L20 LP part 6
description
Transcript of L20 LP part 6
![Page 1: L20 LP part 6](https://reader035.fdocuments.us/reader035/viewer/2022062301/56815177550346895dbfb246/html5/thumbnails/1.jpg)
L20 LP part 6
• Homework• Review• Postoptimality Analysis• Summary
1
![Page 2: L20 LP part 6](https://reader035.fdocuments.us/reader035/viewer/2022062301/56815177550346895dbfb246/html5/thumbnails/2.jpg)
H19
2
![Page 3: L20 LP part 6](https://reader035.fdocuments.us/reader035/viewer/2022062301/56815177550346895dbfb246/html5/thumbnails/3.jpg)
H19 cont’d
3
![Page 4: L20 LP part 6](https://reader035.fdocuments.us/reader035/viewer/2022062301/56815177550346895dbfb246/html5/thumbnails/4.jpg)
Two-phase Simplex Method
• Phase I - finds a feasible basic solution• Phase II- finds an optimal feasible basic
solution, if it exists.
4
Use artificial variables!
![Page 5: L20 LP part 6](https://reader035.fdocuments.us/reader035/viewer/2022062301/56815177550346895dbfb246/html5/thumbnails/5.jpg)
Transforming Process1. Convert Max to Min, i.e. Min f(x) = Min -F(x)2. Convert negative bj to positive, mult by(-1)3. Add slack variables4. Add surplus variables5. Add artificial var’s for “=” and or “≥”
constraints6. Create artificial cost function,
5
1 2( )art art artwx x x
![Page 6: L20 LP part 6](https://reader035.fdocuments.us/reader035/viewer/2022062301/56815177550346895dbfb246/html5/thumbnails/6.jpg)
Problem 8.58 (from lecture)
6
1 2
1 2 3 4 5 6
1 2 3 4 5 6
1 2 3 4 5 6
( ) 4. .
1 2 1 0 0 0 52 1 0 0 1 0 41 1 0 1 0 1 1
0i
Min f x xs tx x x x x xx x x x x xx x x x x xx
x
5 6
5 1 2
6 1 2 4
1 2 1 2 4
1 2 3 4
4 (2 )1 ( )
(4 2 ) (1 )5 3 0 0 1
Minimize w x xx x xx x x x
Min w x x x x xMin w x x x x
1 2
1 2
1 2
1 2
1 2
( ) 4. .
2 52 4
1, 0
Max F x xs tx xx xx xx x
x
![Page 7: L20 LP part 6](https://reader035.fdocuments.us/reader035/viewer/2022062301/56815177550346895dbfb246/html5/thumbnails/7.jpg)
H19 cont’d
7
Phase ISimplex Tableaurow basic x1 x2 x3 x4 x5 x6 b b/a_pivot
a x3 1 2 1 0 0 0 5 5/1b x5 2 1 0 0 1 0 4 4/2c x6 1 -1 0 -1 0 1 1 1/1d cost -1 -4 0 0 0 0 0e art cost -3 0 0 1 0 0 -5
Simplex Tableauoperations row basic x1 x2 x3 x4 x5 x6 b b/a_pivot
a-(1)h f x3 0 3 1 1 0 -1 4 4/3b-(2)h g x5 0 3 0 2 1 -2 2 2/3
h=c h x1 1 -1 0 -1 0 1 1 negd-(-1)h i cost 0 -5 0 -1 0 1 1e-(-3)h j art cost 0 -3 0 -2 0 3 -2
![Page 8: L20 LP part 6](https://reader035.fdocuments.us/reader035/viewer/2022062301/56815177550346895dbfb246/html5/thumbnails/8.jpg)
H19 cont’d
8
Simplex Tableauoperations row basic x1 x2 x3 x4 x5 x6 b b/a_pivot
f-(3)l k x3 0 0 1 -1 -1 1 2g/3 l x2 0 1 0 0.66667 0.33333 -0.6667 0.66667
h-(-1)l m x1 1 0 0 -0.3333 0.33333 0.33333 1.66667i-(-5)l n cost 0 0 0 2.33333 1.66667 -2.3333 4.33333 f=-4.333j-(-3)l o art cost 0 0 0 0 1 1 0 w=0
1 2
1 2
1 2
1 2
1 2
( ) 4. .
2 52 4
1, 0
Max F x xs tx xx xx xx x
x
![Page 9: L20 LP part 6](https://reader035.fdocuments.us/reader035/viewer/2022062301/56815177550346895dbfb246/html5/thumbnails/9.jpg)
Simplex Method Identifies:
• global solutions, if they exist• multiple solutions• unbounded problems• degenerate problems• infeasible problems
9
![Page 10: L20 LP part 6](https://reader035.fdocuments.us/reader035/viewer/2022062301/56815177550346895dbfb246/html5/thumbnails/10.jpg)
Postoptimality Analysis
10
What happens to our results if: The inputs change? The conditions of the business objective change ?The system/factory parameters change
![Page 11: L20 LP part 6](https://reader035.fdocuments.us/reader035/viewer/2022062301/56815177550346895dbfb246/html5/thumbnails/11.jpg)
What-if analyses
11
Problem formulation: …. Make assumptions about Resources/capacities, Price coefficientsCost coefficients
Also what would happen if conditions should change such as :Environmental aspectsConditions of useMarket conditionsProduction capabilities
Determine Strategies to handle:In case things changeTo account for competitive reactionsConsider worst case possibilities
![Page 12: L20 LP part 6](https://reader035.fdocuments.us/reader035/viewer/2022062301/56815177550346895dbfb246/html5/thumbnails/12.jpg)
Sensitivity Analyses
12
how sensitive are the:a. optimal value (i.e. f(x) and b. optimal solution (i.e. x)
… to the parameters (i.e. assumptions) in our model?
![Page 13: L20 LP part 6](https://reader035.fdocuments.us/reader035/viewer/2022062301/56815177550346895dbfb246/html5/thumbnails/13.jpg)
Model parameters
13
1 1 2 2
11 1 1 1
21 1 2 2
1 1
( ). .
0, 10, 1
n n
n n
n n
m mn n m
i
j
Min f c x c x c xs ta x a x ba x a x b
a x a x b
b i to mx j to n
x
( ). .Min fs t
Tx c x
Ax bb 0x 0
Consider your abc’s, i.e. A, b and c
![Page 14: L20 LP part 6](https://reader035.fdocuments.us/reader035/viewer/2022062301/56815177550346895dbfb246/html5/thumbnails/14.jpg)
Recall during Formulation
14
Constraints usually arise from :Laws of nature
(e.g. F=ma)
Laws of economics(e.g. profit=revenues-costs)
Laws of man(e.g. max work week = 40 hrs)
How accurate are our assumptions?
![Page 15: L20 LP part 6](https://reader035.fdocuments.us/reader035/viewer/2022062301/56815177550346895dbfb246/html5/thumbnails/15.jpg)
Any approximations in our formulation?
15
1 1 2 2
11 1 1 1
21 1 2 2
1 1
( ). .
0, 10, 1
n n
n n
n n
m mn n m
i
j
Min f c x c x c xs ta x a x ba x a x b
a x a x b
b i to mx j to n
x
( ). .Min fs t
Tx c x
Ax bb 0x 0
![Page 16: L20 LP part 6](https://reader035.fdocuments.us/reader035/viewer/2022062301/56815177550346895dbfb246/html5/thumbnails/16.jpg)
Abc’s of sensitivity analyses
16
( ). .Min fs t
Tx c x
Ax bb 0x 0
Let’s look at “b” first, i.e. “resource limits”
![Page 17: L20 LP part 6](https://reader035.fdocuments.us/reader035/viewer/2022062301/56815177550346895dbfb246/html5/thumbnails/17.jpg)
Recall Relaxing constraints(i.e. adding more resources)
17
* ( )
* ( )
ii i
ij j
f fυ
b bf f
ue e
x*
x*
The instantaneous rate of change in the objective function with respect to relaxing a constraint IS the LaGrange multiplier!
Constraint Variation Sensitivity TheoremFrom LaGrange theory
![Page 18: L20 LP part 6](https://reader035.fdocuments.us/reader035/viewer/2022062301/56815177550346895dbfb246/html5/thumbnails/18.jpg)
Simplex LaGrange Multipliers
18
the right side paramter of the th constraintthe LaGrange multiplier of the th constraint
* ( )( )
i
i
i i i i new oldi i
e iy i
f fy f y e y e e
e e
x*
Constraint Type≤ = ≥slack either surplus
c’ column “regular” artificial artificial
0iy iy 0iy
Find the multipliers in the final tableau (right side)
![Page 19: L20 LP part 6](https://reader035.fdocuments.us/reader035/viewer/2022062301/56815177550346895dbfb246/html5/thumbnails/19.jpg)
Prob 8.58
19
Simplex Tableau slack surplus art'l art'loperations row basic x1 x2 x3 x4 x5 x6 b b/a_pivot
f-(3)l k x3 0 0 1 -1 -1 1 2g/3 l x2 0 1 0 0.666667 0.333333 -0.66667 0.666667
h-(-1)l m x1 1 0 0 -0.33333 0.333333 0.333333 1.666667i-(-5)l n cost 0 0 0 2.333333 1.666667 -2.33333 4.333333 f=-4.333j-(-3)l o art cost 0 0 0 0 1 1 0 w=0
1 2
1 2 3 4 5 6
1 2 3 4 5 6
1 2 3 4 5 6
( ) 4. .
1 2 1 0 0 0 52 1 0 0 1 0 41 1 0 1 0 1 1
0i
Min f x xs tx x x x x xx x x x x xx x x x x xx
x1 2
1 2
1 2
1 2
1 2
( ) 4. .
2 52 4
1, 0
Max F x xs tx xx xx xx x
x
“=“ “≥”reg. cols art’l col’s
1 3 1 1
1 5 2 2
2 6 3 3
(" "), : 0, 0(" "), : 0, 5 / 3(" "), : 0, 7 / 3
( )i i i new old
g x y yh x y yg x y yf y e y e e
![Page 20: L20 LP part 6](https://reader035.fdocuments.us/reader035/viewer/2022062301/56815177550346895dbfb246/html5/thumbnails/20.jpg)
Excel “shadow prices”
20
Variable CellsFinal Reduced Objective Allowable Allowable
Cell Name Value Cost Coefficient Increase Decrease$C$13 x1 1.666666667 0 -1 1E+30 7$C$14 x2 0.666666667 0 -4 3.5 1E+30
ConstraintsFinal Shadow Constraint Allowable Allowable
Cell Name Value Price R.H. Side Increase Decrease$C$16 g1 3 0 5 1E+30 2$C$17 h1 4 -1.666666667 4 2 2$C$18 g2 1 2.333333333 1 1 2
If Minimizing w/Excel…Reverse sign of LaGrange Multipliers!
Variable CellsFinal Reduced Objective Allowable Allowable
Cell Name Value Cost Coefficient Increase Decrease$C$13 x1 1.666666667 0 1 7 1E+30$C$14 x2 0.666666667 0 4 1E+30 3.5
ConstraintsFinal Shadow Constraint Allowable Allowable
Cell Name Value Price R.H. Side Increase Decrease$C$16 g1 3 0 5 1E+30 2$C$17 h1 4 1.666666667 4 2 2$C$18 g2 1 -2.333333333 1 1 2
If Maximizing w/Excel… signs are the same as our tableaus
![Page 21: L20 LP part 6](https://reader035.fdocuments.us/reader035/viewer/2022062301/56815177550346895dbfb246/html5/thumbnails/21.jpg)
Let’s minimize f even further
21
1 1 2 2 3 3
1 2 3
2
3
1
(0) (5 / 3) ( 7 / 3)1
1(0) (5 / 3)(1) ( 7 / 3)( 1)12 / 3 4
f y e y e y ef e e eeef ef
Increase/decrease ei to reduce f(x)
![Page 22: L20 LP part 6](https://reader035.fdocuments.us/reader035/viewer/2022062301/56815177550346895dbfb246/html5/thumbnails/22.jpg)
Excel solution
22
Objective Cell (Min)Cell Name Original Value Final Value
$C$15 c -4.333333333 -8.333333333
Variable CellsCell Name Original Value Final Value Integer
$C$13 x1 1.666666667 1.666666667 Contin$C$14 x2 0.666666667 1.666666667 Contin
ConstraintsCell Name Cell Value Formula Status Slack
$C$16 g1 5 $C$16<=$E$16 Binding 0$C$17 h1 5 $C$17=$E$17 Binding 0$C$18 g2 0 $C$18>=$E$18 Binding 0
Objective Cell (Min)Cell Name Original Value Final Value
$C$15 c -4.333333333 -4.333333333
Variable CellsCell Name Original Value Final Value Integer
$C$13 x1 1.666666667 1.666666667 Contin$C$14 x2 0.666666667 0.666666667 Contin
ConstraintsCell Name Cell Value Formula Status Slack
$C$16 g1 3 $C$16<=$E$16 Not Binding 2$C$17 h1 4 $C$17=$E$17 Binding 0$C$18 g2 1 $C$18>=$E$18 Binding 0
![Page 23: L20 LP part 6](https://reader035.fdocuments.us/reader035/viewer/2022062301/56815177550346895dbfb246/html5/thumbnails/23.jpg)
Abc’s
• Changes in cost coefficients, c• Changes in coefficient matrix A
Often times it’s simpler to re-run LP Solver
23
![Page 24: L20 LP part 6](https://reader035.fdocuments.us/reader035/viewer/2022062301/56815177550346895dbfb246/html5/thumbnails/24.jpg)
Ranging-limits on changes
• RHS parameters, b• Cost coefficients, c • A coefficient matrix
Yes, formulas are available… often times it’s much easier to just re-run your LP Solver!
24
![Page 25: L20 LP part 6](https://reader035.fdocuments.us/reader035/viewer/2022062301/56815177550346895dbfb246/html5/thumbnails/25.jpg)
Summary• Simplex method determines:
Multiple solutions (think c’)Unbounded problems (think pivot aij<0)Degenerate Solutions (think bj=0)Infeasible problems (think w≠0)
• Sensitivity Analyses add important value to your LP solutions, can provide “strategies”
• Sensitivity is as simple as Abc’s• Constraint variation sensitivity theorem can
answer simple resource limits questions
25