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(Max
problem )
If they are BV, its coefficients =?
Author:Zhang Zhihai, Dept. of Industrial Engineering, Tsinghua University, 100084, Beijing, China
13
Example 1: Compute the optimal tableau
Author:Zhang Zhihai, Dept. of Industrial Engineering, Tsinghua University, 100084, Beijing, China
3
Solution:
Solution: x1=number of soldiers produced each week x2=number of trains produced each week
the current basis remain optimal ? <=b1<= ? b1= 100+D 2x1+x2= 100+D x1+x2=
Author:Zhang Zhihai, Dept. of Industrial Engineering, Tsinghua University, 100084, Beijing, China
15
4.3 Sensitivity Analysis
(
[
)
]
Author:Zhang Zhihai, Dept. of Industrial Engineering, Tsinghua University, 100084, Beijing, China
11
Tableau
z C BV B 1b B 1 NX NBV C NBV X NBV = 0 z + C BV B 1 N C NBV X NBV = C BV B 1b s.t. BX BV + NX NBV = b X BV + B 1 NX NBV = B 1b X BV , X NBV ≥ 0
Author:Zhang Zhihai, Dept. of Industrial Engineering, Tsinghua University, 100084, Beijing, China
8
Max Problem New optimal z-value=(old optimal z-value)+(Constraint i's shadown price) △bi Min Problem New optimal z-value=(old optimal z-value)-(Constraint i's shadown price) △bi
max z = x1 + 4 x2 s.t. x1 + 2 x2 ≤ 6 2 x1 + x2 ≤ 8 BV = {x2 , s2 } x1 , x2 ≥ 0
2 0 B= 1 1
Z X BV X NBV RHS 1 0 CBV B1N CNBV CBV B1b 0 Im B1N B1b
14
Author:Zhang Zhihai, Dept. of Industrial Engineering, Tsinghua University, 100084, Beijing, China
10
4.2 Some Important Formulas
max(min) z = c1 x1 + c2 x2 + s.t. a11 x1 + a12 x2 + a21 x1 + a22 x2 + am1 x1 + an 2 x2 + xi ≥ 0(i = 1,2, + cn xn + a1n xn = b1 + a2 n xn = b2 + amn xn = bm
4
max
z = 3x1 + 2 x2 x1 + x2 + s2 = 80 x1 + s3 = 40 x1 , x2 , s1 , s2 , s3 ≥ 0
s.t. 2 x1 + x2 + s1 = 100
A B C D
s1,x2,s3 x1,x2,s3 x1,x2,s2
Author:Zhang Zhihai, Dept. of Industrial Engineering, Tsinghua University, 100084, Beijing, China
the current basis remain optimal x2=-C/2 x1+constant/2 ? <=C<= ?
Author:Zhang Zhihai, Dept. of Industrial Engineering, Tsinghua University, 100084, Beijing, China
NBV
Coefficient of excess variable ei in optimal row 0 = ith element of C BV B 1
(
)
Coefficient of artificial variable ai in optimal row 0 = ith element of C BV B 1 + M
Author:Zhang Zhihai, Dept. of Industrial Engineering, Tsinghua University, 100084, Beijing, China
9
Importance of Sensitivity Analysis:
Author:Zhang Zhihai, Dept. of Industrial Engineering, Tsinghua University, 100084, Beijing, China
max z = 3 x1 + 2 x2
subject to( s.t.) 2 x1 + x2 ≤ 100 x1 + x2 ≤ 80 x1 ≤ 40
x1
≥0 x2 ≥ 0
Optimal Solution: z=180, x1=20, x2=60
Author:Zhang Zhihai, Dept. of Industrial Engineering, Tsinghua University, 100084, Beijing, China
7
Shadow Prices
Shadow Prices for the ith constraint of an LP to be the amount by which the optimal z-value is improved—increased in a max problem and decreased in min problem –if the rhs of the ith constraint is increased by 1
Author:Zhang Zhihai, Dept. of Industrial Engineering, Tsinghua University, 100084, Beijing, China
2
4.1 A Graphical Introduction to Sensitivity Analysis
6
Effect of a Change in a RHS on the LP's Optimal Solution
max z = 3 x1 + 2 x2 x1 + x2 + s2 = 80 x1 + s3 = 40 x1 , x2 , s1 , s2 , s3 ≥ 0 s.t. 2 x1 + x2 + s1 = 100
5
Effect of a Change in an Objective Function Coefficient
max
z = 3 x1 + 2 x2 x1 + x2 + s2 = 80 x1 + s3 = 40 x1 , x2 , s1 , s2 , s3 ≥ 0
s.t. 2 x1 + x2 + s1 = 100
z = C BV X BV + C NBV X NBV s.t. BX BV + NX NBV = b X BV , X NBV ≥ 0
n)
z C BV B 1b B 1 NX NBV C NBV X NBV = 0 z + C BV B 1 N C NBV X NBV = C BV B 1b s.t. BX BV + NX NBV = b X BV + B 1 NX NBV = B 1b X BV , X NBV ≥ 0
Operations Research (1) Research(I)
Dept. of Industrial Engineering
Chapter 4 Sensitivity Analysis and Duality
22:56
1
Context
4.1 A Graphical Introduction to Sensitivity Analysis 4.2 Some Important Formulas 4.3 Sensitivity Analysis 4.4 Sensitivity Analysis When More Than One Parameter is Changed: The 100% Rule 4.5 Finding the Dual of an LP 4.6 Economic Interpretation of the Dual Problem
