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Quality-guaranteed Dubins Path Planning for USV Based on Mixed-integer Piecewise linear Programming for Addressing the Extended Minimum-time Intercept Problem 认领 引用 被引量：1
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作者 Xing Zhou Kelin Zhu +3 位作者 Shuang Liu Zhaoqing Li Wenxin Zhang Kang Du 《哈尔滨工程大学学报（英文版）》 CSCD 2026年第1期216-227,共12页
During the use of robotics in applications such as antiterrorism or combat,a motion-constrained pursuer vehicle,such as a Dubins unmanned surface vehicle(USV),must get close enough(within a prescribed zero or positive... During the use of robotics in applications such as antiterrorism or combat,a motion-constrained pursuer vehicle,such as a Dubins unmanned surface vehicle(USV),must get close enough(within a prescribed zero or positive distance)to a moving target as quickly as possible,resulting in the extended minimum-time intercept problem(EMTIP).Existing research has primarily focused on the zero-distance intercept problem,MTIP,establishing the necessary or sufficient conditions for MTIP optimality,and utilizing analytic algorithms,such as root-finding algorithms,to calculate the optimal solutions.However,these approaches depend heavily on the properties of the analytic algorithm,making them inapplicable when problem settings change,such as in the case of a positive effective range or complicated target motions outside uniform rectilinear motion.In this study,an approach employing a high-accuracy and quality-guaranteed mixed-integer piecewise-linear program(QG-PWL)is proposed for the EMTIP.This program can accommodate different effective interception ranges and complicated target motions(variable velocity or complicated trajectories).The high accuracy and quality guarantees of QG-PWL originate from elegant strategies such as piecewise linearization and other developed operation strategies.The approximate error in the intercept path length is proved to be bounded to h2/(4√2),where h is the piecewise length. 展开更多
关键词 Minimum-time intercept problem Dubins vehicle Mixed-integer piecewise-linear program Linearization Approximate error trigonometric function USV
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Mixed-integer Linear Programming Based Distribution Network Reconfiguration Model Considering Reliability Enhancement 认领 引用 被引量：2
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作者 Junpeng Zhu Yi Zhou +3 位作者 Xiaofeng Dong Li Zhou Qiong Zhu Yue Yuan 《CSEE Journal of Power and Energy Systems》 SCIE EI CSCD 2025年第3期1336-1346,共11页
With the reform of the power system further deepening,the reliance on electricity and importance attached to the reliable power supply are increasing year by year,and the establishment of a high resilient power system... With the reform of the power system further deepening,the reliance on electricity and importance attached to the reliable power supply are increasing year by year,and the establishment of a high resilient power system has considerable economic,environmental and social benefits.Reconfiguring the network is one of the well-known tactics to enhance reliability.Accordingly,this paper proposes a reconfiguration method of distribution network considering the enhancement of reliability,which reconfigures the network structure both under normal operation conditions and outage scenarios,and considers factors such as power loss,load distribution and voltage quality considered in conventional reconfiguration methods.In this paper,the reliability assessment is integrated into the process of distribution network reconfiguration by using binary variables to represent the operating state of switchable devices.Based on the concept of fictitious fault flows,the reliability indices of distribution network are linearized expressed,and the network loss is reduced by minimizing the voltage deviation.A mixed integer linear programming(MILP)model is established for distribution network reconfiguration problem,which can guarantee the global optimal solution with high solution efficiency.Finally,the applicability and effectiveness of the proposed method are verified by numerical tests on a 54-node test system. 展开更多
关键词 Distribution network reconfiguration fictitious fault flows and mixed-integer linear programming reliability enhancement
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Open-loop control of combustion instabilities in a full-scale annular ramjet combustor using linear genetic programming 认领 引用
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作者 Jianguo TAN Zheng XU +2 位作者 Yao LIU Dongdong ZHANG Yi HOU 《Chinese Journal of Aeronautics》 SCIE EI CAS CSCD 2026年第2期20-28,共9页
The operational demands of a wide range significantly exacerbate combustion instability issues within ramjet combustor.To suppress combustion oscillations,an open-loop control system utilizing Linear Genetic Programmi... The operational demands of a wide range significantly exacerbate combustion instability issues within ramjet combustor.To suppress combustion oscillations,an open-loop control system utilizing Linear Genetic Programming(LGP)has been developed for a full-scale annular ramjet combustor.The LGP is used to generate control laws that include multi-frequency forcing.These laws are then transformed into square waves to actuate the solenoid valve,which modulates the kerosene supply for open-loop control.The results show that the duty cycle has little effect on instability amplitude,whereas an increase in frequency leads to a remarked reduction in combustion amplitude.After five generations evolvements,the pressure amplitude is reduced by 40.6% under the optimal control law generated by LGP.Furthermore,the machine learning process is depicted using a proximity map of control law similarity,with the search pathway visualized by the steepest descent.All individuals go forward to the upper left corner of the map with the evolution process,terminating at the optimal individual of the fifth generation. 展开更多
关键词 Annular ramjet combustor Combustion instabilities Linear genetic programming Machine learning Open-loop control
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Global Optimization Algorithm for Minimizing Linear Fractional Programming 认领 引用
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作者 ZHAO Peng SHEN Pei-ping ZHONG Zhe-wei 《Chinese Quarterly Journal of Mathematics》 2026年第1期50-59,共10页
In this paper,we study a class of Linear Fractional Programming on a nonempty bounded set,called the Problem(LFP),and design a branch and bound algorithm to find the global optimal solution of the problem(LFP).First,w... In this paper,we study a class of Linear Fractional Programming on a nonempty bounded set,called the Problem(LFP),and design a branch and bound algorithm to find the global optimal solution of the problem(LFP).First,we convert the problem(LFP)to the equivalent problem(EP2).Secondly,by applying the linear relaxation technique to the problem(EP2),the linear relaxation programming problem(LRP2Y)was obtained.Then,the overall framework of the algorithm is given,and the convergence and complexity of the algorithm are analyzed.Finally,experimental results are listed to illustrate the effectiveness of the algorithm. 展开更多
关键词 Global optimization Linear Fractional Programming Branch and bound algorithm Linear relaxation
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A mixed-integer linear programming approach for robust state estimation 认领 引用 被引量：4
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作者 Yanbo CHEN Jin MA 《Journal of Modern Power Systems and Clean Energy》 SCIE EI 2014年第4期366-373,共8页
In this paper,a mixed integer linear programming(MILP)formulation for robust state estimation(RSE)is proposed.By using the exactly linearized measurement equations instead of the original nonlinear ones,the existingmi... In this paper,a mixed integer linear programming(MILP)formulation for robust state estimation(RSE)is proposed.By using the exactly linearized measurement equations instead of the original nonlinear ones,the existingmixed integer nonlinear programming formulation for RSE is converted to a MILP problem.The proposed approach not only guarantees to find the global optimum,but also does not have convergence problems.Simulation results on a rudimentary 3-bus system and several IEEE standard test systems fully illustrate that the proposed methodology is effective with high efficiency. 展开更多
关键词 State estimation Robustness Leverage point Mathematical programming Mixed integer linear programming(MILP)
Computation of an Emptiable Minimal Siphon in a Subclass of Petri Nets Using Mixed-Integer Programming 认领 引用 被引量：5
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作者 Shouguang Wang Wenli Duo +4 位作者 Xin Guo Xiaoning Jiang Dan You Kamel Barkaoui MengChu Zhou 《IEEE/CAA Journal of Automatica Sinica》 SCIE EI CSCD 2021年第1期219-226,共8页
Deadlock resolution strategies based on siphon control are widely investigated.Their computational efficiency largely depends on siphon computation.Mixed-integer programming(MIP)can be utilized for the computation of ... Deadlock resolution strategies based on siphon control are widely investigated.Their computational efficiency largely depends on siphon computation.Mixed-integer programming(MIP)can be utilized for the computation of an emptiable siphon in a Petri net(PN).Based on it,deadlock resolution strategies can be designed without requiring complete siphon enumeration that has exponential complexity.Due to this reason,various MIP methods are proposed for various subclasses of PNs.This work proposes an innovative MIP method to compute an emptiable minimal siphon(EMS)for a subclass of PNs named S4PR.In particular,many particular structural characteristics of EMS in S4 PR are formalized as constraints,which greatly reduces the solution space.Experimental results show that the proposed MIP method has higher computational efficiency.Furthermore,the proposed method allows one to determine the liveness of an ordinary S4PR. 展开更多
关键词 Automated manufacturing systems deadlocks discrete event system mixed-integer programming(MIP) Petri nets(PN) siphons
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Exponential distribution-based genetic algorithm for solving mixed-integer bilevel programming problems 认领 引用 被引量：4
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作者 Li Hecheng Wang Yuping 《Journal of Systems Engineering and Electronics》 SCIE EI 2008年第6期1157-1164,共8页
Two classes of mixed-integer nonlinear bilevel programming problems are discussed. One is that the follower＇s functions are separable with respect to the follower＇s variables, and the other is that the follower＇s f... Two classes of mixed-integer nonlinear bilevel programming problems are discussed. One is that the follower＇s functions are separable with respect to the follower＇s variables, and the other is that the follower＇s functions are convex if the follower＇s variables are not restricted to integers. A genetic algorithm based on an exponential distribution is proposed for the aforementioned problems. First, for each fixed leader＇s variable x, it is proved that the optimal solution y of the follower＇s mixed-integer programming can be obtained by solving associated relaxed problems, and according to the convexity of the functions involved, a simplified branch and bound approach is given to solve the follower＇s programming for the second class of problems. Furthermore, based on an exponential distribution with a parameter λ, a new crossover operator is designed in which the best individuals are used to generate better offspring of crossover. The simulation results illustrate that the proposed algorithm is efficient and robust. 展开更多
关键词 mixed-integer nonlinear bilevel programming genetic algorithm exponential distribution optimalsolutions
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Improved Polytope Generation for CRC-Aided Adaptive Linear Programming Polar Decoding 认领 引用
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作者 Xie Mutong Du Zhongze +2 位作者 Zou Guoxue Tian Lin Yuan Jinhong 《China Communications》 SCIE EI CSCD 2025年第12期124-136,共13页
Linear programming(LP)decoding is a classic decoding method for linear block codes,and has attracted recent researches because its potential in joint channel processing.However,for polar codes,LP decoders has long bee... Linear programming(LP)decoding is a classic decoding method for linear block codes,and has attracted recent researches because its potential in joint channel processing.However,for polar codes,LP decoders has long been outperformed by CRCaided successive cancellation list(CA-SCL)decoders.To increase the competitiveness of 5G NR LP polar decoding,it is possible to gain performance improvements by exploiting the cyclic redundancy check(CRC)setup.In this paper,we propose a combined scheme of reduced sparsified factor graph-sparsified CRC(RSFG-SCRC)and augmented generator matrix-CRC(AGM-CRC),for polytope generation in adaptive linear programming(ALP)decoder for 5G polar codes.Augmented generator matrix(AGM)polytope and improved maximum cycle strategy-auxiliary node pairs 4(MCS-ANP-4)algorithm are proposed,to make efficient use of CRC constraints and minimize the constraint size for the decoder.Numerical simulations show that adaptive linear programming decoders with our proposed RSFG-SCRC and AGM-CRC polytopes can achieve significantly better block error rate(BLER)performance than a benchmark CA-SCL-8 decoder especially in harsh low-to-medium SNR regions. 展开更多
关键词 codeword polytope cyclic redundancy check 5G NR linear programming decoding polar code
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Solution for integer linear bilevel programming problems using orthogonal genetic algorithm 认领 引用 被引量：10
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作者 Hong Li Li Zhang Yongchang Jiao 《Journal of Systems Engineering and Electronics》 SCIE EI CSCD 2014年第3期443-451,共9页
An integer linear bilevel programming problem is firstly transformed into a binary linear bilevel programming problem, and then converted into a single-level binary implicit programming. An orthogonal genetic algorith... An integer linear bilevel programming problem is firstly transformed into a binary linear bilevel programming problem, and then converted into a single-level binary implicit programming. An orthogonal genetic algorithm is developed for solving the binary linear implicit programming problem based on the orthogonal design. The orthogonal design with the factor analysis, an experimental design method is applied to the genetic algorithm to make the algorithm more robust, statistical y sound and quickly convergent. A crossover operator formed by the orthogonal array and the factor analysis is presented. First, this crossover operator can generate a smal but representative sample of points as offspring. After al of the better genes of these offspring are selected, a best combination among these offspring is then generated. The simulation results show the effectiveness of the proposed algorithm. 展开更多
关键词 integer linear bilevel programming problem integer optimization genetic algorithm orthogonal experiment design
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Nonlinear Model-Based Process Operation under Uncertainty Using Exact Parametric Programming 认领 引用 被引量：4
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作者 Vassilis M.Charitopoulos Lazaros G.Papageorgiou Vivek Dua 《Engineering》 SCIE EI CAS 2017年第2期202-213,共12页
In the present work,two new,(multi-)parametric programming(mp-P)-inspired algorithms for the solutionof mixed-integer nonlinear programming(MINLP)problems are developed,with their main focus being onprocess synthesis ... In the present work,two new,(multi-)parametric programming(mp-P)-inspired algorithms for the solutionof mixed-integer nonlinear programming(MINLP)problems are developed,with their main focus being onprocess synthesis problems.The algorithms are developed for the special case in which the nonlinearitiesarise because of logarithmic terms,with the first one being developed for the deterministic case,and thesecond for the parametric case(p-MINLP).The key idea is to formulate and solve the square system of thefirst-order Karush-Kuhn-Tucker(KKT)conditions in an analytical way,by treating the binary variables and/or uncertain parameters as symbolic parameters.To this effect,symbolic manipulation and solution tech-niques are employed.In order to demonstrate the applicability and validity of the proposed algorithms,twoprocess synthesis case studies are examined.The corresponding solutions are then validated using state-of-the-art numerical MINLP solvers.For p-MINLP,the solution is given by an optimal solution as an explicitfunction of the uncertain parameters. 展开更多
关键词 Parametric programming Uncertainty Process synthesis Mixed-integer nonlinear programming Symbolic manipulation
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Linear goal programming approach to obtaining the weights of intuitionistic fuzzy ordered weighted averaging operator 认领 引用 被引量：4
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作者 Yejun Xu Chao Huang +1 位作者 Qingli Da Xinwang Liu 《Journal of Systems Engineering and Electronics》 SCIE EI 2010年第6期990-994,共5页
The multiple attribute decision making problems are studied, in which the information about attribute weights is partly known and the attribute values take the form of intuitionistic fuzzy numbers. The operational law... The multiple attribute decision making problems are studied, in which the information about attribute weights is partly known and the attribute values take the form of intuitionistic fuzzy numbers. The operational laws of intuitionistic fuzzy numbers are introduced, and the score function and accuracy function are presented to compare the intuitionistic fuzzy numbers. The intuitionistic fuzzy ordered weighted averaging （IFOWA） operator which is an extension of the well-known ordered weighted averaging （OWA） operator is investigated to aggregate the intuitionistic fuzzy information. In order to determine the weights of intuitionistic fuzzy ordered weighted averaging operator, a linear goal programming procedure is proposed for learning the weights from data. Finally, an example is illustrated to verify the effectiveness and practicability of the developed method. 展开更多
关键词 intuitionistic fuzzy set intuitionistic fuzzy ordered weighted averaging（IFOWA） operator linear goal programming weights multiple attribute decision making（MADM）.
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Efficient Algorithm for K-Barrier Coverage Based on Integer Linear Programming 认领 引用 被引量：16
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作者 Yanhua Zhang Xingming Sun Baowei Wang 《China Communications》 SCIE CSCD 2016年第7期16-23,共8页
Barrier coverage of wireless sensor networks is an important issue in the detection of intruders who are attempting to cross a region of interest.However,in certain applications,barrier coverage cannot be satisfied af... Barrier coverage of wireless sensor networks is an important issue in the detection of intruders who are attempting to cross a region of interest.However,in certain applications,barrier coverage cannot be satisfied after random deployment.In this paper,we study how mobile sensors can be efficiently relocated to achieve k-barrier coverage.In particular,two problems are studied:relocation of sensors with minimum number of mobile sensors and formation of k-barrier coverage with minimum energy cost.These two problems were formulated as 0–1 integer linear programming(ILP).The formulation is computationally intractable because of integrality and complicated constraints.Therefore,we relax the integrality and complicated constraints of the formulation and construct a special model known as RELAX-RSMN with a totally unimodular constraint coefficient matrix to solve the relaxed 0–1 ILP rapidly through linear programming.Theoretical analysis and simulation were performed to verify the effectiveness of our approach. 展开更多
关键词 k-barrier coverage linear programming wireless sensor networks
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An adaptive genetic algorithm for solving bilevel linear programming problem 认领 引用 被引量：2
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作者 王广民 王先甲 +1 位作者 万仲平 贾世会 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2007年第12期1605-1612,共8页
Bilevel linear programming, which consists of the objective functions of the upper level and lower level, is a useful tool for modeling decentralized decision problems. Various methods are proposed for solving this pr... Bilevel linear programming, which consists of the objective functions of the upper level and lower level, is a useful tool for modeling decentralized decision problems. Various methods are proposed for solving this problem. Of all the algorithms, the ge- netic algorithm is an alternative to conventional approaches to find the solution of the bilevel linear programming. In this paper, we describe an adaptive genetic algorithm for solving the bilevel linear programming problem to overcome the difficulty of determining the probabilities of crossover and mutation. In addition, some techniques are adopted not only to deal with the difficulty that most of the chromosomes maybe infeasible in solving constrained optimization problem with genetic algorithm but also to improve the efficiency of the algorithm. The performance of this proposed algorithm is illustrated by the examples from references. 展开更多
关键词 bilevel linear programming genetic algorithm fitness value adaptive operator probabilities crossover and mutation
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A Green Mixed Integer Linear Programming Model for Optimization of Byproduct Gases in Iron and Steel Industry 认领 引用 被引量：12
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作者 Hai-ning KONG 《Journal of Iron and Steel Research International》 SCIE CAS CSCD 2015年第8期681-685,共5页
Byproduct gas is an important secondary energy in iron and steel industry, and its optimization is vital to cost reduction. With the development of iron and steel industry to be more eco-friendly, it is necessary to c... Byproduct gas is an important secondary energy in iron and steel industry, and its optimization is vital to cost reduction. With the development of iron and steel industry to be more eco-friendly, it is necessary to construct an integrated optimized system, taking economics, energy consumption and environment into consideration. Therefore, the environmental cost caused by pollutants discharge should be factored in total cost when optimizing byproduct gas distribution. A green mixed integer linear programming （MILP） model for the optimization of byproduct gases was established to reduce total cost, including both operation cost and environmental cost. The operation cost included penalty for gas deviation, costs of fuel and water consumption, holder booster trip penalty, and so forth; while the environmental cost consisted of penalties for both direct and indirect pollutants discharge. Case study showed that the proposed model brought an optimum solution and 2.2% of the total cost could be reduced compared with previous one. 展开更多
关键词 green mixed integer linear programming environmental cost optimization iron and steel industry byproduct gas
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Prediction of mode I fracture toughness of rock using linear multiple regression and gene expression programming 认领 引用 被引量：11
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作者 Bijan Afrasiabian Mosleh Eftekhari 《Journal of Rock Mechanics and Geotechnical Engineering》 SCIE CSCD 2022年第5期1421-1432,共12页
Prediction of mode I fracture toughness(KIC) of rock is of significant importance in rock engineering analyses. In this study, linear multiple regression(LMR) and gene expression programming(GEP)methods were used to p... Prediction of mode I fracture toughness(KIC) of rock is of significant importance in rock engineering analyses. In this study, linear multiple regression(LMR) and gene expression programming(GEP)methods were used to provide a reliable relationship to determine mode I fracture toughness of rock. The presented model was developed based on 60 datasets taken from the previous literature. To predict fracture parameters, three mechanical parameters of rock mass including uniaxial compressive strength(UCS), Brazilian tensile strength(BTS), and elastic modulus(E) have been selected as the input parameters. A cluster of data was collected and divided into two random groups of training and testing datasets.Then, different statistical linear and artificial intelligence based nonlinear analyses were conducted on the training data to provide a reliable prediction model of KIC. These two predictive methods were then evaluated based on the testing data. To evaluate the efficiency of the proposed models for predicting the mode I fracture toughness of rock, various statistical indices including coefficient of determination(R2),root mean square error(RMSE), and mean absolute error(MAE) were utilized herein. In the case of testing datasets, the values of R2, RMSE, and MAE for the GEP model were 0.87, 0.188, and 0.156,respectively, while they were 0.74, 0.473, and 0.223, respectively, for the LMR model. The results indicated that the selected GEP model delivered superior performance with a higher R2value and lower errors. 展开更多
关键词 Mode I fracture Toughness Critical stress intensity factor Linear multiple regression(LMR) Gene expression programming(GEP)
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Global convergent algorithm for the bilevel linear fractional-linear programming based on modified convex simplex method 认领 引用 被引量：2
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作者 Guangmin Wang Bing Jiang +1 位作者 Kejun Zhu Zhongping Wan 《Journal of Systems Engineering and Electronics》 SCIE EI 2010年第2期239-243,共5页
A global convergent algorithm is proposed to solve bilevel linear fractional-linear programming, which is a special class of bilevel programming. In our algorithm, replacing the lower level problem by its dual gap equ... A global convergent algorithm is proposed to solve bilevel linear fractional-linear programming, which is a special class of bilevel programming. In our algorithm, replacing the lower level problem by its dual gap equaling to zero, the bilevel linear fractional-linear programming is transformed into a traditional sin- gle level programming problem, which can be transformed into a series of linear fractional programming problem. Thus, the modi- fied convex simplex method is used to solve the infinite linear fractional programming to obtain the global convergent solution of the original bilevel linear fractional-linear programming. Finally, an example demonstrates the feasibility of the proposed algorithm. 展开更多
关键词 bilevel linear fractional-linear programming convex simplex method dual problem.
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Smoothing Newton Algorithm for Linear Programming over Symmetric Cones 认领 引用 被引量：3
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作者 刘晓红 倪铁 《Transactions of Tianjin University》 EI CAS 2009年第3期216-221,共6页
By using the theory of Euclidean Jordan algebras,based on a new class of smoothing functions,the QiSun-Zhou's smoothing Newton algorithm is extended to solve linear programming over symmetric cones(SCLP).The algor... By using the theory of Euclidean Jordan algebras,based on a new class of smoothing functions,the QiSun-Zhou's smoothing Newton algorithm is extended to solve linear programming over symmetric cones(SCLP).The algorithm is globally convergent under suitable assumptions. 展开更多
关键词 linear programming symmetric cone Euclidean Jordan algebra smoothing algorithm
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A non-monotone Phase-1 method in linear programming 认领 引用 被引量：7
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作者 潘平奇 李炜 《Journal of Southeast University(English Edition)》 EI CAS 2003年第3期293-296,共4页
To gain superior computational efficiency, it might be necessary to change the underlying philosophy of the simplex method. In this paper, we propose a Phase-1 method along this line. We relax not only the conventiona... To gain superior computational efficiency, it might be necessary to change the underlying philosophy of the simplex method. In this paper, we propose a Phase-1 method along this line. We relax not only the conventional condition that some function value increases monotonically, but also the condition that all feasible variables remain feasible after basis change in Phase-1. That is, taking a purely combinatorial approach to achieving feasibility. This enables us to get rid of ratio test in pivoting, reducing computational cost per iteration to a large extent. Numerical results on a group of problems are encouraging. 展开更多
关键词 linear programming Phase-1 ratio-test-free pivoting rule
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A New Method for Achieving an Initial Regular Solution of a Linear Programming 认领 引用 被引量：3
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作者 梁平 孙艳华 +1 位作者 魏德宾 张相斌 《Northeastern Mathematical Journal》 2008年第1期31-34,共4页
A method is provided for finding an initial regular solution of a linear programming in this paper. The key to this method is to solve an auxiliary linear programming instead of to introduce any artificial variable or... A method is provided for finding an initial regular solution of a linear programming in this paper. The key to this method is to solve an auxiliary linear programming instead of to introduce any artificial variable or constraint. Compared with the traditional method of achieving the regular solution by introducing an artificial constraint, it has advantages of saving the memories and little computational efforts. 展开更多
关键词 initial regular solution auxiliary linear programming artificial variable artificial constrain
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Method for solving fully fuzzy linear programming problems using deviation degree measure 认领 引用 被引量：1
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作者 Haifang Cheng Weilai Huang Jianhu Cai 《Journal of Systems Engineering and Electronics》 SCIE EI CSCD 2013年第5期793-799,共7页
A new fully fuzzy linear programming(FFLP)problem with fuzzy equality constraints is discussed.Using deviation degree measures,the FFLP problem is transformed into a crisp 6-parametric linear programming(LP)problem.Gi... A new fully fuzzy linear programming(FFLP)problem with fuzzy equality constraints is discussed.Using deviation degree measures,the FFLP problem is transformed into a crisp 6-parametric linear programming(LP)problem.Giving the value of deviation degree in each constraint,the 6-fuzzy optimal solution of the FFLP problem can be obtained by solving this LP problem.An algorithm is also proposed to find a balance-fuzzy optimal solution between two goals in conflict:to improve the values of the objective function and to decrease the values of the deviation degrees.A numerical example is solved to illustrate the proposed method. 展开更多
关键词 fully fuzzy linear programming(FFLP) fuzzy equality constraint triangular fuzzy number deviation degree.
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