小婧研学（75）：运筹学单纯形法的灵敏度分析与对偶2

摘要：灵敏度分析：研究线性规划模型中参数（目标函数系数、约束条件右端常数、技术系数）变化对最优解和最优值的影响。比如目标函数系数改变时，若在一定范围内，基变量和最优解不变，目标函数值可能变化；超出范围则需重新计算确定新解。约束条件右端常数变化会使资源数量改变，若在

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1. 灵敏度分析：研究线性规划模型中参数（目标函数系数、约束条件右端常数、技术系数）变化对最优解和最优值的影响。比如目标函数系数改变时，若在一定范围内，基变量和最优解不变，目标函数值可能变化；超出范围则需重新计算确定新解。约束条件右端常数变化会使资源数量改变，若在一定范围内，基变量类型不变但取值和目标函数值会变，超范围则需重新求解。

1. Sensitivity analysis: study the influence of changes in parameters (objective function coefficients, constraint right-end constants, technical coefficients) on the optimal solution and optimal values in the linear programming model. For example, when the objective function coefficient changes, if the base variable and the optimal solution remain unchanged within a certain range, the value of the objective function may change. If the range is exceeded, it needs to be recalculated to determine the new solution. If the type of the base variable remains unchanged but the value of the value and the objective function change within a certain range, the value of the base variable will change, and the solution needs to be re-solved if it is out of range.

2. 对偶：任何线性规划问题都有对偶问题，二者紧密相关。原问题求目标函数最大化，对偶问题求目标函数最小化；原问题约束条件个数对应对偶问题变量个数，原问题变量个数对应对偶问题约束条件个数。对偶问题最优解的影子价格有经济含义，反映资源边际价值。而且对偶问题的对偶是原问题，原问题和对偶问题最优值相等（强对偶性），还可利用对偶单纯形法求解线性规划问题，从对偶可行基出发迭代找最优解。

2. Duality: Any linear programming problem has a duality problem, and the two are closely related. The objective function is maximized for the original problem, and the objective function is minimized for the dual problem. The number of constraints in the original problem is the number of variables in response to the dual problem, and the number of variables in the original problem is the number of constraints in the dual problem. The shadow price of the optimal solution of the dual problem has economic implications and reflects the marginal value of resources. Moreover, the duality of the dual problem is the original problem, and the optimal value of the original problem and the dual problem are equal (strong duality), and the dual simplex method can also be used to solve the linear programming problem, and the optimal solution can be iteratively found from the duality feasible basis.

3. 两者关系：灵敏度分析可借助对偶理论进行。例如，影子价格是对偶问题最优解，通过分析其变化能了解原问题约束条件右端常数改变对目标函数的影响。对偶问题的解能为原问题灵敏度分析提供依据，确定参数变化时最优解和最优值是否改变及如何改变。

3. Relationship between the two: Sensitivity analysis can be performed with the help of dual theory. For example, the shadow price is the optimal solution to the dual problem, and the influence of the change of the right-end constant of the constraints of the original problem on the objective function can be understood by analyzing its changes. The solution of the dual problem can provide a basis for the sensitivity analysis of the original problem, and determine whether and how the optimal solution and optimal value change when the parameters change.

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部分参考内容来自百度