Multi-Objective Optimization Algorithm for Grouping Decision Variables Based on Extreme Point Pareto Frontier

The existing algorithms for solving multi-objective optimization problems fall into three main categories:Decomposition-based,dominance-based,and indicator-based.Traditional multi-objective optimization problemsmainly focus on objectives,treating decision variables as a total variable to solve the problem without consideringthe critical role of decision variables in objective optimization.As seen,a variety of decision variable groupingalgorithms have been proposed.However,these algorithms are relatively broad for the changes of most decisionvariables in the evolution process and are time-consuming in the process of finding the Pareto frontier.To solvethese problems,a multi-objective optimization algorithm for grouping decision variables based on extreme pointPareto frontier(MOEA-DV/EPF)is proposed.This algorithm adopts a preprocessing rule to solve the Paretooptimal solution set of extreme points generated by simultaneous evolution in various target directions,obtainsthe basic Pareto front surface to determine the convergence effect,and analyzes the convergence and distributioneffects of decision variables.In the later stages of algorithm optimization,different mutation strategies are adoptedaccording to the nature of the decision variables to speed up the rate of evolution to obtain excellent individuals,thusenhancing the performance of the algorithm.Evaluation validation of the test functions shows that this algorithmcan solve the multi-objective optimization problem more efficiently.

Large-Scale Multi-Objective Optimization Algorithm Based on Weighted Overlapping Grouping of Decision Variables

The large-scale multi-objective optimization algorithm(LSMOA),based on the grouping of decision variables,is an advanced method for handling high-dimensional decision variables.However,in practical problems,the interaction among decision variables is intricate,leading to large group sizes and suboptimal optimization effects;hence a large-scale multi-objective optimization algorithm based on weighted overlapping grouping of decision variables(MOEAWOD)is proposed in this paper.Initially,the decision variables are perturbed and categorized into convergence and diversity variables;subsequently,the convergence variables are subdivided into groups based on the interactions among different decision variables.If the size of a group surpasses the set threshold,that group undergoes a process of weighting and overlapping grouping.Specifically,the interaction strength is evaluated based on the interaction frequency and number of objectives among various decision variables.The decision variable with the highest interaction in the group is identified and disregarded,and the remaining variables are then reclassified into subgroups.Finally,the decision variable with the strongest interaction is added to each subgroup.MOEAWOD minimizes the interactivity between different groups and maximizes the interactivity of decision variables within groups,which contributed to the optimized direction of convergence and diversity exploration with different groups.MOEAWOD was subjected to testing on 18 benchmark large-scale optimization problems,and the experimental results demonstrate the effectiveness of our methods.Compared with the other algorithms,our method is still at an advantage.

A vague-set-based fuzzy multi-objective decision making model for bidding purchase

A vague-set-based fuzzy multi-objective decision making model is developed for evaluating bidding plans in a bid- ding purchase process. A group of decision-makers (DMs) first independently assess bidding plans according to their experience and preferences, and these assessments may be expressed as linguistic terms, which are then converted to fuzzy numbers. The resulting decision matrices are then transformed to objective membership grade matrices. The lower bound of satisfaction and upper bound of dissatisfaction are used to determine each bidding plan’s supporting, opposing, and neutral objective sets, which together determine the vague value of a bidding plan. Finally, a score function is employed to rank all bidding plans. A new score function based on vague sets is introduced in the model and a novel method is presented for calculating the lower bound of sat- isfaction and upper bound of dissatisfaction. In a vague-set-based fuzzy multi-objective decision making model, different valua- tions for upper and lower bounds of satisfaction usually lead to distinct ranking results. Therefore, it is crucial to effectively contain DMs’ arbitrariness and subjectivity when these values are determined.

Fuzzy Multi-Objective Decision Model of Supplier Selection with Preference Information

Supplier selection is a multi-objective decision problem, which must be considered many objectives, some objectives are qualitative, and others are quantitative. Meanwhile, manufacturer has preference for different suppliers. In this paper, a new multi-objective decision model with preference information of supplier is established. A practical example of supplier selection problem utilizing this model is studied. The result demonstrates the feasibility and effectiveness of the methods proposed in the paper.

Improved Fuzzification Method for Multi-Objective Decision-Making and Its Application in Evaluation of Highway Planning

A new fuzzification method for multi-objective decision-making and selective sorting is proposed on the basis of the fuzzy consistent relation, and the specific algorithm is presented. The method is applied to the evaluation of highway planning of Zhanjiang city. To decrease the subjectivity in the process of decision-making, the LOWA operator is introduced, and a discussion on how to select appropriate weights involved in multi-objective sorting is made. It is concluded that it is feasible to apply the fuzzy consistent relation to multi-objective decision-making analysis, and the improved fuzzication method is workable.

Attribute Reduction of Hybrid Decision Information Systems Based on Fuzzy Conditional Information Entropy

The presence of numerous uncertainties in hybrid decision information systems(HDISs)renders attribute reduction a formidable task.Currently available attribute reduction algorithms,including those based on Pawlak attribute importance,Skowron discernibility matrix,and information entropy,struggle to effectively manages multiple uncertainties simultaneously in HDISs like the precise measurement of disparities between nominal attribute values,and attributes with fuzzy boundaries and abnormal values.In order to address the aforementioned issues,this paper delves into the study of attribute reduction withinHDISs.First of all,a novel metric based on the decision attribute is introduced to solve the problem of accurately measuring the differences between nominal attribute values.The newly introduced distance metric has been christened the supervised distance that can effectively quantify the differences between the nominal attribute values.Then,based on the newly developed metric,a novel fuzzy relationship is defined from the perspective of“feedback on parity of attribute values to attribute sets”.This new fuzzy relationship serves as a valuable tool in addressing the challenges posed by abnormal attribute values.Furthermore,leveraging the newly introduced fuzzy relationship,the fuzzy conditional information entropy is defined as a solution to the challenges posed by fuzzy attributes.It effectively quantifies the uncertainty associated with fuzzy attribute values,thereby providing a robust framework for handling fuzzy information in hybrid information systems.Finally,an algorithm for attribute reduction utilizing the fuzzy conditional information entropy is presented.The experimental results on 12 datasets show that the average reduction rate of our algorithm reaches 84.04%,and the classification accuracy is improved by 3.91%compared to the original dataset,and by an average of 11.25%compared to the other 9 state-of-the-art reduction algorithms.The comprehensive analysis of these research results clearly indicates that our algorithm is highly effective in managing the intricate uncertainties inherent in hybrid data.

Method for triangular fuzzy multiple attribute decision making based on two-dimensional density operator method

Aiming at the triangular fuzzy(TF)multi-attribute decision making(MADM)problem with a preference for the distribution density of attribute(DDA),a decision making method with TF number two-dimensional density(TFTD)operator is proposed based on the density operator theory for the decision maker(DM).Firstly,a simple TF vector clustering method is proposed,which considers the feature of TF number and the geometric distance of vectors.Secondly,the least deviation sum of squares method is used in the program model to obtain the density weight vector.Then,two TFTD operators are defined,and the MADM method based on the TFTD operator is proposed.Finally,a numerical example is given to illustrate the superiority of this method,which can not only solve the TF MADM problem with a preference for the DDA but also help the DM make an overall comparison.

A decision support system for satellite layout integrating multi-objective optimization and multi-attribute decision making

A decision support system, including a multi-objective optimization framework and a multi-attribute decision making approach is proposed for satellite equipment layout. Firstly, given three objectives (to minimize the C.G. offset, the cross moments of inertia and the space debris impact risk), we develop a threedimensional layout optimization model. Unlike most of the previous works just focusing on mass characteristics of the system, a space debris impact risk index is developed. Secondly, we develop an efficient optimization framework for the integration of computer-aided design (CAD) software as well as the optimization algorithm to obtain the Pareto front of the layout optimization problem. Thirdly, after obtaining the candidate solutions, we present a multi-attribute decision making approach, which integrates the smart Pareto filter and the correlation coefficient and standard deviation (CCSD) method to select the best tradeoff solutions on the optimal Pareto fronts. Finally, the framework and the decision making approach are applied to a case study of a satellite platform.

Multiple attribute decision making method based on trapezoid fuzzy linguistic variables

The problem of multiple attribute decision making under fuzzy linguistic environments, in which decision makers can only provide their preferences （attribute values）in the form of trapezoid fuzzy linguistic variables（TFLV）, is studied. The formula of the degree of possibility between two TFLVs is defined, and some of its characteristics are studied. Based on the degree of possibility of fuzzy linguistic variables, an approach to ranking the decision alternatives in multiple attribute decision making with TFLV is developed. The trapezoid fuzzy linguistic weighted averaging （TFLWA） operator method is utilized to aggregate the decision information, and then all the alternatives are ranked by comparing the degree of possibility of TFLV. The method can carry out linguistic computation processes easily without loss of linguistic information, and thus makes the decision results reasonable and effective. Finally, the implementation process of the proposed method is illustrated and analyzed by a practical example.

Land Evaluation Method Based on Decision Tree Produced by C4.5 and Fuzzy Decision

[Objective]The aim was to overcome the shortage of being difficult to build land evaluation model when the impact factors had continuous value in the traditional land evaluation process,as well as to improve the intelligibility of the land evaluation knowledge.[Method] The land evaluation method combining classification rule extracted by C4.5 algorithm with fuzzy decision was proposed in this study.[Result] The result of Second General Soil Survey of Guangdong Province had demonstrated that the method was convenient to extract classification rules,and by using only 100 rules,quantity correct rate 86.67% and area correct rate 84.80% of land evaluation could be obtained.[Conclusions] The use of C4.5 algorithm to obtain the rules,combined with fuzzy decision algorithm to build classifiers had got satisfactory results,which provided a practical algorithm for the land evaluation.

Aggregation operators on intuitionistic trapezoidal fuzzy number and its application to multi-criteria decision making problems

Intuitionistic trapezoidal fuzzy numbers and their operational laws are defined. Based on these operational laws, some aggregation operators, including intuitionistic trapezoidal fuzzy weighted arithmetic averaging operator and weighted geometric averaging operator are proposed. Expected values, score function, and accuracy function of intuitionitsic trapezoidal fuzzy numbers are defined. Based on these, a kind of intuitionistic trapezoidal fuzzy multi-criteria decision making method is proposed. By using these aggregation operators, criteria values are aggregated and integrated intuitionistic trapezoidal fuzzy numbers of alternatives are attained. By comparing score function and accuracy function values of integrated fuzzy numbers, a ranking of the whole alternative set can be attained. An example is given to show the feasibility and availability of the method.

Fuzzy data envelopment analysis approach based on sample decision making units

The conventional data envelopment analysis （DEA） measures the relative efficiencies of a set of decision making units with exact values of inputs and outputs. In real-world prob- lems, however, inputs and outputs typically have some levels of fuzziness. To analyze a decision making unit （DMU） with fuzzy input/output data, previous studies provided the fuzzy DEA model and proposed an associated evaluating approach. Nonetheless, numerous deficiencies must still be improved, including the α- cut approaches, types of fuzzy numbers, and ranking techniques. Moreover, a fuzzy sample DMU still cannot be evaluated for the Fuzzy DEA model. Therefore, this paper proposes a fuzzy DEA model based on sample decision making unit （FSDEA）. Five eval- uation approaches and the related algorithm and ranking methods are provided to test the fuzzy sample DMU of the FSDEA model. A numerical experiment is used to demonstrate and compare the results with those obtained using alternative approaches.

Extension of TOPSIS for fuzzy multi-attribute decision making problem based on experimental analysis

This paper is concerned with a technique for order performance by similarity to ideal solution（TOPSIS） method for fuzzy multi-attribute decision making,in which the information about attribute weights is partly known and the attribute values take form of triangular fuzzy numbers.Considering the fact that the triangular fuzzy TOPSIS results yielded by different distance measures are different from others,a comparative analysis of triangular fuzzy TOPSIS ranking from each distance measure is illustrated with discussion on standard deviation.By applying the most reasonable distance,the deviation degrees between attribute values are measured.A linear programming model based on the maximal deviation of weighted attribute values is established to obtain the attribute weights.Therefore,alternatives are ranked by using TOPSIS method.Finally,a numerical example is given to show the feasibility and effectiveness of the method.

Novel combinatorial algorithm for the problems of fuzzy grey multi-attribute group decision making

To study the fuzzy and grey information in the problems of multi-attribute group decision making, the basic concepts of both fuzzy grey numbers and grey interval numbers are given firstly, then a new model of fuzzy grey multi-attribute group decision making based on the theories of fuzzy mathematics and grey system is presented. Furthermore, the grey interval relative degree and deviation degree is defined, and both the optimistic algorithm of the grey interval relational degree and the algorithm of deviation degree minimization for solving this new model are also given. Finally, a decision making example to demonstrate the feasibility and rationality of this new method is given, and the results by using these two algorithms are uniform.

Entropy-based procedures for intuitionistic fuzzy multiple attribute decision making

The class of multiple attribute decision making （MADM） problems is studied, where the attribute values are intuitionistic fuzzy numbers, and the information about attribute weights is completely unknown. A score function is first used to calculate the score of each attribute value and a score matrix is constructed, and then it is transformed into a normalized score matrix. Based on the normalized score matrix, an entropy-based procedure is proposed to derive attribute weights. Furthermore, the additive weighted averaging operator is utilized to fuse all the normalized scores into the overall scores of alternatives, by which the ranking of all the given alternatives is obtained. This paper is concluded by extending the above results to interval-valued intuitionistic fuzzy set theory, and an illustrative example is also provided.

Time-Variant Reliability-Based Multi-Objective Fuzzy Design Optimization for Anti-Roll Torsion Bar of EMU

Although various types of anti-roll torsion bars have been developed to inhibit excessive roll angle of the electric multiple unit(EMU)car body,it is critical to ensure the reliability of structural design due to the complexity of the problems involving time and uncertainties.To address this issue,amulti-objective fuzzy design optimization model is constructed considering time-variant stiffness and strength reliability constraints for the anti-roll torsion bar.A hybrid optimization strategy combining the design of experiment(DoE)sampling and non-linear programming by quadratic lagrangian(NLPQL)is presented to deal with the design optimization model.To characterize the effect of time on the structural performance of the torsion bar,the continuous-time model combined with Ito lemma is proposed to establish the time-variant stiffness and strength reliability constraints.Fuzzy mathematics is employed to conduct uncertainty quantification for the design parameters of the torsion bar.A physical programming approach is used to improve the designer’s preference and to make the optimization results more consistent with engineering practices.Moreover,the effectiveness of the proposed method has been validated by comparing with current methods in a practical engineering case.

Multi-objective Fuzzy Optimization Algorithm for Separation-Recycle System

Separation-recycle system is an important part in chemical process, and its optimization is a multiobjective problem. In this paper the process optimization procedure is proposed. The fuzzy optimization algorithm with the concept of relative importance degree (RID) is utilized to transfer multi-objective optimization (MO-O) model into a single-objective optimization (SO-O) framework. The treatment of process condensate in synthesisa mmonia plant is taken as example to illustrate the optimization procedure, and the satisfactory result demonstrates feasibility and effectiveness of the suggested method.

Fuzzy Comprehensive Evaluation for Decision Making of Water Saving Irrigation System

A model of fuzzy comprehensive evaluation for water saving irrigation system (WSIS) decision making is proposed based on establishing an index system affected by six kinds of basic factors including qualitative and quantitative indexes. The object function of WSIS is set up by using the concept of fuzzy membership degree, it is to transform characteristic vector matrix into unify membership matrix and extending the least square method to the least of weighted distance square. The optimum weighted membership degree and the inferior weighted membership degree are used to solve the object function. This method effective solves the problem of classify for fuzzy attributive indexes and the problem of optimum for the set of different attributive indexes. A case study shows that the fuzzy comprehensive evaluation model is reasonable and effective in decision making for water saving irrigation system planning.

Group Decision Making Based Fuzzy Pattern Recognition Model for Lectotype Optimization of Offshore Platforms 1

This paper develops a fuzzy pattern recognition model for group decision making to solve the problem of lectotype optimization of offshore platforms. The lack of data and the inexact or incomplete information for criteria are the main cause of uncertainty in the evaluation process, therefore it is necessary to integrate the judgments from different decision makers with different experience, knowledge and preference. This paper first uses a complementary principle based pairwise comparison method to obtain the subjective weight of the criteria from each decision maker. A fuzzy pattern recognition model is then developed to integrate the judgments from all the decision makers and the information from the criteria, under the supervision of the subjective weights. Finally a case study is given to show the efficiency and robustness of the proposed model.

Multi-objectives fuzzy optimization model for ship form demonstration based on information entropy

Selecting optimization ship form scheme is an important content in the process of concept design of ship. Multi-objective fuzzy decision-making model for ship form demonstration is set up according to the fuzzy pattern-recognition theory. Weight coefficients of each target of ship form scheme are determined by information entropy and individual subjective partiality. This model is used to select the optimal ship form scheme, the example shows that the model is exact and the resuh is credible. It can provide a reference for choosing the optimization scheme of ship form.