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1、Chapter 5. What-If Analysis for Linear Programming,第五章. 线性规划的what-if分析,线性规划再认识,最优解一般只是针对某一特定的数学模型，是实际问题的一个粗略的抽象 线性规划的目的：对未来进行各种各样的假设，在这些假设下，测试各种管理方法可能产生的结果，而通过对各种结果的深入分析来指导管理者作出最终的决策 模型参数的获取是困难的，往往只能得到粗略的估计值。估计不准确会造成什么影响？,线性规划的what-if分析,什么是 what-if分析?,线性规划的what-if分析,线性规划的what-if分析是在求得基本模型的最优解之后进行的，其
2、目的就是要分析模型参数的变动会给当前最优解和最优值带来什么影响。具体包括： 目标函数系数的变动 约束函数系数的变动 约束边界值的变动,线性规划之 what-if分析,线性规划的what-if分析,如果模型参数在一定范围内变动不会改变最优解，那么管理者就能够接受该参数的估计值 如果模型参数估计值微小的变动都会改变最优解(这样的参数叫做敏感性参数)，管理者就会要求对这一估计值进行重新定义，使其更加精确,线性规划的what-if分析,what-if分析的基本作用： 确定参数估计值必须精确到怎样的程度，才能避免得出错误的最优解。找出敏感性参数 经过敏感(灵敏)性分析(分析单参数变动对结果的影响)之后，
3、即使不求解，也可以表明模型参数的变化是否会改变最优解 当模型特定的参数反映管理政策决策时，what-if分析可以表明改变这些决策对结果的影响，从而有效指导管理者作出最终的决策,Continuing the Wyndor Case Study (Section 5.2)(继续韦恩德公司的案例研究) Changes in One Objective Function Coefficient (Section 5.3)(更改一个目标函数系数) Simultaneous Changes in Objective Function Coefficients (Section 5.4)(同时改变多个目标函
4、数系数),Table of Contents (主要内容),Single Changes in a Constraint (Section 5.5)(个别更改某个约束) Simultaneous Changes in the Constraints (Section 5.6)(同时更改多个约束),Table of Contents (主要内容),Wyndor (Before What-If Analysis),Using the Spreadsheet to do Sensitivity Analysis,The profit per door has been revised from $3
5、00 to $200. No change occurs in the optimal solution. (每扇门的收益从300美金变成200美金，最优解并未因此发生变化),Using the Spreadsheet to do Sensitivity Analysis,The profit per door has been revised from $300 to $500. No change occurs in the optimal solution. (每扇门的收益从300美金变成500美金，最优解并未因此发生变化),Using the Spreadsheet to do Sen
6、sitivity Analysis,The profit per door has been revised from $300 to $1000. No change occurs in the optimal solution. (每扇门的收益从300美金变成1000美金，最优解发生了变化),Using Solver Table to do Sensitivity Analysis,Using Solver Table to do Sensitivity Analysis,最优域,保持最优解不变的模型参数允许变化的范围称为最优域 有了最优域模型参数发生变化时，就不需要重新建模与求解，可直接
7、判断模型参数的变化是否会影响初始的最优解,线性规划之 what-if分析,Using the Sensitivity Report to Find the Allowable Range,Graphical Insight into the Allowable Range,The two dashed lines that pass through the solid constraint boundary lines are the objective function lines when PD (the unit profit for doors) is at an endpoint o
8、f its allowable range, 0 PD 750. 图中穿过约束边界线(实线)的两条虚线是PD (门的单位收益)在可变范围0 PD 750的两个端点(即PD =0和PD =750)时的目标函数线。,Graphical Insight into the Allowable Range,多目标函数系数的电子表格敏感性分析,The profit per door has been revised from $300 to $450. The profit per window has been revised from $500 to $400. No change occurs in
9、 the optimal solution. (单位门和窗的收益分别由300美金变为450美金、500美金变为400美金，最优解并未因此发生变化),Using the Spreadsheet to do Sensitivity Analysis,The profit per door has been revised from $300 to $600. The profit per window has been revised from $500 to $300. No change occurs in the optimal solution. (单位门和窗的收益分别由300美金变为60
10、0美金、500美金变为300美金，最优解发生了变化),Using Solver Table to do Sensitivity Analysis,Using Solver Table to do Sensitivity Analysis,The 100 Percent Rule,The 100 Percent Rule for Simultaneous Changes in Objective Function Coefficients: If simultaneous changes are made in the coefficients of the objective function
11、, calculate for each change the percentage of the allowable change (increase or decrease) for that coefficient to remain within its allowable range. If the sum of the percentage changes does not exceed 100 percent, the original optimal solution definitely will still be optimal. (If the sum does exce
12、ed 100 percent, then we cannot be sure.),The 100 Percent Rule,目标函数系数同时变动的百分之百法则：如果目标函数系数同时变动，计算出每一系数变动量占该系数允许变化范围的百分比，然后将各个系数的变动百分比相加，如果所得的和不超过100%，最优解不会改变；如果超过100%，则不能确定最优解是否会改变。,多目标函数系数的电子表格敏感性分析,单位门和窗的收益分别由300美金变为450美金、500美金变为400美金，最优解并未因此发生变化,Graphical Insight into 100 Percent Rule,The estimates
13、 of the unit profits for doors and windows change to PD = $525 and PW = $350, which lies at the edge of what is allowed by the 100 percent rule.,Graphical Insight into 100 Percent Rule,When the estimates of the unit profits for doors and windows change to PD = $150 and PW = $250 (half their original
14、 values), the graphical method shows that the optimal solution still is (D, W) = (2, 6) even though the 100 percent rule says that the optimal solution might change. 当门和窗的单位收益的估计值变成PD = $150和PW = $250时(初始值的一半)，图形法显示最优解依然是(D, W) = (2, 6)，尽管百分百规则认为最优解可能会发生变化。,Graphical Insight into 100 Percent Rule,敏感
15、性分析方法小结,电子表格直接变动：适合检测参数的一组变动值 使用Solver Table：可检测多组变动值，但参数个数不能超过两个 图形分析法：适应于至多两个参数的变动分析 Solver报告：没有限制，但只提供可变单元格和约束边界值的单参数变动分析 百分百法则：没有限制，但超过100%则无法判断,单约束边界值的电子表格敏感性分析,The hours available in plant 2 have been increased from 12 to 13. The total profit increases by $150 per week. (工厂2的可用时间从12小时增加至13小时，总
16、收益每周可增加150美金),Using the Spreadsheet to do Sensitivity Analysis,The hours available in plant 2 have been further increased from 13 to 18. The total profit increases by $750 per week ($150 per hour added in plant 2). (工厂2的可用时间进一步从13小时增加至18小时，总收益每周可增加750美金，即工厂2每小时可增加150美金),The hours available in plant 2 have been further increased from 18 to 20. The total profit does not increase any further. (工厂2的可用时间进一步从18小时增加至20小时，总收益不再增加),Using Solver Table to do Sensitivity Analysis,
