taylor introms11ge ppt 03

7/25/2019 Taylor Introms11GE Ppt 03

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3-1Copyright 2013 Pearson Education

Solving Linear

ProgrammingModels

Chapter 3


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Chapter Topics

Computer Solution

Sensitivity Analysis


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Early linear programming used lengthymanual mathematical solution procedurecalled the Simplex Method (See web siteModule A!

Steps o" the Simplex Method ha#e beenprogrammed in so"tware pac$ages designed"or linear programming problems!

Many such pac$ages a#ailable currently!

%sed extensi#ely in business and go#ernment! &ext "ocuses on Excel Spreadsheets and 'M

"or indows!

Computer Solution


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3-)Copyright 2013 Pearson Education

Beaver Creek Pottery ExampleExcel Spreadsheet ata Screen !"o# $%

Exhi&it

Clic$ on *+ata,

tab to in#o$e*Sol#er!,

C./01+./011 .-E.

4-E4

C4/01+4/011

+ecision#ariable5

bowls(x1016

mugs(x2011

7b8ecti#e"unctionC)/01+)/

011


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Beaver Creek Pottery Example)Solver* Parameter Screen!+ o# $%

Exhi&it '(+ Solver

7b8ecti#e

"unction

+ecision#ariables Clic$ on *Add,

to add modelcontraints!

C6*B10+D6*B11


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3-.Copyright 2013 Pearson EducationExhi&it '(' La&or constr

Beaver Creek Pottery ExampleAdding Model Constraints !' o# $%

C./01+./011

)


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Beaver Creek Pottery Example)Solver* Settings !, o# $%

Exhi&it '(, Solution scree

Slack S1= 0

and S2= 0

Slac$5S

1

and

S2


8/403-?Copyright 2013 Pearson Education Exhi&it '($ Ans-er repo

Beaver Creek Pottery ExampleSolution Screen !$ o# $%


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Linear Programming Pro&lem.Standard /orm

Standard form reuires all #ariables in theconstraint euations to appear on the le"t o" theineuality (or euality and all numeric #alues to beon the right-hand side!

ExamplesB x3 x1 x2 must be con#erted to x3- x1 - x2

x1(x2 x3 2 becomes x1 2 (x2 x3

and then x1- 2x2- 2x3



Beaver Creek Pottery Example0M #or 1indo-s !" o# ,%

Exhibit 3!. +ata entry

Setnumber o"constraint

s anddecision#ariables!

Clic$ here whenDnished!



Beaver Creek PotteryExample0M #or 1indo-s !+ o#

,%

Exhibit 3!4 +ata table



Beaver Creek Pottery Example0M #or 1indo-s !' o# ,%

Exhibit 3!? Model solutio



Beaver Creek Pottery Example0M #or 1indo-s !, o# ,%

Exhibit 3!@ raphical sol


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Sensitivity analysis determines the eect on theoptimal solution o" changes in parameter valueso" the ob8ecti#e "unction and constraint euations!

Changes may be reactions to anticipateduncertainties in the parameters or to new orchanged in"ormation concerning the model!

Beaver Creek Pottery ExampleSensitivity Analysis!" o# ,%



Maximi2e 3 4 5,6x"

7

5$6x+su&8ect to. x"7 +x+ ,6

,x"7 'x+"+6

x"9 x+ 6

/igure '(":ptimal solution

point

Beaver Creek Pottery ExampleSensitivity Analysis!+ o# ,%


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Maximi2e 3 4 5"66x"

7

5$6x+su&8ect to. x"7 +x+ ,6

,x"7 'x+"+6

x"9 x+ 6

/igure '(+ Changing the x"o&8ective #unctioncoe;icient

Beaver Creek Pottery ExampleChange x":&8ective /unction

Coe;icient !' o# ,%



Maximi2e 3 4 5,6x"

7

5"66x+su&8ect to. x"7 +x+ ,6

,x"7 'x+"+6

x"9 x+ 6

/igure '(' Changing the x+o&8ective #unctioncoe;icient

Beaver Creek Pottery ExampleChange x+:&8ective /unction

Coe;icient !, o# ,%


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&he sensitivity range "or an ob8ecti#e "unctioncoeicient is the range of values o#er which thecurrent optimal solution point will remainoptimal!

&he sensiti#ity range "or the xicoeicient is

designated as ci!

:&8ective /unction Coe;icientSensitivity


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ob8ecti#e "unction F G)x1G9x2 sensiti#ity

range "orBx1B 29 c1..!.4

x2B 3

c2

?

/igure '(, etermining the sensitivity range

:&8ective /unction Coe;icientSensitivity



Minimi2e 3 4 5=x"7 5'x+su&8ect to.

+x"7 ,x+"=

,x"7 'x++, x"9 x+6

sensitivity ranges.

, c"6 c+ ,($

:&8ective /unction Coe;icient/ertili2er Cost Minimi2ationExample !' o# '%

Kigure 3!9 KertiliLer exampleB sensiti#ity range"or c1

:&8 i i C ;i i



Exhibit 3!1 Sol#er results s

:&8ective /unction Coe;icient



Exhi&it '("+



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Changes in Constraint 0uantity>aluesSensitivity ottery example!Consider a change in which the managercan increase labor hours "rom ) to .!

Ch i C t i t 0 tit



Changes in Constraint 0uantity>alues?ncreasing the La&or Constraint !+

o# ,%

Maximi2e 3 4 5,6x"7

5$6x+su&8ect to. x"7 +x+ 7 s"

4 ,6,x"7 'x+ 7

s+4 "+6 x"9 x+ 6

Kigure 3!. Jncreasing the labor constraint

7riginal"ormulation with) hours labor

Ch i C t i t 0 tit


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Changes in Constraint 0uantity>aluesSensitivity


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Changes in Constraint 0uantity>aluesSensitivity


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3-2?Copyright 2013 Pearson Education Exhibit 3!13

Constraint 0uantity >alue l < &


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Exhibit3!1)

Constraint 0uantity >alue


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Changing indi#idual constraint parameters

Adding new constraints

Adding new #ariables

:ther /orms o# Sensitivity AnalysisTopics !" o# ,%

:ther /orms o# Sensitivity Analysis


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:ther /orms o# Sensitivity AnalysisChanging a Constraint Parameter !+o# ,%

Maximi2e 3 4 5,6x"7

5$6x+su&8ect to. x"7 +x+,6

,x"7 'x+

"+6 x"9 x+ 6

/igure '( Changing the x"

coe;icient in the

la&or constraint



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Adding a ne- constraint to Beaver Creek Model.

6(+6x"7 6("6x+ $ hours #or packaging

:riginal solution. +,&o-ls9 mugs9 5"9'=6 proDt

Exhi&it '("$

:ther /orms o# Sensitivity AnalysisAdding a e- Constraint !' o# ,%

Constraint E8 G8

added to Solver

Added constraint

or pac!aging



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Adding a new #ariable to the 0ea#er Cree$ model; x3;"or a third product; cups

MaximiLe F G)x1 9x2 3x3

sub8ect toB

x1 2x2 1!2x3) hr o" labor

)x1 3x2 2x312 lb o" clay

x1; x2; x3

Sol#ing model shows that change has no eect on theoriginal solution (i!e!; the model is not sensiti#e to thischange!

:ther /orms o# Sensitivity AnalysisAdding a e- >aria&le !, o# ,%

Shado- Prices !ual >aria&le


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+eDned as the marginal value o" oneadditional unit o" resource!

&he sensitivity range "or a constraintuantity #alue is also the range o#erwhich the shadow price is valid!

Shado- Prices !ual >aria&le>alues%

Excel Sensitivity


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Maximi2e 3 4 5,6x"7 5$6x+

su&8ect to.x"7 +x+ ,6 hr o# la&or

,x"7 'x+ "+6 l& o# clay

x"9 x+6

Exhi&it '("=

Excel Sensitivity


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Excel Sensitivity


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&wo airplane partsB no!1 and no! 2! &hree manu"acturing stagesB stamping; drilling;

Dnishing!

+ecision #ariablesB x1(number o" part no! 1 to

produce x2(number o" part no! 2 toproduce

ModelB MaximiLe F G.9x1 @1x2

sub8ect toB )x1 4!9x219 (stamping;hr .!2x1 )!@x2 @ (drilling; hr @!1x1 )!1x211 (Dnishing; hr x1; x2

Example Pro&lemPro&lem Statement !" o# '%

Example Pro&lem


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Maximi2e 3 4 5=$6x"7

5"6x+su&8ect to. ,x"7 @($x+ "6$

=(+x"7 ,(x+ 6

("x"7 ,("x+ ""6

x"9 x+6

s" 4 69 s+ 4 69 s' 4 ""('$ hr

,$('' c" "9"$"(,'

("6 " "'@(@=

Example Pro&lemFraphical Solution !+ o# '%

Example Pro&lem


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Example Pro&lemExcel Solution !' o# '%


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taylor introms11ge ppt 03

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