A method of minimizing rankings inconsistency is proposed for a decision-making problem with rankings of alternatives given by multiple decision makers according to multiple criteria. For each criteria, at first, the ...A method of minimizing rankings inconsistency is proposed for a decision-making problem with rankings of alternatives given by multiple decision makers according to multiple criteria. For each criteria, at first, the total inconsistency between the rankings of all alternatives for the group and the ones for every decision maker is defined after the decision maker weights in respect to the criteria are considered. Similarly, the total inconsistency between their final rankings for the group and the ones under every criteria is determined after the criteria weights are taken into account. Then two nonlinear integer programming models minimizing respectively the two total inconsistencies above are developed and then transformed to two dynamic programming models to obtain separately the rankings of all alternatives for the group with respect to each criteria and their final rankings. A supplier selection case illustrated the proposed method, and some discussions on the results verified its effectiveness. This work develops a new measurement of ordinal preferences’ inconsistency in multi-criteria group decision-making （MCGDM） and extends the cook-seiford social selection function to MCGDM considering weights of criteria and decision makers and can obtain unique ranking result.展开更多
In the constrained reentry trajectory design of hypersonic vehicles, multiple objectives with priorities bring about more difficulties to find the optimal solution. Therefore, a multi-objective reentry trajectory opti...In the constrained reentry trajectory design of hypersonic vehicles, multiple objectives with priorities bring about more difficulties to find the optimal solution. Therefore, a multi-objective reentry trajectory optimization (MORTO) approach via generalized varying domain (GVD) is proposed. Using the direct collocation approach, the trajectory optimization problem involving multiple objectives is discretized into a nonlinear multi-objective programming with priorities. In terms of fuzzy sets, the objectives are fuzzified into three types of fuzzy goals, and their constant tolerances are substituted by the varying domains. According to the principle that the objective with higher priority has higher satisfactory degree, the priority requirement is modeled as the order constraints of the varying domains. The corresponding two-side, single-side, and hybrid-side varying domain models are formulated for three fuzzy relations respectively. By regulating the parameter, the optimal reentry trajectory satisfying priorities can be achieved. Moreover, the performance about the parameter is analyzed, and the algorithm to find its specific value for maximum priority difference is proposed. The simulations demonstrate the effectiveness of the proposed method for hypersonic vehicles, and the comparisons with the traditional methods and sensitivity analysis are presented.展开更多
The application of multiple UAVs in complicated tasks has been widely explored in recent years.Due to the advantages of flexibility,cheapness and consistence,the performance of heterogeneous multi-UAVs with proper coo...The application of multiple UAVs in complicated tasks has been widely explored in recent years.Due to the advantages of flexibility,cheapness and consistence,the performance of heterogeneous multi-UAVs with proper cooperative task allocation is superior to over the single UAV.Accordingly,several constraints should be satisfied to realize the efficient cooperation,such as special time-window,variant equipment,specified execution sequence.Hence,a proper task allocation in UAVs is the crucial point for the final success.The task allocation problem of the heterogeneous UAVs can be formulated as a multi-objective optimization problem coupled with the UAV dynamics.To this end,a multi-layer encoding strategy and a constraint scheduling method are designed to handle the critical logical and physical constraints.In addition,four optimization objectives:completion time,target reward,UAV damage,and total range,are introduced to evaluate various allocation plans.Subsequently,to efficiently solve the multi-objective optimization problem,an improved multi-objective quantum-behaved particle swarm optimization(IMOQPSO)algorithm is proposed.During this algorithm,a modified solution evaluation method is designed to guide algorithmic evolution;both the convergence and distribution of particles are considered comprehensively;and boundary solutions which may produce some special allocation plans are preserved.Moreover,adaptive parameter control and mixed update mechanism are also introduced in this algorithm.Finally,both the proposed model and algorithm are verified by simulation experiments.展开更多
In military service joint operations, when there are more operational forces, more multifarious materials are consumed, the support is more complex and fuzzy, the deployment of personnel is more rapid, and the support...In military service joint operations, when there are more operational forces, more multifarious materials are consumed, the support is more complex and fuzzy, the deployment of personnel is more rapid, and the support provided by wartime military material support powers can be more effective. When the principles,requirements, influencing factors and goals of military material support forces are deployed in wartime, an evaluation indicator system is established. Thus, a new combined empowerment method based on an analytic hierarchy process（AHP） is developed to calculate the subjective weights, and the rough entropy method is used to calculate the objective weights. Combination weights can be obtained by calculating the weight preference coefficient error, which is determined by combining the cooperative game method and the minimum deviation into objectives. This approach can determine the grey relation projection coefficient and synthesize the measure scheme superiority to finally optimize the deployment plan using the grey relation projection decision-making method. The results show that the method is feasible and effective;it can provide a more scientific and practical decision-making basis for the military material support power deployment in wartime.展开更多
With the fast growth of Chinese economic, more and more capital will be invested in environmental projects. How to select the environmental investment projects （alternatives） for obtaining the best environmental qua...With the fast growth of Chinese economic, more and more capital will be invested in environmental projects. How to select the environmental investment projects （alternatives） for obtaining the best environmental quality and economic benefits is an important problem for the decision makers. The purpose of this paper is to develop a decision-making model to rank a finite number of alternatives with several and sometimes conflicting criteria. A model for ranking the projects of municipal sewage treatment plants is proposed by using exports＇ information and the data of the real projects. And, the ranking result is given based on the PROMETHEE method. Furthermore, by means of the concept of the weight stability intervals （WSI）, the sensitivity of the ranking results to the size of criteria values and the change of weights value of criteria are discussed. The result shows that some criteria, such as “proportion of benefit to project cost”, will influence the ranking result of alternatives very strong while others not. The influence are not only from the value of criterion but also from the changing the weight of criterion. So, some criteria such as “proportion of benefit to project cost” are key critera for ranking the projects. Decision makers must be cautious to them.展开更多
To make the dynamic assembly reliability analysis more effective for complex machinery of multi-object multi-discipline(MOMD),distributed collaborative extremum response surface method(DCERSM)was proposed based on ext...To make the dynamic assembly reliability analysis more effective for complex machinery of multi-object multi-discipline(MOMD),distributed collaborative extremum response surface method(DCERSM)was proposed based on extremum response surface method(ERSM).Firstly,the basic theories of the ERSM and DCERSM were investigated,and the strengths of DCERSM were proved theoretically.Secondly,the mathematical model of the DCERSM was established based upon extremum response surface function(ERSF).Finally,this model was applied to the reliability analysis of blade-tip radial running clearance(BTRRC)of an aeroengine high pressure turbine(HPT)to verify its advantages.The results show that the DCERSM can not only reshape the possibility of the reliability analysis for the complex turbo machinery,but also greatly improve the computational speed,save the computational time and improve the computational efficiency while keeping the accuracy.Thus,the DCERSM is verified to be feasible and effective in the dynamic assembly reliability(DAR)analysis of complex machinery.Moreover,this method offers an useful insight for designing and optimizing the dynamic reliability of complex machinery.展开更多
As a new-style stochastic algorithm, the electromagnetism-like mechanism（EM） method gains more and more attention from many researchers in recent years. A novel model based on EM（NMEM） for multiobjective optimizat...As a new-style stochastic algorithm, the electromagnetism-like mechanism（EM） method gains more and more attention from many researchers in recent years. A novel model based on EM（NMEM） for multiobjective optimization problems is proposed, which regards the charge of all particles as the constraints in the current population and the measure of the uniformity of non-dominated solutions as the objective function. The charge of the particle is evaluated based on the dominated concept, and its magnitude determines the direction of a force between two particles. Numerical studies are carried out on six complex test functions and the experimental results demonstrate that the proposed NMEM algorithm is a very robust method for solving the multiobjective optimization problems.展开更多
To increase the robustness of the optimization solutions of the mixed-flow pump,the impeller was firstly indirectly parameterized based on the 2D blade design theory.Secondly,the robustness of the optimization solutio...To increase the robustness of the optimization solutions of the mixed-flow pump,the impeller was firstly indirectly parameterized based on the 2D blade design theory.Secondly,the robustness of the optimization solution was mathematically defined,and then calculated by Monte Carlo sampling method.Thirdly,the optimization on the mixed-flow pump′s impeller was decomposed into the optimal and robust sub-optimization problems,to maximize the pump head and efficiency and minimize the fluctuation degree of them under varying working conditions at the same time.Fourthly,using response surface model,a surrogate model was established between the optimization objectives and control variables of the shape of the impeller.Finally,based on a multi-objective genetic optimization algorithm,a two-loop iterative optimization process was designed to find the optimal solution with good robustness.Comparing the original and optimized pump,it is found that the internal flow field of the optimized pump has been improved under various operating conditions,the hydraulic performance has been improved consequently,and the range of high efficient zone has also been widened.Besides,with the changing of working conditions,the change trend of the hydraulic performance of the optimized pump becomes gentler,the flow field distribution is more uniform,and the influence degree of the varia-tion of working conditions decreases,and the operating stability of the pump is improved.It is concluded that the robust optimization method proposed in this paper is a reasonable way to optimize the mixed-flow pump,and provides references for optimization problems of other fluid machinery.展开更多
With the development of central-private enterprises integration,selecting suitable key suppliers are able to provide core components for smart complex equipment.We consider selecting suitable key suppliers from matchi...With the development of central-private enterprises integration,selecting suitable key suppliers are able to provide core components for smart complex equipment.We consider selecting suitable key suppliers from matching perspective,for it not only satisfies natural development of smart complex equipment,it is also a good implementation of equipment project in central-private enterprises integration context.In in this paper,we carry out two parts of research,one is evaluation attributes based on comprehensive analysis,and the other is matching process between key suppliers and core components based on the matching attribute.In practical analysis process,we employ comprehensive evaluated analysis methods to acquire relevant attributes for the matching process that follows.In the analysis process,we adopt entropy-maximum deviation method(MDM)-decision-making trial and evaluation laboratory(DEMATEL)-technique for order preference by similarity to an ideal solution(TOPSIS)to obtain a comprehensive analysis.The entropy-MDM is applied to get weight value,DEMATEL is utilized to obtain internal relations,and TOPSIS is adopted to get ideal evaluated solution.We consider aggregating two types of evaluation information according to similarities of smart complex equipment based on the combination between geometric mean and arithmetic mean.Moreover,based on the aforementioned attributes and generalized power Heronian mean operator,we aggregate preference information to acquire relevant satisfaction degree,then combine the constructed matching model to get suitable key supplier.Through comprehensive analysis of selecting suitable suppliers,we know that two-sided matching and information aggregation can provide more research perspectives for smart complex equipment.Through analysis for relevant factors,we find that leading role and service level are also significant for the smart complex equipment development process.展开更多
In order to reduce both the weight of vehicles and the damage of occupants in a crash event simultaneously, it is necessary to perform a multi-objective optimal design of the automotive energy absorbing components. Mo...In order to reduce both the weight of vehicles and the damage of occupants in a crash event simultaneously, it is necessary to perform a multi-objective optimal design of the automotive energy absorbing components. Modified non-dominated sorting genetic algorithm II(NSGA II) was used for multi-objective optimization of automotive S-rail considering absorbed energy(E), peak crushing force(Fmax) and mass of the structure(W) as three conflicting objective functions. In the multi-objective optimization problem(MOP), E and Fmax are defined by polynomial models extracted using the software GEvo M based on train and test data obtained from numerical simulation of quasi-static crushing of the S-rail using ABAQUS. Finally, the nearest to ideal point(NIP)method and technique for ordering preferences by similarity to ideal solution(TOPSIS) method are used to find the some trade-off optimum design points from all non-dominated optimum design points represented by the Pareto fronts. Results represent that the optimum design point obtained from TOPSIS method exhibits better trade-off in comparison with that of optimum design point obtained from NIP method.展开更多
Optimization of an automotive body structure faces the difficulty of having too many design variables and a too large design search space. A simplified model of body-in-prime(BIP) can solve this difficulty by reducing...Optimization of an automotive body structure faces the difficulty of having too many design variables and a too large design search space. A simplified model of body-in-prime(BIP) can solve this difficulty by reducing the number of design variables. In this study, to achieve lighter weight and higher stiffness, the simplified model of BIP was developed and combined with an optimization procedure;consequently, optimal designs of automotive body B-pillar were produced. B-pillar was divided into four quarters and each quarter was modelled by one simplified beam. In the optimization procedure, depth, width, and thickness of the simplified beams were considered as the design variables.Weight, bending and torsional stiffness were also considered as objective functions. The optimization procedure is composed of six stages: designing the experiments, calculating grey relational grade, calculating signal-to noise ratio,finding an optimum design using Taguchi grey relational analysis, performing sensitivity analysis using analysis of variance(ANOVA) and performing non-dominated sorting and multi-criteria decision making. The results show that the width of lower B-pillar has the highest effect(about 55%) and the obtained optimum design point could reduce the weight of B-pillar by about 40% without reducing the BIP stiffness by more than 1.47%.展开更多
基金supported by the National Natural Science Foundation of China (60904059 60975049)+1 种基金the Philosophy and Social Science Foundation of Hunan Province (2010YBA104)the National High Technology Research and Development Program of China (863 Program)(2009AA04Z107)
文摘A method of minimizing rankings inconsistency is proposed for a decision-making problem with rankings of alternatives given by multiple decision makers according to multiple criteria. For each criteria, at first, the total inconsistency between the rankings of all alternatives for the group and the ones for every decision maker is defined after the decision maker weights in respect to the criteria are considered. Similarly, the total inconsistency between their final rankings for the group and the ones under every criteria is determined after the criteria weights are taken into account. Then two nonlinear integer programming models minimizing respectively the two total inconsistencies above are developed and then transformed to two dynamic programming models to obtain separately the rankings of all alternatives for the group with respect to each criteria and their final rankings. A supplier selection case illustrated the proposed method, and some discussions on the results verified its effectiveness. This work develops a new measurement of ordinal preferences’ inconsistency in multi-criteria group decision-making （MCGDM） and extends the cook-seiford social selection function to MCGDM considering weights of criteria and decision makers and can obtain unique ranking result.
基金supported by the Natural Science Foundation of Tianjin(12JCZDJC30300)the Research Foundation of Tianjin Key Laboratory of Process Measurement and Control(TKLPMC-201613)the State Scholarship Fund of China
文摘In the constrained reentry trajectory design of hypersonic vehicles, multiple objectives with priorities bring about more difficulties to find the optimal solution. Therefore, a multi-objective reentry trajectory optimization (MORTO) approach via generalized varying domain (GVD) is proposed. Using the direct collocation approach, the trajectory optimization problem involving multiple objectives is discretized into a nonlinear multi-objective programming with priorities. In terms of fuzzy sets, the objectives are fuzzified into three types of fuzzy goals, and their constant tolerances are substituted by the varying domains. According to the principle that the objective with higher priority has higher satisfactory degree, the priority requirement is modeled as the order constraints of the varying domains. The corresponding two-side, single-side, and hybrid-side varying domain models are formulated for three fuzzy relations respectively. By regulating the parameter, the optimal reentry trajectory satisfying priorities can be achieved. Moreover, the performance about the parameter is analyzed, and the algorithm to find its specific value for maximum priority difference is proposed. The simulations demonstrate the effectiveness of the proposed method for hypersonic vehicles, and the comparisons with the traditional methods and sensitivity analysis are presented.
基金Project(61801495)supported by the National Natural Science Foundation of China
文摘The application of multiple UAVs in complicated tasks has been widely explored in recent years.Due to the advantages of flexibility,cheapness and consistence,the performance of heterogeneous multi-UAVs with proper cooperative task allocation is superior to over the single UAV.Accordingly,several constraints should be satisfied to realize the efficient cooperation,such as special time-window,variant equipment,specified execution sequence.Hence,a proper task allocation in UAVs is the crucial point for the final success.The task allocation problem of the heterogeneous UAVs can be formulated as a multi-objective optimization problem coupled with the UAV dynamics.To this end,a multi-layer encoding strategy and a constraint scheduling method are designed to handle the critical logical and physical constraints.In addition,four optimization objectives:completion time,target reward,UAV damage,and total range,are introduced to evaluate various allocation plans.Subsequently,to efficiently solve the multi-objective optimization problem,an improved multi-objective quantum-behaved particle swarm optimization(IMOQPSO)algorithm is proposed.During this algorithm,a modified solution evaluation method is designed to guide algorithmic evolution;both the convergence and distribution of particles are considered comprehensively;and boundary solutions which may produce some special allocation plans are preserved.Moreover,adaptive parameter control and mixed update mechanism are also introduced in this algorithm.Finally,both the proposed model and algorithm are verified by simulation experiments.
基金supported by the Education Science Fund of the Military Science Institute of Beijing,China(2015JY320)
文摘In military service joint operations, when there are more operational forces, more multifarious materials are consumed, the support is more complex and fuzzy, the deployment of personnel is more rapid, and the support provided by wartime military material support powers can be more effective. When the principles,requirements, influencing factors and goals of military material support forces are deployed in wartime, an evaluation indicator system is established. Thus, a new combined empowerment method based on an analytic hierarchy process（AHP） is developed to calculate the subjective weights, and the rough entropy method is used to calculate the objective weights. Combination weights can be obtained by calculating the weight preference coefficient error, which is determined by combining the cooperative game method and the minimum deviation into objectives. This approach can determine the grey relation projection coefficient and synthesize the measure scheme superiority to finally optimize the deployment plan using the grey relation projection decision-making method. The results show that the method is feasible and effective;it can provide a more scientific and practical decision-making basis for the military material support power deployment in wartime.
基金Shanghai Leading Academic Discipline Project (T0502)Shanghai Municipal Educational Commission Project (05EZ32).
文摘With the fast growth of Chinese economic, more and more capital will be invested in environmental projects. How to select the environmental investment projects （alternatives） for obtaining the best environmental quality and economic benefits is an important problem for the decision makers. The purpose of this paper is to develop a decision-making model to rank a finite number of alternatives with several and sometimes conflicting criteria. A model for ranking the projects of municipal sewage treatment plants is proposed by using exports＇ information and the data of the real projects. And, the ranking result is given based on the PROMETHEE method. Furthermore, by means of the concept of the weight stability intervals （WSI）, the sensitivity of the ranking results to the size of criteria values and the change of weights value of criteria are discussed. The result shows that some criteria, such as “proportion of benefit to project cost”, will influence the ranking result of alternatives very strong while others not. The influence are not only from the value of criterion but also from the changing the weight of criterion. So, some criteria such as “proportion of benefit to project cost” are key critera for ranking the projects. Decision makers must be cautious to them.
基金Project(51175017)supported by the National Natural Science Foundation of ChinaProject(YWF-12-RBYJ-008)supported by the Innovation Foundation of Beihang University for PhD Graduates,ChinaProject(20111102110011)supported by the Research Fund for the Doctoral Program of Higher Education of China
文摘To make the dynamic assembly reliability analysis more effective for complex machinery of multi-object multi-discipline(MOMD),distributed collaborative extremum response surface method(DCERSM)was proposed based on extremum response surface method(ERSM).Firstly,the basic theories of the ERSM and DCERSM were investigated,and the strengths of DCERSM were proved theoretically.Secondly,the mathematical model of the DCERSM was established based upon extremum response surface function(ERSF).Finally,this model was applied to the reliability analysis of blade-tip radial running clearance(BTRRC)of an aeroengine high pressure turbine(HPT)to verify its advantages.The results show that the DCERSM can not only reshape the possibility of the reliability analysis for the complex turbo machinery,but also greatly improve the computational speed,save the computational time and improve the computational efficiency while keeping the accuracy.Thus,the DCERSM is verified to be feasible and effective in the dynamic assembly reliability(DAR)analysis of complex machinery.Moreover,this method offers an useful insight for designing and optimizing the dynamic reliability of complex machinery.
基金supported by the National Natural Science Foundation of China(60873099)the Fundamental Research Funds for the Central Universities(2011QNA29)
文摘As a new-style stochastic algorithm, the electromagnetism-like mechanism（EM） method gains more and more attention from many researchers in recent years. A novel model based on EM（NMEM） for multiobjective optimization problems is proposed, which regards the charge of all particles as the constraints in the current population and the measure of the uniformity of non-dominated solutions as the objective function. The charge of the particle is evaluated based on the dominated concept, and its magnitude determines the direction of a force between two particles. Numerical studies are carried out on six complex test functions and the experimental results demonstrate that the proposed NMEM algorithm is a very robust method for solving the multiobjective optimization problems.
基金National Natural Science Foundation of China(51609107)Open Subject of Provincial and Ministerial Discipline Platform of Xihua University(szjj2018-123)。
文摘To increase the robustness of the optimization solutions of the mixed-flow pump,the impeller was firstly indirectly parameterized based on the 2D blade design theory.Secondly,the robustness of the optimization solution was mathematically defined,and then calculated by Monte Carlo sampling method.Thirdly,the optimization on the mixed-flow pump′s impeller was decomposed into the optimal and robust sub-optimization problems,to maximize the pump head and efficiency and minimize the fluctuation degree of them under varying working conditions at the same time.Fourthly,using response surface model,a surrogate model was established between the optimization objectives and control variables of the shape of the impeller.Finally,based on a multi-objective genetic optimization algorithm,a two-loop iterative optimization process was designed to find the optimal solution with good robustness.Comparing the original and optimized pump,it is found that the internal flow field of the optimized pump has been improved under various operating conditions,the hydraulic performance has been improved consequently,and the range of high efficient zone has also been widened.Besides,with the changing of working conditions,the change trend of the hydraulic performance of the optimized pump becomes gentler,the flow field distribution is more uniform,and the influence degree of the varia-tion of working conditions decreases,and the operating stability of the pump is improved.It is concluded that the robust optimization method proposed in this paper is a reasonable way to optimize the mixed-flow pump,and provides references for optimization problems of other fluid machinery.
文摘With the development of central-private enterprises integration,selecting suitable key suppliers are able to provide core components for smart complex equipment.We consider selecting suitable key suppliers from matching perspective,for it not only satisfies natural development of smart complex equipment,it is also a good implementation of equipment project in central-private enterprises integration context.In in this paper,we carry out two parts of research,one is evaluation attributes based on comprehensive analysis,and the other is matching process between key suppliers and core components based on the matching attribute.In practical analysis process,we employ comprehensive evaluated analysis methods to acquire relevant attributes for the matching process that follows.In the analysis process,we adopt entropy-maximum deviation method(MDM)-decision-making trial and evaluation laboratory(DEMATEL)-technique for order preference by similarity to an ideal solution(TOPSIS)to obtain a comprehensive analysis.The entropy-MDM is applied to get weight value,DEMATEL is utilized to obtain internal relations,and TOPSIS is adopted to get ideal evaluated solution.We consider aggregating two types of evaluation information according to similarities of smart complex equipment based on the combination between geometric mean and arithmetic mean.Moreover,based on the aforementioned attributes and generalized power Heronian mean operator,we aggregate preference information to acquire relevant satisfaction degree,then combine the constructed matching model to get suitable key supplier.Through comprehensive analysis of selecting suitable suppliers,we know that two-sided matching and information aggregation can provide more research perspectives for smart complex equipment.Through analysis for relevant factors,we find that leading role and service level are also significant for the smart complex equipment development process.
文摘In order to reduce both the weight of vehicles and the damage of occupants in a crash event simultaneously, it is necessary to perform a multi-objective optimal design of the automotive energy absorbing components. Modified non-dominated sorting genetic algorithm II(NSGA II) was used for multi-objective optimization of automotive S-rail considering absorbed energy(E), peak crushing force(Fmax) and mass of the structure(W) as three conflicting objective functions. In the multi-objective optimization problem(MOP), E and Fmax are defined by polynomial models extracted using the software GEvo M based on train and test data obtained from numerical simulation of quasi-static crushing of the S-rail using ABAQUS. Finally, the nearest to ideal point(NIP)method and technique for ordering preferences by similarity to ideal solution(TOPSIS) method are used to find the some trade-off optimum design points from all non-dominated optimum design points represented by the Pareto fronts. Results represent that the optimum design point obtained from TOPSIS method exhibits better trade-off in comparison with that of optimum design point obtained from NIP method.
文摘Optimization of an automotive body structure faces the difficulty of having too many design variables and a too large design search space. A simplified model of body-in-prime(BIP) can solve this difficulty by reducing the number of design variables. In this study, to achieve lighter weight and higher stiffness, the simplified model of BIP was developed and combined with an optimization procedure;consequently, optimal designs of automotive body B-pillar were produced. B-pillar was divided into four quarters and each quarter was modelled by one simplified beam. In the optimization procedure, depth, width, and thickness of the simplified beams were considered as the design variables.Weight, bending and torsional stiffness were also considered as objective functions. The optimization procedure is composed of six stages: designing the experiments, calculating grey relational grade, calculating signal-to noise ratio,finding an optimum design using Taguchi grey relational analysis, performing sensitivity analysis using analysis of variance(ANOVA) and performing non-dominated sorting and multi-criteria decision making. The results show that the width of lower B-pillar has the highest effect(about 55%) and the obtained optimum design point could reduce the weight of B-pillar by about 40% without reducing the BIP stiffness by more than 1.47%.
