L20 LP part 6

• Homework• Review• Postoptimality Analysis• Summary

1

H19

2

H19 cont’d

3

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!

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

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

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

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

Simplex Method Identifies:

• global solutions, if they exist• multiple solutions• unbounded problems• degenerate problems• infeasible problems

9

Postoptimality Analysis

10

What happens to our results if: The inputs change? The conditions of the business objective change ?The system/factory parameters change

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

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?

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

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?

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

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”

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

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)

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

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

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)

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

Abc’s

• Changes in cost coefficients, c• Changes in coefficient matrix A

Often times it’s simpler to re-run LP Solver

23

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

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