1、Operations Research(I) Dept. of Industrial EngineeringChapter 4 Sensitivity Analysis and Duality Operations Research (1)* 1Author:Zhang Zhihai, Dept. of Industrial Engineering, Tsinghua University, 100084, Beijing, ChinaContextn 4.1 A Graphical Introduction to Sensitivity Analysisn 4.2 Some Importan
2、t Formulasn 4.3 Sensitivity Analysisn 4.4 Sensitivity Analysis When More Than One Parameter is Changed: The 100% Rulen 4.5 Finding the Dual of an LPn 4.6 Economic Interpretation of the Dual Problem 2Author:Zhang Zhihai, Dept. of Industrial Engineering, Tsinghua University, 100084, Beijing, China4.1
3、A Graphical Introduction to Sensitivity Analysisn Giapettos Woodcarving Example:Types of toys SoldierTrainPrice $27 $21Raw material $10 $9Variable labor and overhead costs$14 $10Labor:carpentry 1hour 1hourLabor:finishing 2hours1hourn Available resourceCarpentry hours:80hoursn Trains:unlimited; Soldi
4、ers: =40n Objective:Maximize weekly profit3Author:Zhang Zhihai, Dept. of Industrial Engineering, Tsinghua University, 100084, Beijing, ChinaSolution:nx1=number of soldiers produced each weeknx2=number of trains produced each weekSolution:Optimal Solution: z=180, x1=20, x2=604Author:Zhang Zhihai, Dep
5、t. of Industrial Engineering, Tsinghua University, 100084, Beijing, ChinaA s1,x2,s3B x1,x2,s3C x1,x2,s2D5Author:Zhang Zhihai, Dept. of Industrial Engineering, Tsinghua University, 100084, Beijing, ChinaEffect of a Change in an Objective Function Coefficientx2=-C/2 x1+constant/2? =C= ?the current bas
6、is remain optimal6Author:Zhang Zhihai, Dept. of Industrial Engineering, Tsinghua University, 100084, Beijing, ChinaEffect of a Change in a RHS on the LPs Optimal Solutionthe current basis remain optimal? =b1= ?b1= 100+D2x1+x2= 100+Dx1+x2=80x1= 20+Dx2=60-D7Author:Zhang Zhihai, Dept. of Industrial Eng
7、ineering, Tsinghua University, 100084, Beijing, ChinaShadow PricesShadow Prices for the ith constraint of an LP to be the amount by which the optimal z-value is improvedincreased in a max problem and decreased in min problem if the rhs of the ith constraint is increased by 18Author:Zhang Zhihai, Dep
8、t. of Industrial Engineering, Tsinghua University, 100084, Beijing, Chinan Max Problemn New optimal z-value=(old optimal z-value)+(Constraint is shadown price) bin Min Problemn New optimal z-value=(old optimal z-value)-(Constraint is shadown price) bi9Author:Zhang Zhihai, Dept. of Industrial Engineering, Tsinghua University, 100084, Beijing, ChinaImportance of Sensitivity Analysis:10
