Output program, output method, and information processing device

Abstract

To support selection of an appropriate solution.SOLUTION: An output program causes a computer to execute acquisition process, fitting processing, calculation processing, and output processing. The acquisition process acquires a plurality of solutions on a Pareto front. The fitting processing fits a Bezier simplex with respect to the plurality of acquired solutions. The calculation processing calculates a gradient on the Bezier simplex at a plurality of data points on the fitted Bezier simplex. The output processing outputs evaluation information on robustness of a solution corresponding to the data point based on the calculated gradient, about each of the plurality of generated data points.SELECTED DRAWING: Figure 10

Description

Technical Field

[0001] Embodiments of the present invention relate to an output program, an output method, and an information processing apparatus.

Background Art

[0002] Conventionally, in the field of design such as aircraft and engines, design values may be selected based on solutions (design candidates) obtained by solving multi-objective optimization problems. A multi-objective optimization problem is a problem of simultaneously optimizing a plurality of objective functions, and there is not always a single optimal solution that gives an appropriate trade-off relationship for each objective function. Therefore, when selecting a design value from the solutions obtained by solving a multi-objective optimization problem, the goal is to obtain an optimal trade-off surface (Pareto front) obtained when a plurality of solutions are plotted in a multi-dimensional space.

Prior Art Documents

Non-Patent Documents

[0003]

Non-Patent Document 1

Summary of the Invention

Problems to be Solved by the Invention

[0004] However, when there are a large number of objective functions, data (solutions) is sparsely distributed due to the curse of dimensionality. When the solutions are sparsely distributed in this way, it is difficult to determine the robustness against value fluctuations due to noise or the like by simply plotting a plurality of solutions on the Pareto front. Therefore, there is a problem that it is difficult to select a solution that is robust against noise.

[0005] One aspect of this is the aim to provide an output program, output method, and information processing device that can assist in selecting an appropriate solution. [Means for solving the problem]

[0006] One proposed solution involves an output program that has the computer perform the following processes: data acquisition, fitting, generation, calculation, and output. The data acquisition process retrieves multiple solutions on the Pareto front. The fitting process fits a Bézier simplice to the acquired solutions. The calculation process calculates the gradient on the Bézier simplice at multiple data points on the fitted Bézier simplice. The output process outputs robustness evaluation information for each of the generated data points, based on the calculated gradient of the solution corresponding to that data point. [Effects of the Invention]

[0007] It can help in selecting the appropriate solution. [Brief explanation of the drawing]

[0008] [Figure 1] Figure 1 is an explanatory diagram illustrating the selection of design candidates (solutions) from the results of multi-objective optimization. [Figure 2] Figure 2 is an explanatory diagram illustrating the problems that arise when the Pareto front is not visualized. [Figure 3] Figure 3 is a block diagram showing an example of the functional configuration of an information processing device according to an embodiment. [Figure 4] Figure 4 is an explanatory diagram illustrating an example of input information. [Figure 5] Figure 5 is an explanatory diagram illustrating that the Pareto front is singular. [Figure 6] Figure 6 is an explanatory diagram illustrating an example of a single Bézier frost. [Figure 7] Figure 7 is an explanatory diagram illustrating the fitting of a single Bézier prism to a Pareto front. [Figure 8] Figure 8 is an explanatory diagram illustrating an example of fitting a single Bézier finial. [Figure 9] Figure 9 is an explanatory diagram illustrating an example of data point generation. [Figure 10] Figure 10 is an explanatory diagram illustrating an example of display output. [Figure 11] Figure 11 is an explanatory diagram illustrating an example of a visualization image. [Figure 12] Figure 12 is an explanatory diagram illustrating an example of a visualization image. [Figure 13] Figure 13 is a flowchart showing an example of the operation of the information processing device according to the embodiment. [Figure 14] Figure 14 is an explanatory diagram illustrating an example of a computer configuration. [Modes for carrying out the invention]

[0009] The output program, output method, and information processing device according to the embodiments will be described below with reference to the drawings. Components having the same function in the embodiments are denoted by the same reference numerals, and redundant descriptions are omitted. The output program, output method, and information processing device described in the following embodiments are merely examples and do not limit the embodiments. Furthermore, the following embodiments may be combined as appropriate within the limits of consistency.

[0010] Figure 1 is an explanatory diagram illustrating the selection of design candidates (solutions) from the results of multi-objective optimization. As shown in Figure 1, in the design field, the user selects a desired solution from among the solutions A1 and A2 (design candidates) on the Pareto front (dotted line in the figure) obtained by solving a multi-objective optimization problem and selects it as the design value. In the illustrated example, solutions A1 and A2 are obtained by solving optimization problems for two objective functions related to evaluation index (1) and evaluation index (2).

[0011] In selecting a solution, evaluation metrics (1) and (2) may contain noise, and users have requested that we select a solution that is robust to this noise.

[0012] For example, for Solution A1, when the value of Evaluation Index (2) deviates slightly, the value of Evaluation Index (1) deteriorates significantly. Therefore, it is not a solution that is very robust against noise. In contrast, for Solution A2, even if one evaluation index deviates slightly, the other evaluation index is not affected much. Therefore, it is a solution that is robust against noise.

[0013] FIG. 2 is an explanatory diagram for explaining the problems when the Pareto front cannot be visualized. As shown in FIG. 2, when there are a large number of objective functions, due to the curse of dimensionality, the data (solutions) are sparsely distributed. Therefore, in a diagram simply plotting and visualizing the solutions, it is difficult to understand how much the other evaluation index is affected by the deviation of one evaluation index, and it is difficult to evaluate the robustness of the solutions.

[0014] In the information processing apparatus according to the embodiment, a Bezier unit is fitted to a plurality of solutions on the Pareto front, and a plurality of data points are generated on the Bezier unit. Next, in the information processing apparatus according to the embodiment, the gradient on the Bezier unit at the generated plurality of data points is calculated, and for each of the generated plurality of data points, evaluation information on the robustness of the solution corresponding to the data point based on the calculated gradient is output. Thereby, in the information processing apparatus according to the embodiment, the robustness of the solution can be easily determined, and it becomes possible to support the selection of an appropriate solution.

[0015] FIG. 3 is a block diagram showing a functional configuration example of the information processing apparatus according to the embodiment. As shown in FIG. 3, the information processing apparatus 1 includes a communication unit 10, an input unit 20, a display unit 30, a storage unit 40, and a control unit 50. As this information processing apparatus 1, for example, a PC (Personal Computer) or the like can be applied.

[0016] The communication unit 10 receives various data from an external device via a network. The communication unit 10 is an example of a communication device. For example, the communication unit 10 may receive a part or all of the input information 41 described later from an external device.

[0017] The input unit 20 is an input device that inputs various types of information to the control unit 50 of the information processing device 1. The input unit 20 is compatible with keyboards, mice, touch panels, etc. For example, the input unit 20 receives some or all of the input information 41, which will be described later, in response to input operations from the user.

[0018] The display unit 30 is a display device that displays information output from the control unit 50. For example, the display unit 30 displays the processing results of the information processing device 1.

[0019] The memory unit 40 stores input information 41 and calculation information 42. The memory unit 40 supports semiconductor memory elements such as RAM (Random Access Memory) and flash memory, as well as storage devices such as HDD (Hard Disk Drive).

[0020] Input information 41 is information about input to the information processing device 1. For example, input information 41 may be the calculation result of a multi-objective optimization problem and may include multiple solutions on a Pareto front.

[0021] Figure 4 is an explanatory diagram illustrating an example of input information 41. As shown in Figure 4, the input information 41 includes multiple solutions 41a on a Pareto front that have solved the optimization problem of three objective functions (evaluation metrics) f1, f2, and f3.

[0022] Regarding evaluation metrics, for example, in a two-objective optimization problem concerning the wing shape of an aircraft, the metrics would include the aerodynamic drag coefficient and structural weight. In the case of a three-objective optimization problem concerning the environmental emissions of a diesel engine, the evaluation metrics would include NOx (nitrogen oxides), SFC (saturated fuel consumption rate), and Soot.

[0023] The calculation information 42 consists of various data obtained by the calculation processing of the information processing device 1. For example, the calculation information 42 includes results obtained by processing by the fitting unit 52, the data point generation unit 53, the gradient calculation unit 54, etc., based on the input information 41. Specifically, the calculation information 42 includes a single Bézier curve, multiple data points on a single Bézier curve, and gradient information indicating the gradient on a single Bézier curve (details will be described later).

[0024] The control unit 50 includes an acquisition unit 51, a fitting unit 52, a data point generation unit 53, a gradient calculation unit 54, and an output unit 55. The control unit 50 is implemented by hardwired logic such as a CPU (Central Processing Unit), GPU (Graphics Processing Unit), ASIC (Application Specific Integrated Circuit), or FPGA (Field Programmable Gate Array).

[0025] The acquisition unit 51 is a processing unit that acquires multiple solutions on a Pareto front that have been solved for a multi-objective optimization problem to be analyzed, which are input via the communication unit 10 or the input unit 20. The multiple solutions on the Pareto front are obtained by applying a multi-objective optimization algorithm, such as a genetic algorithm. The acquisition unit 51 stores the acquired information on the multiple solutions on the Pareto front (such as coordinate values) as input information 41 in the storage unit 40.

[0026] The fitting unit 52 is a processing unit that fits the Bézier curves to the multiple Bézier curves 42a on the Pareto front acquired by the acquisition unit 51. Here, a Bézier curve refers to a Bézier curve defined using multiple control points, generalized to a higher dimension.

[0027] Pareto fronts that appear in reality are often simplicous. Figure 5 is an explanatory diagram illustrating the simplicity of Pareto fronts.

[0028] As shown in Figure 5, the Pareto front can be represented as a simple curved "number of objective functions - 1" dimensional triangle. In this case, the solution to a problem that optimizes some of the objective functions consists of the vertices, edges, faces, etc., of the triangle. Such simple multi-objective optimization problems are known to frequently occur in various fields, such as the design of airplanes and diesel engines, facility layout problems, net exchange economies, and water cycle models.

[0029] Therefore, the fitting unit 52 can fit multiple solutions on the Pareto front by using a Bézier simplistic, taking into account the boundaries of the solutions corresponding to the vertices, edges, faces, etc., of the triangles mentioned above. Specifically, the fitting unit 52 performs fitting using an M-1 dimensional Bézier simplistic of degree D, defined by equation (1) below.

[0030]

number

[0031] In equation (1), b(t) corresponds to a Bézier simplicial (mapping). t is a parameter of an M-dimensional real vector. (D d) are multinomial coefficients. d p is a monomial of degree D (multiple exponents). d Δ is a control point of an M-dimensional real vector. M-1 represents an M-1 dimensional simplex. The number of control points in a Bézier simplex is determined by its order D and dimension M.

[0032] Figure 6 is an explanatory diagram illustrating an example of a Bézier uniform. As shown in Figure 6, the shaded Bézier uniform is defined by multiple control points (p). The number of these control points (p) is determined by the order (D) and dimension (M) of the Bézier uniform.

[0033] The illustrated example shows a Bézier simplice with D=3 and M=3. The subscripts of each control point p(i,j,k) (circles in Figure 6) are non-negative integers, satisfying i+j+k=D=3. In particular, p(3,0,0), p(0,3,0), and p(0,0,3) are control points corresponding to the vertices of the triangle representing the Bézier simplice. In fitting the Bézier simplice in the illustrated example, the estimation is made to match the solutions obtained by optimizing the first, second, and third objective functions, respectively.

[0034] The fitting unit 52 estimates the coordinates (vector values) of each control point, for example, using an inductive skeleton estimation method. The inductive skeleton estimation method is a technique that estimates the control points sequentially, starting from those that define the low-dimensional simple (skeleton), and the number of control points adjusted at one time does not depend on the number of objective functions. Therefore, the inductive skeleton estimation method can suppress the number of control points adjusted at one time, even when approximating a high-dimensional simple.

[0035] Figure 7 is an explanatory diagram illustrating the fitting of a single Bézier curve to a Pareto front. As shown in Figure 7, the fitting unit 52 first estimates the vertices (S1). Specifically, the fitting unit 52 estimates p(3,0,0) to match the optimized solution for the first objective function, p(0,3,0) for the second objective function, and p(0,0,3) for the third objective function.

[0036] Next, the fitting unit 52 estimates the edges (S2). Specifically, with the control points p(3,0,0), p(0,3,0), and p(0,0,3) that represent the vertices of the triangle fixed, the fitting unit 52 estimates the control points (dotted lines in S2) that determine the shape of the sides of the triangle.

[0037] Next, the fitting unit 52 estimates the surface (S3). Specifically, with the control points corresponding to the vertices and edges of the triangle fixed, the fitting unit 52 estimates the control points (dotted lines in S3) that define the shape of the triangular surface.

[0038] Figure 8 is an explanatory diagram illustrating an example of fitting a single Bézier curve. As shown in Figure 8, the fitting unit 52 fits the single Bézier curve 42a by estimating each control point (p) based on multiple solutions 41a on the Pareto front. Then, the fitting unit 52 stores information representing the Bézier curve 42a fitted by such estimation (such as the coordinate values ​​of multiple control points) in the calculation information 42.

[0039] The data point generation unit 53 is a processing unit that generates multiple data points on the Bézier curve 42a fitted by the fitting unit 52.

[0040] Figure 9 is an explanatory diagram illustrating an example of data point generation. As shown in Figure 9, the data point generation unit 53 generates multiple data points 42b in a grid pattern on a single Bézier curve 42a defined by multiple control points (p). The data point generation unit 53 stores information (e.g., coordinate values) of the generated data points 42b in the calculation information 42.

[0041] Here, the spacing of the grid-like data points 42b may be set in advance by the user. In this way, the information processing device 1 can complement sparse solutions by generating multiple data points 42b on a single Bézier curve 42a fitted to multiple solutions 41a on the Pareto front.

[0042] The gradient calculation unit 54 is a processing unit that calculates the gradient on the Bézier simplice 42a at multiple data points 42b generated by the data point generation unit 53. Specifically, the gradient calculation unit 54 calculates the gradient (length of the gradient vector) of the Bézier simplice 42a by calculating the total differential of the Bézier simplice 42a, and determines the gradient at each of the multiple data points 42b. The gradient calculation unit 54 stores the information indicating the gradient at each of the multiple data points 42b (length of the gradient vector) in the calculation information 42.

[0043] Here, the definition of a Bézier simplex 42a of degree D in M ​​dimensions is given by equation (2) below.

[0044]

number

[0045] Here, the parameter t m The gradient vector obtained by taking the partial derivative with respect to this is given by equation (3) below.

[0046]

number

[0047] Using these equations (2) and (3), the parameter t = (t1, t2, ..., t M The total differential of the simplex Bézier 42a in ) is calculated as shown in equation (4) below.

[0048]

number

[0049] The output unit 55 is a processing unit that outputs evaluation information on the robustness of the solution corresponding to each of the multiple data points 42b generated by the data point generation unit 53, based on the gradient calculated by the gradient calculation unit 54.

[0050] The magnitude of the gradient of the Bézier simplex 42a fitted to the Pareto front indicates its robustness against value fluctuations due to noise, etc. For example, data points 42b where the gradient of the Bézier simplex 42a is large will experience large value fluctuations due to noise, etc., and the nearby solution 41a will be less robust. Conversely, data points 42b where the gradient of the Bézier simplex 42a is small will experience small value fluctuations due to noise, etc., and the nearby solution 41a will be more robust.

[0051] For example, the output unit 55 lists and outputs, as evaluation information, the value (magnitude of the gradient) corresponding to the gradient (length of the gradient vector) on the Bézier simplex 42a for each of the multiple data points 42b. Alternatively, the output unit 55 may generate a visualization image plotting the multiple data points 42b in the objective function space, and display each of these multiple data points 42b from the display unit 30 in a display mode corresponding to the evaluation information.

[0052] Figure 10 is an explanatory diagram illustrating an example of the display output. As shown in Figure 10, the output unit 55 visualizes and displays the objective function space obtained by fitting a Bezier simplex 42a to a point (solution 41a) on the Pareto front.

[0053] Specifically, the output unit 55 generates a visualization image plotting the Bézier figure 42a and the 42b on the calculation information 42 in a two-dimensional or three-dimensional space corresponding to two or three objective functions selected from four or more objective functions (evaluation metrics). Then, the output unit 55 displays the multiple data points 42b generated on the Bézier figure 42a in a display mode corresponding to the magnitude of the gradient calculated by the gradient calculation unit 54.

[0054] For example, the output unit 55 displays data points 42ba, which are located in a region where the gradient is smaller than a predetermined threshold and is robust against noise (value fluctuations), with a dark gradient. Conversely, the output unit 55 displays data points 42bb, which are located in a region where the gradient is larger than a predetermined threshold and is not robust against noise (value fluctuations), with a light gradient.

[0055] This allows users to easily verify the robustness of solution 41a by observing how data points 42b located near solution 41a are displayed.

[0056] Figures 11 and 12 are explanatory diagrams illustrating an example of a visualization image. As shown in Figure 11, in the visualization image G1 of the objective function space of the three evaluation indices (f1, f2, f3), the degree of robustness against noise (value fluctuations) may be indicated by the intensity of the data points 42b. For example, the output unit 55 displays the data points 42b more intensely if the gradient is smaller than a predetermined threshold and the robustness is strong. Conversely, the output unit 55 displays the data points 42b less intensely if the gradient is larger than a predetermined threshold and the robustness is weak.

[0057] Furthermore, as shown in Figure 12, in the visualization image G2 in the objective function space of the three evaluation indices (f1, f2, f3), the size of the data points 42b may indicate whether or not the system is robust to noise (value fluctuations) (strength of robustness). For example, the output unit 55 displays the data points 42b larger if the gradient is smaller than a predetermined threshold and the robustness is strong. Conversely, the output unit 55 displays the data points 42b smaller if the gradient is larger than a predetermined threshold and the robustness is weak.

[0058] Next, the operation of the information processing device 1 will be described in detail. Figure 13 is a flowchart showing an example of the operation of the information processing device 1 according to the embodiment.

[0059] As shown in Figure 13, when processing begins, the acquisition unit 51 receives input of points (multiple solutions 41a) on the Pareto front that have been solved for the multi-objective optimization problem to be analyzed (S10).

[0060] Next, the fitting unit 52 fits the Bézier curve 42a to the points on the Pareto front acquired by the acquisition unit 51 (S11). Then, the data point generation unit 53 generates multiple data points 42b on the Bézier curve 42a in a grid pattern (S12).

[0061] Next, the gradient calculation unit 54 performs loop processing for each data point 42b on the Bézier uniform 42a generated by the data point generation unit 53 (S13-S15). Specifically, the gradient calculation unit 54 calculates the total differential of the Bézier uniform 42a and calculates the length of the gradient vector of the tangent plane at the data point 42b (S14).

[0062] Next, the output unit 55 receives the specification of evaluation indicators to be displayed from among a number of objective functions (evaluation indicators) via operation input from the input unit 20 (S16). In specifying these evaluation indicators, the output unit 55 accepts two specifications when visualizing in two dimensions, and three specifications when visualizing in three dimensions.

[0063] Next, the output unit 55 generates a visualization image plotting the Bézier figure 42a and the 42b on the calculation information 42 in a two-dimensional or three-dimensional space of the specified evaluation index. Then, the output unit 55 visualizes and displays the length of the gradient vector of each data point 42b on the Bézier figure 42a, i.e., the evaluation information (S17), and terminates the process.

[0064] As described above, the information processing device 1 obtains multiple solutions on the Pareto front. The information processing device 1 fits a Bézier simplex 42a to the multiple obtained solutions. The information processing device 1 generates multiple data points 42b on the fitted Bézier simplex 42a. The information processing device 1 calculates the gradient on the Bézier simplex 42a at the generated data points 42b. For each of the generated data points 42b, the information processing device 1 outputs evaluation information on the robustness of the solution corresponding to the data point 42b, based on the calculated gradient. This allows the user of the information processing device 1 to easily determine the robustness of the solution based on the output evaluation information. In this way, the information processing device 1 can assist the user in selecting an appropriate solution.

[0065] Furthermore, the information processing device 1 displays each of the multiple data points 42b in a display mode corresponding to the evaluation information in the visualization image G1 in which the multiple data points 42b are plotted. This allows the user of the information processing device 1 to easily confirm the robustness of the solution corresponding to the data points 42b based on the display mode of the data points 42b plotted in the visualization image G1.

[0066] Furthermore, the information processing device 1 displays and outputs the strength of the robustness of the solution corresponding to the evaluation information in the form of the intensity of the data points 42b or the size of the data points 42b. This allows the user of the information processing device 1 to easily confirm the robustness of the solution corresponding to the data points 42b based on the intensity of the data points 42b or the size of the data points 42b.

[0067] It should be noted that the components of each illustrated device do not necessarily have to be physically configured as shown. In other words, the specific forms of distribution and integration of each device are not limited to those shown, and all or part of them can be functionally or physically distributed and integrated in any unit according to various loads and usage conditions.

[0068] Furthermore, the various processing functions of the acquisition unit 51, fitting unit 52, data point generation unit 53, gradient calculation unit 54, and output unit 55 performed in the control unit 50 of the information processing device 1 may be executed in whole or in part on a CPU (or a microcomputer such as an MPU or MCU (Micro Controller Unit)). It goes without saying that the various processing functions may also be executed in whole or in part on a program analyzed and executed by a CPU (or a microcomputer such as an MPU or MCU), or on wired logic hardware. Additionally, the various processing functions performed in the information processing device 1 may be executed collaboratively by multiple computers through cloud computing.

[0069] Incidentally, the various processes described in the above embodiment can be realized by executing a pre-prepared program on a computer. Therefore, below we will describe an example of a computer configuration (hardware) that executes a program having the same functions as the above embodiment. Figure 14 is an explanatory diagram illustrating an example of a computer configuration.

[0070] As shown in Figure 14, the computer 200 includes a CPU 201 that performs various calculations, an input device 202 that accepts data input, a monitor 203, and a speaker 204. The computer 200 also includes a media reader 205 that reads programs and the like from a storage medium, an interface device 206 for connecting to various devices, and a communication device 207 for communicating with external devices via wired or wireless connections. The information processing device 1 includes a RAM 208 for temporarily storing various information and a hard disk drive 209. The various parts (201 to 209) within the computer 200 are connected to a bus 210.

[0071] The hard disk drive 209 stores a program 211 for executing various processes in the functional configuration described in the above embodiment (for example, the acquisition unit 51, the fitting unit 52, the data point generation unit 53, the gradient calculation unit 54, and the output unit 55). The hard disk drive 209 also stores various data 212 that the program 211 refers to. The input device 202 receives, for example, operation information from the operator. The monitor 203 displays, for example, various screens operated by the operator. The interface device 206 is connected to, for example, a printing device. The communication device 207 is connected to a communication network such as a LAN (Local Area Network) and exchanges various information with external devices via the communication network.

[0072] The CPU 201 reads the program 211 stored in the hard disk drive 209, loads it into the RAM 208, and executes it, thereby performing various processes related to the above-mentioned functional configuration (for example, the acquisition unit 51, the fitting unit 52, the data point generation unit 53, the gradient calculation unit 54, and the output unit 55). Note that the program 211 does not have to be stored in the hard disk drive 209. For example, the computer 200 may read and execute the program 211 stored in a storage medium that it can read. Examples of storage media that the computer 200 can read include portable recording media such as CD-ROMs, DVD discs, USB (Universal Serial Bus) memory, semiconductor memory such as flash memory, and hard disk drives. Alternatively, the program 211 may be stored in a device connected to a public network, the internet, a LAN, etc., and the computer 200 may read and execute the program 211 from there.

[0073] The following additional information is disclosed regarding the embodiments described above.

[0074] (Note 1) Obtain multiple solutions on the Pareto front, A single Bézier curve is fitted to the obtained multiple solutions. The gradient on the Bézier curve at multiple data points on the fitted Bézier curve is calculated. For each of the generated data points, the robustness evaluation information of the solution corresponding to the data point, based on the calculated gradient, is output. An output program characterized by having a computer perform the processing.

[0075] (Note 2) The output process involves displaying each of the multiple data points in the visualization image plotted from the multiple data points in a display manner corresponding to the evaluation information. The output program described in Appendix 1, characterized by the features described herein.

[0076] (Note 3) The output process displays the strength of the robustness of the solution corresponding to the evaluation information in the form of the intensity of the data points or the size of the data points. The output program described in Appendix 2, characterized by the features described herein.

[0077] (Note 4) Obtain multiple solutions on the Pareto front, A single Bézier curve is fitted to the obtained multiple solutions. The gradient on the Bézier curve at multiple data points on the fitted Bézier curve is calculated. For each of the generated data points, the robustness evaluation information of the solution corresponding to the data point, based on the calculated gradient, is output. An output method characterized by the processing being performed by a computer.

[0078] (Note 5) The output process involves displaying each of the multiple data points in the visualization image plotted from the multiple data points in a display manner corresponding to the evaluation information. The output method described in Appendix 4, characterized by the following:

[0079] (Note 6) The output process displays the strength of the robustness of the solution corresponding to the evaluation information in the form of the intensity of the data points or the size of the data points. The output method described in Appendix 5, characterized by the following:

[0080] (Note 7) Obtain multiple solutions on the Pareto front, A single Bézier curve is fitted to the obtained multiple solutions. The gradient on the Bézier curve at multiple data points on the fitted Bézier curve is calculated. For each of the generated data points, the robustness evaluation information of the solution corresponding to the data point, based on the calculated gradient, is output. An information processing apparatus characterized by including a control unit that performs processing.

[0081] (Note 8) The output process involves displaying each of the multiple data points in the visualization image plotted from the multiple data points in a display manner corresponding to the evaluation information. The information processing apparatus described in Appendix 7, characterized by the features described herein.

[0082] (Note 9) The output process displays the strength of the robustness of the solution corresponding to the evaluation information in the form of the intensity of the data points or the size of the data points. The information processing apparatus described in Appendix 8, characterized by the features described herein. [Explanation of Symbols]

[0083] 1…Information Processing Device 10… Communications Department 20...Input section 30...Display section 40...Storage section 41... Input Information 42...Calculation information 42a... Bezier alone 42b, 42ba, 42bb, 42bc, 42bd… data points 50…Control Unit 51…Acquisition part 52…Fitting section 53...Data point generation unit 54…Gradient Calculation Unit 55…Output section 200... Computer 201…CPU 202...Input device 203…Monitor 204...Speaker 205... Media reading device 206… Interface device 207...Communication equipment 208...RAM 209... Hard disk drive 210... Bus 211…Program 212... Various data 41a, A1, A2…Solution G1, G2...Visualization images p...control point

Claims

1. Obtain multiple solutions on the Pareto front, A single Bézier curve is fitted to the obtained multiple solutions. The gradient on the Bézier curve at multiple data points on the fitted Bézier curve is calculated. For each of the generated data points, the robustness evaluation information of the solution corresponding to the data point, based on the calculated gradient, is output. An output program characterized by having a computer perform the processing.

2. The output process involves displaying each of the multiple data points in a visualization image plotted from the multiple data points in a display manner corresponding to the evaluation information. The output program according to feature 1.

3. The output process displays the strength of the robustness of the solution corresponding to the evaluation information in the form of the intensity of the data points or the size of the data points. The output program according to feature 2.

4. Obtain multiple solutions on the Pareto front, A single Bézier curve is fitted to the obtained multiple solutions. The gradient on the Bézier curve at multiple data points on the fitted Bézier curve is calculated. For each of the generated data points, the robustness evaluation information of the solution corresponding to the data point, based on the calculated gradient, is output. An output method characterized by the processing being performed by a computer.

5. Obtain multiple solutions on the Pareto front, A single Bézier curve is fitted to the obtained multiple solutions. The gradient on the Bézier curve at multiple data points on the fitted Bézier curve is calculated. For each of the generated data points, the robustness evaluation information of the solution corresponding to the data point, based on the calculated gradient, is output. An information processing apparatus characterized by including a control unit that performs processing.

Images