ise 626 fall 10 ch 4 simplex sensitivity

8/8/2019 ISE 626 Fall 10 Ch 4 Simplex Sensitivity

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Introduction

WhensolvinganLPproblemweassumethat

valuesof

all

model

coefficients

are

known

withcertainty.

Suchcertaintyrarelyexists.

abouthowsensitivetheoptimalsolutionisto

.


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GeneralFormofa

LinearProgramming

LP

Pro em

or : c1 1 +c2 2 ++cn n

Subjectto: a11X1 +a12X2 ++a1nXn


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ApproachestoSensitivityAnalysis

Changethedataandresolvethemodel!

Sometimesthis

is

the

only

practical

approach.

Solveralsoproducessensitivityreportsthat

cananswer

various

questions


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SolversSensitivityReport

Answersquestionsabout:

canchange

without

changing

the

optimal

solution.

ofchangesinconstrainedresources.

Theim act

on

the

o timal

ob ective

function

value

offorcedchangesindecisionvariables.

Theimpactchangesinconstraintcoefficientswill

haveon

the

optimal

solution.


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BVsObjective

Function

Coefficient

C angingt eco umno aij va ues oraNBV

Changingthe

RHS

of

a

constraint

Addinganewvariable


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When

solving

LP

problems,

be

sure

to

select

the

AssumeLinearModelo tionintheSolverO tions

dialogboxasthisallowsSolvertoprovidemore

sensitivityin ormationt anitcou ot erwise o.


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AnExampleLPProblem

tubs:Aqua

Spas

&

Hydro

Luxes.

Pumps 1 1

Tubing 12feet 16feet

UnitProfit 350 300

Thereare200pumps,1566hoursoflabor,and

2880 eeto tu ingavai a e.


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BlueRidge

Hot

Tubs

Example

MAX:350X1 +300X2 } profit

. .: 1 + 2


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How Changes in Objective Coefficients

X2

250

200

originallevelcurve

150

newop ma so u on

100

or g na op ma so u onnewlevelcurve

50

X1

0

0 50 100 150 200 250


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How Changes in Objective Coefficients

Seefileforsensitivityoptions


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Changesin

O jectiveFunction

Coe icients

AllowableDecrease

columns

for

the

ang ng e s n catet eamounts yw c

anobjective

function

coefficient

can

change

withoutchangingtheoptimalsolution,

assumingallothercoefficientsremain

constant.


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orAllowableDecreasecolumnsforthe

ChangingCellsindicatethatanalternate

.


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ChangesinConstraintRHSValues

Theshadowprice ofaconstraintindicatestheamount

bywhichtheobjectivefunctionvaluechangesgivena

unitincrease in

the

RHS

value

of

the

constraint,

assumingallothercoefficientsremainconstant.

ShadowpricesholdonlywithinRHSchangesfalling

Decreasecolumns.

a owpr ces

or

non n ng

cons ra n s

are

a ways

zero.


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CommentsAboutChanges

n

ons ra n

a ues

theobjective

function

value

as

RHS

values

change.

ChangingaRHSvalueforabindingconstraintalso

changesthe

feasible

region

and

the

optimal

solution

(seegraphonfollowingslide).

Tofindtheoptimalsolutionafterchangingabinding

RHSvalue,

you

must

re

solve

the

problem.


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How Changing an RHS Value Can Change

the Feasible Re ion and O timal SolutionX2

250

200

Supposeavailablelaborhours

increasefrom1,566to1,728.

150 oldoptimalsolution

100

o a orcons ra n

50

newlaborconstraint

X1

0 50 100 150 200 250


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How Changing an RHS Value Can Change

the Feasible Re ion and O timal Solution


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TheMeaningofReducedCosts

Thereducedcostforeachproductequalsitsperunitmarginalprofit minustheperunitvalueofthe

resources

it

consumes

(priced

at

their

shadow

.

OptimalValue

of Optimal

Value

of

TypeofProblem DecisionVariable ReducedCost

atsimplelowerbound =0

atsimplelowerbound >=0

n m za on e ween ower upper oun s =atsimpleupperbound


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KeyPoints I

Theshadowpricesofresourcesequatethe

marg na va ueo t eresourcesconsume w t

themarginalbenefitofthegoodsbeingproduced.

Resourcesinexcesssupplyhaveashadowprice(or

mar inal value of zero.


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KeyPointsII

Thereducedcostofaproductisthedifference

etween tsmarg na pro tan t emarg na va ue

oftheresourcesitconsumes.

Productswhosemarginalprofitsarelessthanthe

mar inal value of the oods re uired for their

productionwillnotbeproducedinanoptimal

.


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AnalyzingChangesin

Q:SupposeaTyphoonLagoonrequiredonly7labor

oursra er an . s nowpro a e o

produceany?

Q:WhatisthemaximumamountoflaborTyphoon

Lagoons

could

require

and

still

be

profitable?

ise 626 fall 10 ch 4 simplex sensitivity

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