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Although departed methods such as dynamic programming, integer programming, and branch bound method can solve small scale problems, the rise in execution time of these methods is exponential with the number of generating units.

虽然在以往的研究中，已有许多方法使用在此问题上，例如动态规划法、整数规划法及支界法等，但是由於这些方法所需的求解时间会随机组数的增加呈指数关系成长，因此都比较适合於小规模系统上使用。

When problem scale is small, we can adopt many classical algorithms such as linear programming integer programming, branch and bound method to solve this problem.

当问题规模较小时，可用线性规划、整数规划、动态规划等经典算法求解。

Moreover, if the optimization problem has certain special feature, the simpler relaxation methods can be provided. For example, considering a generalized geometric programming problem, relevant relaxation linear programming can be obtained using its special structure. Based on the branch and bound method, a linear relaxation algorithm of global optimization is presented for generalized geometric programming, and the convergence of the algorithm is also proven.

此外，当优化问题具有某种特殊性质时，我们能给出更加简便的松弛化方法，如考虑广义几何规划问题，能充分利用其特殊结构，建立了广义几何规划问题的松弛线性规划问题；在分枝定界技术的基础上，提出了一个求广义几何规划全局解的线性松弛算法；证明了算法的收敛性。

The given Branch and Bound method can find out the optimal solution of the problem in small size while the given GA implement can quickly find out the approximate optimal solution of the problem in large size. The given GA implement proves to be very effective by numerical experiments and examples.

对于问题的较小规模情形分支定界法能精确地求得最优解；对于问题的较大规模情形遗传算法能很快地求得近似最优解；算例及大量实验表明用遗传算法来求问题的近似解是成功的。

By means of Heuristic Search, Branch and Bound method and Voronoi Diagram, the algorithm analyzes occlusion on complicated denture models precisely and quickly, which solved the practical problem in the application of VR technologies in dentistry.

通过推广基于分离轴定理的经典碰撞检测算法，借助启发式搜索、分支定界和Voronoi图等手段，算法可以在较短的时间内对复杂牙列模型完成高精度的定量咬合分析任务，解决了虚拟现实技术在口腔医学领域应用中的实际问题。

Second, according to model characteristic, such as non-linear, non-convex, multiple-peaked and so on, the paper discusses limitation of traditional solving algorithm, such as branch and bound method, GBD method and OA method, puts forward penalty function-makeup integer algorithm and genetic algorithm, analyses procedures of genetic algorithm, such as selection, crossover and mutation, brings forward modifications for specific realized problems of genetic algorithm, such as selection of initial points, operator design, adoptive value adjustment, constraint treatment and so on, and works out correlated solving program.

其次针对该模型非线性、非凸、多峰等特点，讨论了分支定界法、GBD法和OA法等传统求解算法的局限性，提出了罚函数一凑整算法和遗传算法的求解思路，对遗传算法的选择、交叉、变异等过程进行分析，对遗传算法初始点选择、算子设计、适应值调整和约束处理等具体实现问题提出了改进措施，并编制了相关的求解程序。

The first model is the optimization problems with s linear objective function subject to a system of fuzzy relation equations and a system of fuzzy relation inequalities, which are solved by using path method and 0-1 integer programming with branch and bound method.

第一种为约束为模糊关系方程和模糊关系不等式的具有线性目标函数的优化问题，采用路径法及0-1整数规划中的分枝定界法对其求解；第二种为约束为模糊关系方程和模糊关系不等式的格线性规划问题，采用直接法、路径法及0-1整数规划法对其求解；第三种为模糊关系不等式为模糊的，即约束中有一个可接受的容差，采用容差法及正负理想法对其求解；第四种是一种新的可能性模糊线性规划问题，采用模糊数大小比较法、"切片-粘贴"法及正负理想法，并设计了随机模拟的遗传算法对其求解；第五种为期望值模型，利用期望值将其转化为清晰约束系数的线性规划问题的方法。

The imprecise polynomial circuit optimization is based on arithmetic transformation method, considering the fixed point and floating point representation condition first, then give a realization of circuits under different accuracy boundary, trade off error with power dissipation and area, applying branch and bound method to find the solution with minimal cost, determine the minimal word length which satisfies the error bound, make use of the automation synthesis approach to get ultimate optimize structure at the same time.

非精确多项式电路优化基于算术变换方法，首先考虑电路的定点和浮点表示情况，给出电路在不同精度界限下的实现，进行误差和功耗、面积的权衡，然后应用分枝定界方法求解出最小代价电路，决定满足误差要求的最小字长，同时利用自动综合优化方法得到最终的优化结构。

One is tradition algorithm,which consists branch and bound method,improving loop method,cupidity algorithm,MST algorithm,MM algorithm,insertion method.Another is modern optimization algorithm which consists simulated annealing algorithm,artificial immune algorithm,genetic algorithm,ant colony algorithm,particle swarm optimization,Tabu Search,Hopfield neural network.The third is algorithm of DNA computing which is designed by authors.

第一种为传统算法，包括分支定界法、改良回路法、贪婪算法、MST算法、MM算法、插入法等；第二种为现代优化算法，包括模拟退火算法、人工免疫算法、遗传算法、蚁群算法、粒子群优化算法、禁忌搜索算法、Hopfield神经网络算法等；第三种为论文提出的DNA计算算法。

An advanced branch and bound method which reduces the computer resolving time enormously and leads the reliability research of large-scale spatial frame structural systems close to practical use is presented on condition that the same computation accuracy is ensured.

导出了大型空间刚架结构系统某梁元断面达到塑性屈服极限时的修正刚度矩阵和相应的等效节点力向量；提出了在保证同等计算精度的条件下，可成倍地减少运算时间的改进分枝限界方法，使大型空间刚架结构系统的可靠性研究向实用化迈进；本文开发出可自动生成失效衡准，自动选择主要失效形式，追踪失效过程的计算机程序。

And Pharaoh spoke to Joseph, saying, Your father and your brothers have come to you.

47：5 法老对约瑟说，你父亲和你弟兄们到你这里来了。

Additionally, the approximate flattening of surface strip using lines linking midpoints on perpendicular lines between geodesic curves and the unconditional extreme value method are discussed.

提出了用测地线方程、曲面上两点间短程线来计算膜结构曲面测地线的方法，同时，采用测地线间垂线的中点连线和用无约束极值法进行空间条状曲面近似展开的分析。