201502271602484 goal programming.ppt

7/21/2019 201502271602484 GOAL PROGRAMMING.ppt

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1 2003 Thomson 2003 Thomson/South-Western/South-WesternSlide

GOAL PROGRAMMING


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Introduction

Most of the optimization problemsconsidered to this point have had a singleobjective

Often! more than one objective can beidentified for a given problemMa"imize Return or Minimize Ris#

Ma"imize Profit or Minimize Pollution $hese objectives often conflict %ith one

another


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Goal Programming &GP'

Most LP problems have hardconstraints that cannot be violated

$here are (!)** labor hoursavailable$here is RM+),!,, available for

projects


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Goal Programming &GP'

o

In some cases! hard constraints are toorestrictive

o-ou have a ma"imum price in mind %henbu.ing a car &this is .our /goal0 or targetprice'

oIf .ou can1t bu. the car for this price.ou1ll li#el. find a %a. to spend more

o 2e use soft constraintsto represent suchgoals or targets %e1d li#e to achieve


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GP3 4"ample (Indah 5each 6otel 4"pansion

Roslan %ants to e"pand the convention center at hishotel in Port 7ic#son

$he t.pes of conference rooms being considered are3

8ize &s9 m' :nit ;ost

8mall >!,,,

Large (!,), RM


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4"ample (

Roslan %ould li#e to add ) small! (, medium and ()large conference rooms

6e %ould also li#e the total e"pansion to be ?)!,,,s9uare meter and to limit the cost to RM(!,,,!,,,


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7efining the 7ecision @ariables

(B number of small rooms to add

?B number of medium rooms to add

>B number of large rooms to add



7efining the GoalsGoal (3 $he e"pansion should include appro"imatel. ) small

conference roomsGoal ?3 $he e"pansion should include appro"imatel. (,

medium conference rooms

Goal >3 $he e"pansion should include appro"imatel. () largeconference rooms

Goal



7efining the Goal ;onstraints

8mall RoomsX d d

1 1 1 5+ =

+

Medium Rooms

X d d2+ =

+

2 2 10

Large Rooms

X d d3 3 3+ = +

15

d di i

+, 0

%here



7efining the Goal ;onstraints&cont1d'

$otal 4"pansion

400X X X d d1 3+ + + = +750 1 050 25 0002 4 4, ,

$otal ;ost &in RM(!,,,s'

18X X X d d1 3+ + + = +33 4515 1 0002 5 5. ,

d di i

+, 0

%here


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Slide

GP Objective Cunctions

$here are numerous objective functions%e could formulate for a GP problem

Minimize the sum of the deviations3

MIN ( )d di ii

++

Problem3 $he deviations measure different things! so %hat does

this objective representD

&choice ('



Slide

GP Objective Cunctions &cont1d'

Minimize the sum of percentagedeviations

MIN

%heretirepresents the target value of goal i( )1

td d

i

i i

i

++ &choice ?'



Slide


Problem3

8uppose the first goal is underachieved

b. ( small room and the fifth goal is

overachieved b. RM?,!,,,

2e underachieve goal ( b. (E)B?,F

2e overachieve goal ) b. ?,!,,,E(!,,,!,,,B ?F



Slide


$his implies being RM?,!,,, overbudget is just as undesirable ashaving one too fe% small rooms

Is this trueD Onl. the decision ma#ercan sa. for sure



Slide


2eights can be used in the previous objectives toallo% the decision ma#er indicate desirable vsundesirable deviations the relative importanceof various goals



Slide


Minimize the %eighted sum of deviations

MIN

( )w d w d i i i ii + +

+

Minimize the %eighted sum of F deviationsMIN ( )

1

t w d w d

i

i i i i

i

+ ++

&choice >'

&choice


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7efining the Objective Assume

It is undesirable to underachieve an. of the firstthree room goals

It is undesirable to overachieve or underachieve the?)!,,, s9 meter e"pansion goal

It is undesirable to overachieve the RM(!,,,!,,,total cost goal


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7efining the Objective

+

+

+

+

+++++ 55

4

4

4

4

3

3

2

2

1

1 d000,000,1

wd

000,25

wd

000,25

wd

15

wd

10

wd

5

w:MIN

Initiall.! %e %ill assume all the above%eights e9ual (


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reatif ;ompan. emplo.s s#illed artisans toproduce cla. bo%ls and mugs %ith authenticMala.sian designs and colors $he t%oprimar. resources used b. the compan. arespecial potter. cla. and s#illed labor

GP3 4AMPL4 ?


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$he compan. desires to #no% ho% man.bo%ls and mugs to produce each da. toma"imize profit $he t%o products havethe follo%ing resource re9uirements forproduction and profit per item produced3

4AMPL4 ?


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Resource re9uirementProducts Labor

&hrEunit'

;la.

gEunit'

Profit

&RMEunit'5o%l ( < ),

$here are


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"(B number of bo%ls to produce

"?B number of mugs to produce

Objective function3 to ma"imize total profit

7ecision variables3

Ma"imize H B


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$he ;omplete LP Model

Ma"imize H B


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Goal Programming

Goal programming ma. be used to solvelinear programs %ith multipleobjectives! %ith each objectivevie%ed as a JgoalJ

In goal programming! diIand diK!deviation variables! are the amounts atargeted goal i is overachieved or

underachieved! respectivel.


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Goal Programming

$he goals themselves are added to theconstraint set %ith diIand diKactingas the surplus and slac# variables

One approach to goal programming is tosatisf. goals in a priorit. se9uence8econdKpriorit. goals are pursued%ithout reducing the firstKpriorit.

goals! etc


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Goal Programming

Cor each priorit. level! the objectivefunction is to minimize the &%eighted'sum of the goal deviations

Previous JoptimalJ achievements of goalsare added to the constraint set so thatthe. are not degraded %hile tr.ing toachieve lesser priorit. goals


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Goal Programming Approach

8tep (3

7ecide the priorit. level of each goal

8tep ?3

7ecide the %eight on each goal If a priorit.level has more than one goal! for each goal idecide the %eight! %i ! to be placed on the

deviation&s'! diI

andEor diK

! from the goal


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8tep >3

8et up the initial linear programMin %(d( %?d?K

st Cunctional ;onstraints!and Goal ;onstraints

8tep


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8tep )3 8et up the ne% linear program

;onsider the ne"tKlo%er priorit. level goalsand formulate a ne% objective function

based on these goals Add a constraintre9uiring the achievement of the ne"tKhigher priorit. level goals to be maintained


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8tep )3 cont

$he ne% linear program might be3

Min %>d>I %


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Recall reatif ;ompan. Problem)!$imi*e + , 40$1 50$2

Sue%t to

40221

+ xx

12034 21 + xx0, 21 xx

No%! let us suppose that instead of having one objective! the compan.has several objectives that are listed belo% in order of importance


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Alter the problem3

$o avoid la.offs! the compan. do not%ant to use fe%er than


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Alter the problem3

5ecause the cla. must be stored in aspecial place so that it does not dr. out!the compan. prefers not to #eep morethan (?, #g on hand each da. &priorit.

>'5ecause high overhead costs results %hen

the plant is #ept open past normal

hours! the compan. %ould li#e tominimize the amount of overtime&priorit.


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Alter the problem3$o avoid la.offs! the compan. do not %ant to use

fe%er than


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Alter the problem3

$he compan. %ould li#e to achieve a satisfactor. profit

of RM (!*,, per da. &priorit. ?'

Ma"imize H B


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Al h bl


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Alter the problem3

5ecause high overhead costs results %hen the plant is #ept

open past normal hours! the compan. %ould li#e tominimize the amount of overtime &priorit.


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38 2003 Thomson 2003 Thomson/South-Western/South-Western Slide

$he ;omplete GP ModelMinimize

8ubject to

),,,( 1321++

dddd

402 1121 =++ +

ddxx

16005040 2221 =++ +

ddxx

120343321

=++ +

ddxx

402 1121 =++ +

ddxx

$h ; l GP M d l


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$he ;omplete GP ModelMinimize

8ubject to

),,,( 1321++

dddd

402 1121 =++ +

ddxx

16005040 2221 =++ + ddxx

12034 3321 =++ +

ddxx

0,,,,,,, 33221121 +++

ddddddxx


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7emonstration :sing 4"cel 8olverand 78 for 2indo%s

;omments About GP


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;omments About GP

GP involves ma#ing tradeKoffs among

the goals until the most satisf.ing

solution is found

GP objective function values should notbe compared because the %eights are

changed in each iteration ;ompare

the solutions

;omments About GP


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;omments About GP

An arbitraril. large %eight %ill

effectivel. change a soft constraint

to a hard constraint

6ard constraints can be place ondeviational variables

8 f G l P i


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8ummar. of Goal Programming

( Identif. the decision variablesin the problem

? Identif. an. hard constraints

in the problem and formulate them inthe usual %a.

> 8tate the goals of the problemalong %ith their target values

8 f G l P i


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= Cormulate an objective that

penalizes the undesirable deviations+ Identif. appropriate %eights forthe objective

8olve the problem(, Inspect the solution to theproblem If the solution is unacceptable!

return to step + and revise the %eightsas needed

C d f th ht


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Cood of thought

One cannot escape the feeling thatthese mathematical formulas have anindependent e"istence and an

intelligence of their o%n! that the. are%iser than %e are! %iser even thantheir discoverers

201502271602484 goal programming.ppt

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