preference range, then d - is less than the IGD+ (H, Z - ), which leads to the second term in formula (10) as a penalty function, increasing the overall PIGD + value.

[0142] To evaluate whether the PIGD + metric meets these properties, we examine its performance for each property, as shown in Table 3. The PIGD + metric effectively addresses the local convergence (GP1) and local diversity (GP2) problems. It can be extended to multi-objectives (GP3). However, as an index based on the improved inverted generational distance, the PIGD + requires knowledge of the PF and does not satisfy (GP4). In addition, the PIGD + can utilize various types of preference information (such as preference points, preference vectors, and preference angles) to handle different types of preferences (GP5). Due to its low computational cost, the PIGD + can effectively solve multi-objective problems (GP6). The PIGD + is independent of other preference methods, allowing it to evaluate different methods separately and make comparisons (GP7). However, it cannot directly evaluate interactive methods (GP9), which indicates potential areas for further improvement. The coverage of the PIGD + for GPs was compared with the metrics designed by some prior methods. Compared with some recently proposed preference metrics, we selected the R-metric, EH-metric, PMOD, and PMDA because they have the desirable properties mentioned in the performance metrics of interactive evolutionary multi-objective optimization algorithms.

[0143] Table 3 Performance of the PIGD + metric and existing preference metrics for each property

[0144]

[0145] Figure 4 represents the preference candidate solution selection process based on the PIGD + metric, where {z 1 , z 2 , z 3 , z 4} is a set of reference points, {s 1 , s 2 , s 3 , s 4} is a set of candidate solutions labeled as the overall P, p represents the preference point, and γ represents the preference angle. The PIGD + value of the overall P is calculated as PIGD + (P, Z) = 0.885. Excluding s 1After that, the result becomes PIGD + (P / s 1 , Z) = 0.674. Similarly, other excluded values are PIGD + (P / s 2 , Z) = 1.118, PIGD + (P / s 3 , Z) = 0.906, PIGD + (P / s 4 , Z) = 0.688. Since the calculation of PIGD + focuses on the optimal solution closest to the preference reference point Z + rather than the entire population. Therefore, the reduced overall P / s i has a PIGD + value greater than that of the complete population's PIGD + value. Thus, when the contribution of s i is negative, it indicates that s is close to the optimal solution of Z i . Conversely, a + positive value indicates that s i is close to the non - preference reference point Z - . As Figure 6 shown, the PIGD + contribution of each candidate solution is calculated as follows: We found that candidate solutions s 2 and s 3 with negative contributions are within the preference range, while s 1 and s 4 with positive contributions are outside the preference range. In summary, the PIGD + index not only evaluates the convergence and diversity of the population within the preference range but also effectively selects the preferred solution based on its contribution value.

[0146] Example 4

[0147] In this example, the PIGA selection process in step S4 of Example 1 is further illustrated:

[0148] The PIGA selection steps are as follows:

[0149] S1: Population classification

[0150] The initialization of PIGA classifies the population R t according to the objective value and its angular relationship with the preference point. First, calculate the objective value FR t of the population R t . For each candidate solution, the angle δ between its corresponding objective vector FR t,i and the preference point p​​​i is determined by the cosine similarity of the dot product normalized by their magnitudes. The angle δ i is used as a measure to evaluate the consistency between the candidate solution and the decision maker's preference. Solutions with δ i ≤γ are assigned to the preferred population R p , while the remaining candidate solutions are in the non-preferred region, denoted as population R f .

[0151] S2: Selection procedure based on PIGD + index

[0152] In the environmental selection stage, the PIGD + metric of the comprehensive preference distance d - and the penalty distance d + is used to strictly evaluate and refine the population. For each solution R t,k , Norm(k) is equal to the PIGD t value of R + minus the PIGD + value when this solution is deleted. Each iteration of calculating Norm(k) generates a set of individuals with negative Norm values, denoted as P 1 , and a set of individuals with positive Norm values, denoted as P 2 . Then the P 1 and P 2 individuals generated by the iteration are deleted from R t , and the population R t is updated. Finally, it is divided into two populations: Rank p represents the solutions with negative contributions, and Rank f represents the solutions with positive contributions. If |Rank p | < N (N is the size of the expected population), then the first N - |Rank p | solutions in Rank p are retained. On the contrary, if |Rank p | > N, then P p is formed by selecting the first N solutions from Rank t+1 . Additionally, when |Rank p | = N, then all the solutions in Rank p are retained.

[0153] Table 4 PIGA selection procedure

[0154]

[0155]

[0156]

[0157] Figure 5 Example of the selection procedure for PIGA. {S 1 ,S 2 ,…,S 10} represents the combined population R t , where p and γ are the preference point and preference angle of the DM, and H is the lowest point of the population. According to the preference reference point (solid dot, Z + ) and the non-preference reference point (hollow dot, Z - ), we divide the population into two parts: the preference sub-population and the non-preference sub-population. In this figure, {S 1 ,S 2 ,…,S 10} is the initialized population. S 1 ,S 6 and S 7 are denoted as R f , which are located outside the preference region, while the remaining candidate solutions are located in the preference region, denoted as R p . According to the definition of the PIGD + metric, the preference distance d + and the penalty distance d - are calculated. In the first selection process of the first-level candidate solutions, S 2 , S 3 , S 4 and S 5 are selected because their PIGD + contribution values are negative, while S 1 and S 6 are excluded because their PIGD + contribution values are positive. S 7 , S 8 , S 9 , S 10 enter the next-level population selection because their contribution values are 0. For the second-level candidate solutions, the contribution values of S8, S9, and S10 are negative PIGD+, and the one with the smallest contribution value, S8, is selected to supplement the next-generation population.

[0160] Example 5

[0161] In this example, the experimental results obtained on two widely used benchmark test suites, ZDT and DTLZ, as well as two real engineering problems, RE22 and RE37, are used to evaluate the performance of the PIGA algorithm (the method of the present invention) and some related algorithms.

[0162] (1) Experimental setup

[0163] The experiments were conducted using MATLAB R2020b on a desktop computer equipped with a 2.50-GHz Intel Core i7-11700 processor and 16 GB of RAM. All algorithms were implemented within the PlatEMO framework. The comparative experiments adopted the following parameter settings:

[0164] 1) The maximum number of iterations was set to maxFE = 1000.

[0165] 2) The population size was fixed at N = 100.

[0166] 3) The number of decision variables was denoted as D = 10.

[0167] 4) The crossover distribution index η c and the mutation distribution index η m were both set to 20, and the corresponding probabilities p c and p m were set to 1.0.

[0168] 5) For PIGA, the preference angle γ was configured as 15°.

[0169] 6) The critical value δ in r-NSGA-II was set to 0.1, while the preference range b in WV-MOEA-P was set to 0.05.

[0170] 7) In the ZDT test problems, the reference point was configured as (0.25, 0.25). For the DTLZ test problems, the reference point was configured as (0.25, 0.25, 0.25).

[0171] 8) In the RE22 test problems, the reference point was configured as (80, 110). For the RE37 test problems, the reference point was configured as (0.25, 0.25, 0.25).

[0172] (2) Performance metrics

[0173] There are many performance metrics in the field of EMOA, such as hyper-volume, R2, IGD, and IGD + metrics. For preference MOPs, we adopted three metrics: the preference-based hyper-volume index (PHI), the R-index based on the inverted generational distance (R-IGD), and the R-index based on the hyper-volume (R-HV) to evaluate the effectiveness of the proposed algorithms.

[0174] PHI metric: PHI combines the preference of the DM through a reference point and divides the hypervolume measurement region into positive and negative components. Its calculation method is as follows:

[0175]

[0176] where P represents the solution set, represents the ideal point, P ＞ represents domination of the solution set, z dy represents the non - ideal point (slightly greater than the lowest point), v ＜ represents the hypervolume between the solutions dominated by and z dy v ＞ represents the hypervolume between the solutions dominated by and z dy and.

[0177] R - metric: The R - metric is a systematic method for evaluating the performance of PBEMOAs by introducing a reference point. It pre - processes the obtained solution set according to the DM's preference. Specifically, the non - dominated solution set within the preference range is moved along the direction specified by the DM's preference, and then traditional metrics (such as IGD and HV) are used to evaluate the processed solution set. The R - metric includes three steps: pre - processing, sorting, and solution transfer.

[0178] (3) Experimental results

[0179] Experiment 1:

[0180] To verify the performance of the method proposed in the present invention, we conducted experiments using two benchmark test suites, ZDT and DTLZ. PIGA was compared with three related PBEMOAs, such as g - NSGA - II, r - NSGA - II, and WV - MOEA - P. Each algorithm was executed 31 times. Tables 5, 6, and 7 respectively give the statistical results of the PHI, R - IGD, and R - HV metrics. The symbol “+” indicates that, after statistical testing, the statistical performance of the compared algorithm is better than that of PIGA, “ - ” indicates that the statistical performance of the PIGA algorithm is better than the related method, and “ = ” indicates that there is no statistically significant difference between the results of PIGA and the results of the compared algorithm. These metrics comprehensively reflect the convergence and diversity of the preference - considered solution set. For each test instance, the best performance is highlighted with a gray background.

[0181] For the ZDT test instances, compared with some related algorithms in Tables 5, 6, and 7, PIGA achieved the best results on ZDT1, ZDT2, and ZDT4. Figure 6 、 7The preference PFs obtained by g-NSGA-II, r-NSGA-II, WV-MOEA-P, and PIGA on the ZDT2 and ZDT4 problems are given. The "five-pointed stars" in the figures are the preference points, and the "dots" are the solution sets obtained by each algorithm for this problem. As shown by these approximate preference PFs, g-NSGA-II converges the solution set to the region of interest of the DM. However, there are still some solutions outside the preference range, indicating that this algorithm generates irrelevant solutions outside the preference region. The solutions obtained by r-NSGA-II, on the other hand, are outside the DM's preference range, as shown in (b) of Figure 6 and (b) of Figure 7 . In addition, Figure 6 (c) of Figure 7 and (c) of

[0182] show that WV-MOEA-P has sufficient room for improvement on the ZDT test problems. Figure 8 and Figure 9 give the results of these algorithms on the DTLZ1 and DTLZ4 problems. The "five-pointed stars" in the figures are the preference points, and the "dots" are the solution sets obtained by each algorithm for this problem. From (a) of Figure 8 , (a) of Figure 9 , (b) of Figure 8 , and (b) of Figure 9 , it can be seen that both g-NSGA-II and r-NSGA-II show suboptimal performance when dealing with three-objective optimization problems. In contrast, WV-MOEA-P performs better, as shown in (c) of Figure 8 and (c) of Figure 9 , but it still produces solutions outside the preference range. Overall, PIGA is consistently superior to other algorithms on the ZDT and DTLZ test problems, generating well-distributed solutions within the preference range specified by the DM without generating redundant solutions outside this range.

[0183] Through qualitative and quantitative analysis of the comprehensive experimental results, PIGA shows superior performance in solving the ZDT and DTLZ test instances, especially in terms of convergence and diversity. Compared with some related algorithms, it is an effective and efficient algorithm when solving complex multi-objective optimization problems and is superior to classical PBEMOAs in most scenarios. In summary, the preference evolutionary multi-objective optimization algorithm based on the improved inverse generational distance metric is suitable for obtaining the preference PF.

[0184] Table 5 Statistical results of the PHI index of multiple algorithms

[0185]

[0186] Table 6 Statistical Results of R-IGD Index for Multiple Algorithms

[0187]

[0188] Table 7 Statistical Results of R-HV Index for Multiple Algorithms

[0189]

[0190] Experiment 2:

[0191] To evaluate the effectiveness of PIGA in practical applications, we conducted a comparative analysis for two real-world problems: the four-bar truss design (RE22) problem and the rocket injector design (RE37) problem. The focus of the RE22 problem is to optimize the structural efficiency and stability of the truss. The two main objectives are to minimize the mass of the truss and reduce the vertical displacement at the nodes. The decision variables are the cross-sectional areas x1, x2, x3, and x4 of the four bars, which directly affect the weight and stiffness of the structure. The RE37 problem aims to optimize the performance and durability of the rocket injector. The goal is to minimize fuel consumption and wall heat flux. The decision variables include the nozzle diameter (x1), the injector angle (x2), and other relevant dimensions (x3 to x7), which are crucial for efficient combustion and thermal management.

[0192] Figure 10 and Figure 11 The experimental results shown illustrate the superior performance of the PIGA algorithm in these engineering design challenges. The "pentagram" in the figure is the preference point, and the "dot" is the solution set obtained by each algorithm for this problem. Specifically, for the RE22 problem ( Figure 10 ), after 30 runs, the approximate preference solutions generated by PIGA converge more effectively to the true Pareto front, while g-NSGA-II and r-NSGA-II both produce some candidate solutions outside the preference range, and WV-MOEA-P fails to converge to the PF in the preference region. For the RE37 problem ( Figure 11 ), PIGA not only achieves a more uniform distribution but also ensures better coverage of the entire objective space. In contrast, the solution sets of g-NSGA-II and r-NSGA-II are unevenly distributed and have poor convergence, while WV-MOEA-P shows competitiveness. The comprehensive experimental results confirm that the PIGA algorithm not only obtains preference solutions theoretically but also can solve complex practical engineering problems and has superior practical performance compared with some related algorithms.

[0193] In summary, the main purpose of PBEMOAs is to find a preferred Pareto front with good convergence and distribution. For the preference-based multi-objective optimization problem, a preference-based evolutionary multi-objective optimization algorithm PIGA based on an improved inverted generational distance metric is proposed. A set of uniformly distributed reference points is generated using the direction angle method. First, coordinate transformation preference construction is used to divide the reference points into preference points and non-preference points. Based on these two components, the preference distance d + and the penalty distance d - are designed to design PIGD + . Therefore, the new metric is designed to guide the selection mechanism to ensure that the solutions are in good agreement with the DM's preferences. Individuals with a negative contribution to the PIGD + metric will be retained, while those with a positive contribution will be discarded. Comparative experiments of PIGA with related algorithms such as r-NSGA-II, g-NSGA-II, and WV-MOEA-P were conducted on ZDT, DTLZ, RE22, and RE37 test instances. The experimental results show that PIGA obtains a PF with uniform distribution and good coverage in the preference region. Future research will focus on incorporating interactivity into PIGA, allowing the reference points to be continuously improved based on the solution information obtained in each iteration.

[0194] Although the embodiments of the present invention have been shown and described above, it can be understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Those of ordinary skill in the art can make changes, modifications, substitutions, and variations to the above embodiments within the scope of the present invention.

Claims

1. A preference evolution multi-objective optimization method based on improved performance indicators, characterized in that: The following steps are involved: S1: Initialization According to the predefined parameters, an initialization population P0 of size N is randomly generated, and a set of uniformly distributed reference points Z is generated according to the direction angle; S2: Preference Construction When DM provides the preference point p and the preference angle γ, the preference is constructed using coordinate transformation and the preference reference point Z is obtained using the transition matrix T. + and the non-preferred reference point Z - ; S3: Offspring Generation Apply simulated binary crossover and polynomial mutation to create a new offspring population Q t , the offspring population Q t and the parent population P t Merge into a new population R t ; S4: PIGA Selection Divide the population into preferred subpopulations and non-preferred subpopulations based on PIGD + Index for the combined population R t Perform hierarchical sorting to find a set of preferred candidate solutions P that meet the decision maker's preferences t+1 .

2. According to claim 1, a preference evolution multi-objective optimization method based on improved performance indicators is characterized in that: In step S2, the specific processing process is as follows: S21: Analysis of preference point p and ideal point Z * the relationship between; S22: When the preferred point p and the ideal point Z * When they do not overlap, that is, p≠Z * When * The vector to the preference point p is normalized to form a unique preference vector r. Based on the preference vector r and the preference angle γ, the corresponding moderate preference basis vector e is calculated. pi , and use the moderate preference basis vector e pi Construct transition matrix T; S23: When the preferred point p and the ideal point Z * When they overlap, that is, p=Z * , the preference vector r is defined as the moderate reference vector. In this case, the preference vector r is not unique. The moderate preference basis vector e is calculated. pi With the extreme reference vector e ci The basis vector e of the corresponding preference region pij , and use the moderate preference basis vector e pi With the extreme preference basis vector e pij Construct transition matrix T; S24: Apply the transition matrix T to the reference point Z for coordinate transformation to obtain the preferred reference point Z + , non-preferred reference point Z - A reference point that remains outside the preferred region before coordinate transformation.

3. The preference evolution multi-objective optimization method based on improved performance index according to claim 2 is characterized in that: In step S22, the moderate preference basis vector e pi The calculation formula is as follows: e pi =e pi +k i (r-e ci )=(1-k i )e ci +k i r Where i = 1, 2, ..., m, m represents the dimension of the target space, k i is a vector (e pi -e ci ) and vector (re ci ), e ci is the extreme reference vector in the target space; Using the moderate preference basis vector e pi The transition matrix T is constructed as follows: T=[e p1 And p2 …And pm ]。 4. The preference evolution multi-objective optimization method based on improved performance index according to claim 2 is characterized in that: In step S23, the moderate preference basis vector e pi The calculation formula is as follows: e pi =e pi +k i (r-e ci )=(1-k i )e ci +k i r Where i = 1, 2, ..., m, m represents the dimension of the target space, k i is a vector (e pi -e ci ) and vector (re ci ), e ci is the extreme reference vector in the target space; Extreme preference basis vector e pij The calculation formula is as follows: Using the moderate preference basis vector e pi and the extreme preference basis vector e pij The transition matrix T is constructed as follows:

5. The preference evolution multi-objective optimization method based on improved performance index according to claim 2 is characterized in that: In step S24, the coordinate transformation formula is as follows: Z′=T·Z Among them, Z′ represents the preferred reference point after the coordinate transformation of the transition matrix T, denoted by Z + .

6. The preference evolution multi-objective optimization method based on improved performance index according to claim 1 is characterized in that: In step S4, PIGD + It is an improved anti-generation distance indicator, and its specific design process is as follows: S401: Define preference distance d + The penalty distance d - , where the preference distance d + Used to quantify the IGD between the preferred subpopulation and the preferred reference point + Indicator, penalty distance d - Used to quantify the IGD between a non-preferred subpopulation and a non-preferred reference point + Indicator, preference distance d + The penalty distance d - The calculation formulas are as follows: d + =IGD + (P + ,Z + ) d - =IGD + (P - {H},Z - ) Among them, H represents the lowest point of the population; S402: Using the defined preference distance d + The penalty distance d - Design PIGD + The indicators are as follows: Among them, P represents the total population, that is, the population R t , including multiple candidate solutions.

7. The method of preference evolution based on improved performance index according to claim 6 is characterized in that: In step S4, the specific process of population classification is as follows: S411: Calculate population R t The target value FR t ; S412: For each candidate solution, determine its corresponding target vector FR t,i The angle δ between the preferred point p i , where the angle δ i A measure used to assess the consistency of candidate solutions with the decision maker's preferences; S413: δ i Candidate solutions with ≤γ are assigned to the preferred population R p , and the remaining candidate solutions are located in the non-preferred region, represented by the population R f , where R p is the preferred subpopulation, R f is a non-preferred subpopulation.

8. The preference evolution multi-objective optimization method based on improved performance index according to claim 7 is characterized in that: In step S4, the first candidate solution P t+1 The acquisition process is as follows: S421: For population R t For each candidate solution in , calculate its normalized value, which is equal to the population R t PIGD + The value minus the PIGD when the solution is deleted + value; S422: Each iteration process of calculating the normalized value will generate an individual with a negative normalized value, denoted as P1, and a group of individuals with a positive normalized value, denoted as P2; S423: P1 generated in this iteration is converted from R p Delete and update the population R p , P2 generated by this iteration is taken from R f Delete and update the population R f , P1 and P2 generated in this iteration are taken from R t Delete and update the population R t ; S424: R t Divided into two populations Rank p ,Rank f ,Rank p Denotes a candidate solution with negative contribution, Rank f Represents a candidate solution with positive contribution; then P1 is incorporated into the population Rank p , merge P2 into the population Rank f , update population Rank p ,Rank f ; S425: When Rank p <N, then keep the population Rank p Top N-Rank p The candidate solution is the preferred candidate solution P t+1 Output, when Rank p > N, then by p Select the first N candidate solutions as the preferred candidate solution P t+1 Output, when Rank p = N, then keep Rank p All candidate solutions in are taken as the preferred candidate solutions P t+1 Output.

9. The application of the preference evolution multi-objective optimization method based on improved performance index according to any one of claims 1 to 8, characterized in that: The proposed method is applied to the four-bar truss design problem to minimize the mass of the truss and reduce the vertical displacements at the nodes.

10. The application of the preference evolution multi-objective optimization method based on improved performance index according to any one of claims 1 to 8, characterized in that: The method is applied to a rocket injector design problem, minimizing fuel consumption and wall heat flux.

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