小婧研学（74）：运筹学单纯形法的灵敏度分析与对偶

摘要：一、灵敏度分析：研究线性规划模型的参数（目标函数系数、约束条件系数以及资源限量等）发生变化时，对最优解和最优值产生的影响。I. Sensitivity Analysis:It studies the impacts on the optimal solutio

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一、灵敏度分析：研究线性规划模型的参数（目标函数系数、约束条件系数以及资源限量等）发生变化时，对最优解和最优值产生的影响。I. Sensitivity Analysis:It studies the impacts on the optimal solution and the optimal value when the parameters of the linear programming model (such as the coefficients of the objective function, the coefficients of the constraint conditions, and the resource limits, etc.) change.

1、目标函数系数的灵敏度分析：主要研究当目标函数中某个变量的系数发生变化时，最优解是否会改变。通过计算该系数的变化范围，使得在这个范围内最优基不变，即最优解的结构不发生变化，只是目标函数值可能会有所改变。

1、Sensitivity Analysis of the Coefficients of the Objective Function: It mainly focuses on whether the optimal solution will change when the coefficient of a certain variable in the objective function changes. By calculating the variation range of this coefficient, within this range, the optimal basis remains unchanged, that is, the structure of the optimal solution does not change, but only the value of the objective function may change.

2、约束条件右端项的灵敏度分析：探讨约束条件中右端项（资源限量）的变化对最优解和目标函数值的影响。分析资源的单位变化所引起的目标函数值的变化量，即影子价格。影子价格反映了资源的稀缺程度和经济价值，可帮助决策者判断是否值得去获取更多的资源。

2、Sensitivity Analysis of the Right-hand Side Terms of the Constraint Conditions: It explores the influences of the changes in the right-hand side terms (resource limits) of the constraint conditions on the optimal solution and the value of the objective function. Analyze the variation amount of the objective function value caused by the unit change of resources, that is, the shadow price. The shadow price reflects the scarcity degree and economic value of resources, which can help decision-makers determine whether it is worthwhile to obtain more resources.

3、约束条件系数矩阵的灵敏度分析：当约束条件系数矩阵中的某个元素发生变化时，可能会导致可行域的形状和大小发生改变，进而影响最优解。需要重新判断最优解的可行性和最优性，可能需要通过重新计算单纯形表等方法来确定新的最优解。

3、Sensitivity Analysis of the Coefficient Matrix of the Constraint Conditions: When an element in the coefficient matrix of the constraint conditions changes, it may lead to changes in the shape and size of the feasible region, and thus affect the optimal solution. It is necessary to rejudge the feasibility and optimality of the optimal solution, and it may be necessary to determine the new optimal solution by recalculating the simplex table and other methods.

二、对偶：：对于每一个线性规划问题，都存在一个与之相对应的对偶问题。原问题和对偶问题在数学模型的结构上存在着特定的对应关系。例如，若原问题是求目标函数的最大值，约束条件为小于等于型；则其对偶问题是求目标函数的最小值，约束条件为大于等于型，并且两者的系数矩阵、变量和约束条件之间有着明确的转换规则。II. Duality:For every linear programming problem, there exists a corresponding dual problem. There is a specific corresponding relationship in the structure of the mathematical models between the original problem and the dual problem. For example, if the original problem is to find the maximum value of the objective function and the constraint conditions are of the less-than-or-equal-to type; then its dual problem is to find the minimum value of the objective function, and the constraint conditions are of the greater-than-or-equal-to type, and there are clear conversion rules between the coefficient matrices, variables, and constraint conditions of the two.

1、弱对偶性：原问题的任意可行解的目标函数值都小于等于对偶问题的任意可行解的目标函数值。这一性质为原问题和对偶问题的最优解提供了一个范围界定，可用于初步判断解的质量。

1、Weak Duality: The objective function value of any feasible solution of the original problem is less than or equal to the objective function value of any feasible solution of the dual problem. This property provides a range definition for the optimal solutions of the original problem and the dual problem, and can be used to initially judge the quality of the solutions.

2、强对偶性：若原问题有最优解，那么对偶问题也有最优解，且两者的最优目标函数值相等。这一性质深刻揭示了原问题与对偶问题之间的内在联系，在实际应用中，可以通过求解对偶问题来间接得到原问题的最优解，或者利用对偶问题的解来对原问题进行分析。

2、Strong Duality: If the original problem has an optimal solution, then the dual problem also has an optimal solution, and the optimal objective function values of the two are equal. This property profoundly reveals the internal connection between the original problem and the dual problem. In practical applications, the optimal solution of the original problem can be indirectly obtained by solving the dual problem, or the solution of the dual problem can be used to analyze the original problem.

3、互补松弛性：在原问题和对偶问题的最优解中，当某个约束条件对应的对偶变量不为零时，该约束条件在原问题中为等式约束；反之，当某个约束条件在原问题中为严格不等式约束时，其对应的对偶变量为零。这一性质可以帮助我们在已知一个问题的最优解时，快速求出另一个问题的最优解，或者用于检验最优解的正确性。

3、Complementary Slackness: In the optimal solutions of the original problem and the dual problem, when the dual variable corresponding to a certain constraint condition is not zero, this constraint condition is an equality constraint in the original problem; conversely, when a certain constraint condition is a strict inequality constraint in the original problem, its corresponding dual variable is zero. This property can help us quickly find the optimal solution of the other problem when the optimal solution of one problem is known, or it can be used to check the correctness of the optimal solution.

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部分参考内容来自百度