05 lp iii handout

7/28/2019 05 LP III Handout

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Linear Programming (LP): Sensitivity Analysis

Benot Chachuat

McMaster UniversityDepartment of Chemical Engineering

ChE 4G03: Optimization in Chemical Engineering

Benot Chachuat (McMaster University) LP: Sensitivity Analysis 4G03 1 / 23

Linear Programming

Problem Statement

max z= x1 + x2

s.t. x

1 +x

2 0x1 2

x2 3

x1, x2 0

Simplex

=Search

Solution

z = 5 Optimal value

x1 + x2 = 1 0 Inactive

x1 = 2 2 Active

x2 = 3 3 Active

x1 = 2 0 Inactive

x2 = 3 0 Inactive

Is there more about the solution that we would like to know?

Mathematical optima do not suffice because such data as costs,

profits, yields, supplies and demands are always uncertain (at the timethe model is solved) Sometimes within a factor of 10!

How can we trust the outcome of imperfectly parameterized models? Maybe a cost and demand controls everything, or has no impact?


Linear Programming

Yes, sensitivity or what if analysis!

Mathematical optima simply provide a best choice of decisionvariables for one fixing of the inputs (nominal values)

Sensitivity analysis then tries to complete the picture by studying how

results would vary with changes in parameter values

Problem Statement

max z= x1 + x2

s.t. x1 + x2 0

x1 2 =

x2 3

x1, x2 0

Example of SensitivityInformation

What is the effect of a changein this parameter on the optimal

cost z

and on the optimal solu-tion point x = (x1 x

2 )?


Linear Programming

Yes, sensitivity or what if analysis!

Mathematical optima simply provide a best choice of decisionvariables for one fixing of the inputs (nominal values)

Sensitivity analysis then tries to complete the picture by studying how

results would vary with changes in parameter values We could always change parameter values and resolve the LP This

is a lot of work for large models! Sensitivity analysis yields useful information with little computation and

helps build insight

Parameter StructureChange Change

Qualitative

Sensitivity

direction of z change Yes Yes

QuantitativeSensitivity

magnitude of change in z limited range of validity

Yes No



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Outline

1 Motivations

2 Qualitative Sensitivity

3 Quantitative Sensitivity

For additional examples, see Rardin (1998), Chapter 7


Qualitative Sensitivity: Relaxing vs. Tightening Constraints

1 Relaxing the constraints of an optimization model either makes theoptimal value better (higher for a maximize, lower for a minimize) orleave it unchanged.

2 Tightening the constraints of an optimization model either makesthe optimal value worse or leave it unchanged.

x2 x2x2

x1 x1x1

(a) Nominal (b) Relaxed (c) Tightened

Warning: Tightening the constraints too much can make the problembecome infeasible!


Qualitative Sensitivity: Relaxing vs. Tightening Constraints

Class Exercise: For each of the following single changes to amaximization problem, indicate the effect on the optimal solution value:

Problem Statement

max z = x1 + x2

s.t. x1 + x2 0

x1 2

x2 3

x1, x2 0

Change the RHS of the 1st constraint to -1

Eliminate the 2nd constraint

Add the constraint 7x1 + 3x2 8

Allow variable x2 to be unrestricted (URS)

Change the 1st constraint to an equality

Change the coefficient of x1 in the LHS ofthe 1st constraint to -2


Change in Feasible Region: Direction of Change

1 The optimal solution value z changes monotonically as the feasibleregion is enlarged (constraint relaxation) or reduced (tightening)

2 The rate of change of z decreases in absolute value as the feasibleregion increases (no matter if active constraints change or not)

nominal optimal

solution point

nominal optimal

solution point

perturbed optimal

solution point

perturbed optimal

solution point

bi bi

zz

(a) without change in active set (b) with change in active setzbi

= i < i

z

bi

= i



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Change in Feasible Region: Direction of ChangeClass Exercise: Sketch the typical qualitative behavior of the optimalsolution value z as the RHS value bi of an active constraint is increasedor decreased from the nominal case; assume that the active set changesover the range of bi See Rardin (1998) for further discussion

a1ix1 + + anixn bi a1ix1 + + anixn bi

bi

bi bi

bi

zz

z zmax.

min.objective

objective


Change in Cost Coefficients: Direction of Change1 Over a limited range of cost coefficients ci, the actual extreme point

x (optimal solution) does not change2 Changes that help the optimal value z help more and more as the

change becomes bigger (no matter if active constraints change or not)

x2 x2

x1x1

nominal=perturbedoptimal point

perturbedoptimal point

cici

zz

(a) without change in active set (b) with change in active setzci

= i >

iz

ci

= i


Change in Cost Coefficients: Direction of Change

Class Exercise: Sketch the typical qualitative behavior of the optimalsolution value z as the cost coefficient ci is increased or decreased fromthe nominal case; assume that the active set changes over the range of ci See Rardin (1998) for further discussion

ci ci

z z

max. objective min. objective


Quantitative Sensitivity1 Quantitative sensitivity analysis is limited to optima with the same

active constraints as the nominal case.2 Quantitative sensitivity analysis provides an exact value of the

sensitivity z

parameter .3 Quantitative sensitivity analysis provides the range of parameter

changes which result in no change to the active constraint.

Marginal Price

The change in optimal solution value z per unit increase in the right-handside bi of a main constraint

nj=1 aijxj , =, bi is called a marginal price

(or shadow price or dual variable)

There is one marginal price i for each main constraint

Constraint i is: =

min. objective i 0 i 0 i URSmax. objective i 0 i 0 i URS



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Calculating Marginal Prices

Consider the partitioning of x, A, c into basic and nonbasiccomponents (at the nominal solution):

x= [xB | x

NB], A= [AB | A

NB], c= [cB | c

NB]

If the active constraints remain identical, a change bi in the RHS ofconstraint i induces basic/nonbasic variable changes as:

xNB = 0

xB = (A

B)1 0 0 bi 0 0

T

Why?

The marginal price i is therefore obtained as:

i = cTB (A

B)1 0 0 bi 0 0

T

Why?


Calculating Marginal Prices

Class Exercise: For the following LP problem, what are the marginalprices for unit changes in the RHS of the first and second constraints?

x1 x2 x3 x4 x5max c 1 1 0 0 0 b

A -1 1 -1 0 0 01 0 0 1 0 20 1 0 0 1 3

B B B B N Nx 2 3 1 0 0 z = 5


Range of Validity of Marginal Prices

How much can the right-hand

side bi of constraint i bechanged without modifying theset of active constraints?

bi

Both the optimal point x and optimal solution value z change, eventhough the active set does not change

On the other hand, the marginal price i does not change when biremains in the allowable range


Sensitivity to Change in Cost Coefficients

In the case that the active constraints remain identical:

A change ci in the costcoefficient of variable xi incursa change in optimal cost as:

z = 0 if xi is nonbasic Why?

z = xi ci if xi is basic

The solution point remainsidentical:

x = 0

The feasible region remainsunaffected

x2

x1

nominal=perturbedoptimal point

ci

z

No change in active set



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Range of Validity of Cost Sensitivity

How much can the cost coefficient ci of variable i be changed withoutmodifying the set of active constraints?

x2

x1

nominaloptimal point

perturbedoptimal points

ci


Computer Outputs and What If Questions

Software products (such as the solvers in GAMS) report:

1 the marginal price and the associated range of validity for everyconstraint

2

the sensitivity to change in cost coefficient and the associated rangeof validity

Outside of this range, the set of active constraints changes What canwe do?

Typical (Constraint) Sensitivity Analysis Output:

Constraint ID Status Slack Marginal Lower Upper

Price Range Range

Max. reflux flow Active 0 3.74 123 47

Max. Pump 7 Inactive 321 0 321 1.0E+30


Answering What If Questions (for Single Parameter)

Class Exercise: Use the table on the preceeding slide to answer thefollowing questions:

What is the effect on the profit of changing the reflux flow rate

capacity by 2.1?

Are the slack and marginal price values consistent for Pump 7?

What does a maximum allowable increase of 1.0E + 30 mean?


Answering What If Questions (for Single Parameter)

Class Exercise: Use the table on the preceeding slide to answer thefollowing questions:

A proposal has been made to increase the capacity of Pump 7.What is your response to the proposal?

What is the effect of reducing the reflux flow by 125?



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The Case of Multiple Parameter ChangesIn certain cases, it is possible to predict the effect of changing:

multiple RHS coefficients bi; or

multiple cost coefficients cj

The 100% Rule (Worst-Case Rule)Consider the percentage of allowable change being made, and add all thesepercentages.

If the sum is no more than 100%, then the current active constraintswill remain identical

If the sum is more than 100%, the current active constraints may notremain optimal

Caution:The 100% rule does not apply to mixed changes in

the constraint RHS coefficients bi and the cost coefficients cj


Answering What If Questions (for Multiple Parameters)

Class Exercise: Answer the following questions based on the data givenin the table:

Constraint ID Status Slack Marginal Lower Upper

Price Range RangeMax. reflux flow Active 0 3.74 123 47

Max. Pump 7 Inactive 321 0 321 1.0E+30

Max. Pump 5 Active 0 1.27 55 23

What is the effect on the profit of changing the reflux flow ratecapacity by 35 and reducing the capacity of Pump 5 by 14 atthe same time?


Quantitative Sensitivity Analysis in a Nutshell

Question that can sometimes be answered with quantitative sensitivityanalysis:

Change in a single RHS coefficient bi or cost coefficient cj

Change in multiple RHS coefficients

Change in multiple cost coefficients

If quantitative sensitivity analysis does not apply, the model has to beresolve for quantitative results:

Single large changes in RHS or cost coefficients

Multiple changes in RHS and cost coefficients simultaneously

Changes in LHS coefficients (A matrix)

max z= cTx

s.t. A x = b

x 0


Sensitivity Analysis The Final Words

We have covered the interpretation of the qualitative and quantitativesensitivity results for LP.

These are critical skills for the engineer!

We have not covered the details regarding the computation ofquantitative sensitivity results.

For an introduction to the dual formulation, see Rardin (1998)


05 lp iii handout

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