lec 9 OT problem formulation.ppt

8/9/2019 lec 9 OT problem formulation.ppt

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Problem Formulation

Dr. Nasir M Mirza

Optimization TechniquesOptimization Techniques

Email: [email protected]


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Optimal Problem Formulation

• it is almost impossible to apply a single formulationprocedure for all engineering design problems.

• Since the objective in a design problem vary fromproduct to product, different techniques need to be

used in different problems.• The purpose of the formulation procedure is to create a

mathematical model of the optimal design problem,

which then can be solved.

• Since an optimization algorithm accepts anoptimization problem in a particular format, every

optimal design problem must be formulated in that

format.


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Hierarchy of Optimal DesinProcess! the "esiner nee"s tochoose the important

"esin #ariables associate" $ith the"esin problem.

! %he formulation ofoptimal "esin problems

in#ol#es otherconsi"erations& such as! constraints& ob'ecti#e

function& an" #ariableboun"s.

! (s sho$n in the )ure&there is usually ahierarchy in the optimal"esin process*

! +e "iscuss all theseaspects in the follo$insubsections.

Need for Optimization

Choose design variables

Formulate constraints

Formulate objective function

Setup variable bounds

Choose an optimization algo

Obtain solutions


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Design variables

! %he formulation of an optimization problem beins $ithsuestion of "esin #ariables& $hich are #arie" "urin theoptimization process.

! ( "esin problem ,usually in#ol#es many "esin parameters&of $hich some are hihly sensiti#e to the proper $or-in ofthe "esin. %hese parameters are calle" "esin #ariables inthe parlance of optimization proce"ures.

! Other not so important/ "esin parameters usually remain)0e" or #ary in relation to the "esin #ariables.

! %here is no rii" ui"eline to choose the parameters $hichmay be important in a problem& because one parameter maybe more important $ith respect to minimizin the o#erallcost of the "esin& $hile it may be insini)cant $ith respect

to ma0imizin the life of the pro"uct.! %hus& the choice of the important parameters larely

"epen"s on the user.! Ho$e#er& the eciency an" spee" of optimization alorithms

"epen" on the number of chosen "esin #ariables.


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Constraints

! Ha#in chosen the "esin #ariables& the ne0t tas- is to i"entifythe constraints associate" $ith the optimization problem.! %he constraints represent some functional relationships amon

the "esin #ariables an" other "esin parameters satisfyincertain physical phenomenon an" certain resource limitations.

! 2ome of these consi"erations re3uire that the "esin remain in

static or "ynamic e3uilibrium. 4n many mechanical an" ci#ilenineerin problems& the constraints are formulate" to satisfystress an" "e5ection limitations.

! Often& a component nee"s to be "esine" in such a $ay that itcan be place" insi"e a )0e" housin& thereby restrictin thesize of the component.

! %here is& ho$e#er& no uni3ue $ay to formulate a constraint inall problems.

! %he nature an" number of constraints to be inclu"e" in theformulation "epen" on the user.


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6onstraints! For e0ample& a mechanical enineerin component "esin problem may

in#ol#e a constraint to restrain the ma0imum stress "e#elope" any$here inthe component to the strenth of the material.

! 4n an irreular,shape" component& there may not e0ist an e0actmathematical e0pression for the ma0imum stress "e#elope" in thecomponent.

!( )nite element simulation soft$are may be necessary

! %here are t$o types of constraints: an ine3uality type or of an e3uality type.

! 4ne3uality constraints state that the functional relationships amon "esin#ariables are either reater than& smaller than& or e3ual to& a resource #alue.

! For e0ample& the stress (a( x)) "e#elope" any$here in a component must besmaller than or e3ual to the allo$able strenth 2allo$able/ of the material.

Mathematically&

a(x) 78 2allo$able


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6onstraints

! E3uality constraints state that the functional relationshipsshoul" e0actly match a resource #alue.

! For e0ample& a constraint may re3uire that the "e5ection

δ0// of a point in the component must be e0actly e3ual to

9 mm& or mathematically& δ0/ 8 5.

! E3uality constraints are usually more "icult to han"le

an"& therefore& nee" to be a#oi"e" $here#er possible.

! 4f the functional relationships of e3uality constraints are

simpler& it may be possible to re"uce the number of

"esin #ariables by usin the e3uality constraints.

! 4n such a case& the e3uality constraints re"uce the

comple0ity of the problem& thereby ma-in it easier for

the optimization alorithms to sol#e the problem.


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6onstraints

! Fortunately& in many enineerin "esin optimizationproblems& it may be possible to rela0 an e3ualityconstraint by inclu"in t$o ine3uality constraints.

! %he δ0/ 8 5 "e5ection e3uality constraint can bereplace" by t$o constraints:

δ0/ ;&

δ 0/ 7


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Objective function

! %he thir" tas- in the formulation proce"ure is to )n" theob'ecti#e function in terms of the "esin #ariables an" otherproblem parameters.

! %he common enineerin ob'ecti#es in#ol#e minimization ofo#erall cost of manufacturin& or minimization of o#erall$eiht of a component*

! Most of the abo#e ob'ecti#es can be 3uanti)e" e0presse" ina mathematical form/.

! %here are some ob'ecti#es that may not be 3uanti)e" easily.! For e0ample& the aesthetic aspect of a "esin& ri"e

characteristics of a car suspension "esin& an" reliability of a"esin are important ob'ecti#es that one may be intereste"in ma0imizin in a "esin&

! =ut the e0act mathematical formulation may not be possible.4n such a case& usually an appro0imatin mathematicale0pression is use".


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Ob'ecti#e Function

! %he ob'ecti#e function can be of t$o types.! Either the ob'ecti#e function is to be ma0imize" or it has to beminimize".

! >nfortunately& the optimization alorithms are usually $ritteneither for minimization problems or for ma0imizationproblems an" not for both.

! (lthouh in some alorithms& some minor structural chanes$oul" enable to perform either minimization or ma0imization&this re3uires e0tensi#e -no$le"e of the alorithm.

! Moreo#er& if an optimization soft$are is use" for thesimulation& the mo"i)e" soft$are nee"s to be compile" beforeit can be use" for the simulation.

! Fortunately& the duality principle helps by allo$in the samealorithm to be use" for minimization or ma0imization $ith aminor chane in the ob'ecti#e function instea" of a chane inthe entire alorithm.


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Ob'ecti#e Function

! 4f the alorithm is"e#elope" for sol#ina minimizationproblem& it can alsobe use" to sol#e ama0imization

problem by simplymultiplyin theob'ecti#e function by,1 an" #ice #ersa.

! For e0ample& consi"erthe ma0imization of

the sinle,#ariablefunction f( x) 8 x 2 (1 , x) sho$n by a soli"line in Fiure 1.?.


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Ob'ecti#e Function

• For eample, consider themaimization of the single!

variable function f( x) " x 2 (1 !

x) shown by a solid line in

Figure #.$.

• The maimum point happens

to be at x* " %.&&'. The

duality principle suggests that

the above problem is

equivalent to minimizing thefunction f(x) " -x 2 (1- x),

which is shown by a dashed

line in Figure #.$.



Ob'ecti#e Function

• The figure shows that theminimum point of the function

f(x) is also at x* " %.&&'. Thus,

the optimum solution remains

the same.

• (ut once we obtain the

optimum solution by

minimizing the function f(x),

we need to calculate theoptimal function value of the

original function f( x) by

multiplying f(x) by !#.



Ob'ecti#e Function

• )fter the above four tas*s are completed, the optimization problemcan be mathematically written in a special format, *nown as

nonlinear programming +-/ format. 0enoting the design variables

as a column vector1 x " (x 1, x 2, ... , x N )T , the objective function as

a scalar quantity f( x), K inequality constraints as g j(x) 2 %, and K

equality constraints as hk(x) " %, we write the - problem1



Developing Optimization Problem

! Formulation of an optimization problem in#ol#esta-in statements& "e)nin eneral oals an"

re3uirements of a i#en acti#ity& an" transcribin

them into a series of $ell,"e)ne" mathematical

statements.

! More precisely& the formulation of an optimization

problem in#ol#es:

1. Selecting one or more optimization variables,

2. hoosing an ob!ective function, and". #dentifying a set of constraints.

! %he ob'ecti#e function an" the constraints must all

be functions of one or more optimization #ariables.

! %he follo$in e0amples illustrate the process.


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Example: Building Design

• To save energy costs for heating and cooling, an architect isconsidering designing a partially buried rectangular building.

• The total floor space needed is 20,000 m2.

• Plot size limits the building plan dimension to 50 m.

• It has already been decided that the ratio beteen the plandimensions must be e!ual to the golden ratio "#.$#%& and that

each story must be '.5 m high.

• The heating and cooling costs are estimated at (#00 per m2 of

the e)posed surface area of the building.• The oner has specified that the annual energy costs should not

e)ceed (225,000.

• *ormulate the problem of determining building dimensions to

minimize cost of e)cavation.


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Optimization variables• *rom the given data and *igure #.#, it is

easy to identify the folloing variables

associated ith the problem+

n = Number of stories;

d = Depth of building below ground;

h = Height of building above ground;

l = Length of building in plan;

w = Width of building in plan.



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Objective Function• The stated design obective is to

minimize e)cavation cost.

• -ssuming the cost of e)cavation to

be proportional to the volume ofe)cavation, the obective function

can be stated as follos+

Minimize f = dlw


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!onstraints• -ll optimization variables are not

independent. ince the height ofeach story is given, the number ofstories and the total height arerelated to each other as follos+

"d # h$%n = &.'

• -lso, the re!uirement that the ratio beteen the plan dimensions must be e!ual to the golden ratio ma/esthe to plan dimensions dependenton each other as follos+

l =(.)(*w


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!onstraints

• The total floor space is e!ual to thearea per floor multiplied by the

number of stories. Thus, the floor

space re!uirement can be e)pressed

as follos+

new > 20,000

• The lot size places the folloinglimits on the plan dimensions+

l ≤ 50 ; w≤ 50


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!onstraints• The energy cost is proportional to the

e)posed building area hich includes

the areas of the e)posed sides and the

roof.• Thus, the energy budget places the

folloing restriction on the design+

(++",hl + ,hw +lw$ ≤ 225,000

• )plicitly state that the design

variables cannot be negative.

• l- w, h- d > 0 ; n > 1 must be an

integer.


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• The complete optimization problemcan be stated as follos+

• *ind (n, l, , h, d) in order to• Minimize f = dlw• Subject to:

(d+h)/n = 3.5

l = 1.618w

nlw ≥ 2: 20,000

l ≤ 50

w ≤ 50

100(2hl +2hw +lw) ≤ 225,000

n ≥ 1

l, W, h, d ≥ 0


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Example: Plant Operation

! ( tire manufacturin plant has the ability topro"uce both ra"ial an" bias,ply automobile tires:Durin the upcomin summer months& they ha#econtracts to "eli#er tires as follo$s.

Date a"ial tires =ias,ply tires

Aune BC 9CCC BCCC

Auly B1


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! %he plant has t$o types of machines& ol"machines an" blac- machines& $ith appropriatemol"s to pro"uce these tires.

! %he follo$in pro"uction hours are a#ailable"urin the summer months:

month ol" machine =lac- machine

Aune CC 19CC

Auly BCC ;CC

(uust 1CCC BCC


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! %he pro"uction rates for each machine type an" tirecombination& in terms of hours per tire& are as follo$s

%ype ol" machine =lac- machine

a"ial C.19 C.1<

=ias,ply .1? C.1;


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• The labor costs of producing tires are (#0.00 per operating hour,regardless of hich machine type is being used or hich tire is being produced.

• The material costs for radial tires are (5.25 per tire and those for bias1 ply tires are (.#5 per tire.

• *inishing, pac/ing and shipping costs are (0.0 per tire.

• The e)cess tires are carried over into the ne)t month but are subectedto an inventory carrying charge of (0.#5 per tire.

• 3holesale prices have been set at (20 per tire for radials and (#5 per

tire for bias1ply.• 4o should the production be scheduled in order to meet the delivery

re!uirements hile ma)imizing profit for the company during thethree1month period


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Optimization ariables

*rom the problem statement, it is clear that the only variables that the productionmanager has control over are the number and type of tires produced on eachmachine type during a given month. Thus the optimization variables are asfollos+

)# 6 7umber of radial tires produced in 8une on the gold machines

)2 6 7umber of radial tires produced in 8uly on the gold machines

)' = 7umber of radial tires produced in -ugust on the gold machines

) 6 7umber of bias1ply tires produced in 8une on the gold machines

)5 = 7umber of bias1ply tires produced in 8uly on the gold machines

)$ 6 7umber of bias1ply tires produced in -ugust on the gold machines

)9 = 7umber of radial tires produced in 8une on the blac/ machines

)% 6 7umber of radial tires produced in 8uly on the blac/ machines

): 6 7umber of radial tires produced in -ugust on the blac/ machines

)#0 6 7umber of bias1ply tires produced in 8une on the blac/ machines

)## = 7umber of bias1ply tires produced in 8uly on the blac/ machines

)#2 6 7umber of bias1ply tires produced in -ugust on the blac/ machines


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Objective Function:

! %he ob'ecti#e of the company is to ma0imize pro)t. %he pro)t is e3ual tothe total re#enue from sales minus all costs associate" $ith thepro"uction& storin& an" shippin.

! $evenue from sales%

R & '2(x1 x2 x" x x* x+) '15(x x5 0< 01C 011 x12)

! -aterial costs%

M & '5.25(x1 x2 x" x x* x+) '.15(x 09 0< 01C011 x12)

! /abor costs%

L 8 G1C C.1901 x2 x") .10(x x* x+) .12(x x5 x0)

.1(x1 011 x12)

! inishing, pac3ing, and shipping costs%

F & '.1(x# x2 x" x x5 0< x x* x+ 01C 011 x12)


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• The inventory1carrying charges are a little difficult toformulate. -ssuming no inventory is carried into or out of thethree summer months, e can determine the e)cess tires

produced as follos+

• )cess tires produced by 8une '0

)#6 ";#


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• =y assumption, there are no e)cess tires left by the end of -ugust'#. -t (0.#5 per tire, the total inventory carrying charges are asfollos+

• Inventory cost+

I> 6 (0.#5"")#


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!onstraints• In a given month, the company must meet the deliverycontracts. @uring 8uly and -ugust, the company can also usethe e)cess inventory to meet the demand. Thus, the deliverycontract constraints can be ritten as follos+

x1 + x7 ≥ 5.000 x4 + x10 ≥ 3,000

x1 + x2 + x7 + x8 ≥ 11,000

x4 + x5 + x10 + x11 ≥ 6,000

x1 + x2 + x3 + x7 + x8 + x9 = 15,000

x4 + x5 + x8 + x10 + x11 + x12 = 11,000• 7ote that the last to constraints are e)pressed as e!ualities to

stay consistent ith the assumption that no inventory is carriedinto eptember.


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!onstraints• The production hours for each machine are limited. Asing +the

time that it ta/es to produce a given tire on a given machine type,these limitations can be e)pressed as follos+

0.15x1 + 0.12x4 ≤ 700

0.15x2 + 0.12x5 ≤ 300

0.15x3 + 0.12x6 ≤ 1,000

0.16x7 + 0.14x10 ≤ 1,500

0.16x8 + 0.l4x11 ≤ 4000.16x9 + 0.14x12 ≤ 300

• The only other re!uirement is that all optimization variablesmust be positive.


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#$e com%&ete o%timiation %'ob&em can be (tate) a( fo&&ow(:

Fin) "/(- /,- * * * , x12 in o')e' to

Maximie: f = &-0'+ # (,.''/( # (,.0/, # (,.*'/& # *.1'/2 # 1.(/' # 1.,'/)# (,.2'/0 # (,.)/* # (,.0'-/1 # *.0'/(+ # *.1/(( # 1.+'/(,

Subject to con(t'aint(:

/( # /0 5 '.+++

/2 # /(+ 5 &-+++ /( # /, # /0 # /* 5 ((-+++

/2 # /' # /(+ # /(( 5 )-+++

/( # /, # /& # /0 # /* # /1 = ('-+++

/2 # /' # /* # /(+ # /(( # /(, = ((-+++

+.('/( # +.(,/2 6 0+++.('/, # +.(,/' 6 &++

+.('/& # +.(,/) 6 (-+++

+.()/0 # +.(2/(+ 6 (-'++

+.()/* # +.l2/(( 6 2++

+.()/1 # +.(2/(, 6 &++


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Example : Data Fitting

• *inding the best functionthat fits a given set ofdata can be formulated as

an optimization problem.

• -s an e)ample, considerfitting a surface to thedata given in the

folloing table+

7oint / 8 zobserved

1 C 1 1.?<

? C.?9 1 ?.1I

B C.9 1 C.<; C.9 1 1.?<

9 1.CC ? 1.J<

< 1.?9 ? 1.;B

1.9C ? 1.?IJ 1.9 ? C.


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• The form of the function is first chosen based on prior/noledge of the overall shape of the data surface.

• *or the e)ample data, consider the folloing general form+

z"9omputed$ = 9( / , # 9, 8, # 9& /8

• The goal no is to determine the best values of coefficients 9( -9, and 9& in order to minimize the sum of s!uares of error

beteen the computed z values and the observed ones.


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O%timiation a'iab&e(:Balues of coefficients c#, c2 , and c3

Objective Function:

Cinimize f 6 D E zobserved ")i , y ) 1 zcomputed ")i , y ) !2

Asing the given numerical data, the obective function can beritten as follos+

f 6 "#.2$ 1 c2 )2 < "2.#: 1 0.0$25c# 1 c2 1 0.25c3 )2 < . . .

< "#.$ F 4c1 F 4c2 F 4c3 )2 or

f 6 #%.9 1 32.8462c1 < 34.2656c12 1 65.58c2 < 96.75c1c2 < 84c22 1

43.425c3


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• The complete optimization problem can be stated as follos+• *ind "9( - 9, - and 9& $ in order to Cinimize+

f = (*.0 : &,.*2),9( # &2.,)')9(, : )'.'*9, # 1).0'9(9, # *29,,

: 2&.2,'9& #01.*0'9(9& # (,&9, 9& #2*.&0'9&,

• This e)ample represents a simple application from a ide field/non as Gegression -nalysis. *or more details refer to many

e)cellent boo/s on the ubect


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Te !tandard Form of an OptimizationProblem

• - large class of situations involving optimization can bee)pressed in the folloing form+

• *ind a vector of optimization variables, / = "/ ( - / , - . . . , / n $; inorder to Cinimize or Ca)imize an obective function,

f"/$ = f"


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"ultiple Objective Functions

• There can be more than one obective to be optimized.• *or e)ample, e may ant to ma)imize profit from an car thate are designing and at the same time minimize the possibilityof damage to the car during a collision.

• hese t8pes of problems are diffi9ult to handle because the

obective functions are often contradictory.• Then a((i-n wei-$t( to each obective function depending ontheir relative importance and then define a composite obectivefunction as a eighted sum of all these functions, as follos+

f "/$ = w( f ("/$ #w, f ,"/$ # . . .

here w( - w, - . . . are suitable eighting factors.• The success of the method clearly )e%en)( on a c&eve' c$oice

of t$e(e wei-$tin- facto'(*


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6lassi)cation of Optimization Problems

• The methods for solving the general form of the optimization problem tend to be 9omple/

• It re!uires considerable numeri9al effort.

• pecial, more efficient methods are available for certainspecials forms of the general problem.

• *or this purpose? the optimization problems are usuallyc&a((ifie) into t$e fo&&owin- t.%e(:

• An9onstrained 7roblems

• Linear 7rogramming "L7$ 7roblems

• @uadrati9 7rogramming "@7$ 7roblems• Nonlinear 7rogramming "NL7$ 7roblems


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>nconstraine" Problems

• These problems have an obCe9tive fun9tion but no 9onstraints.

• The data1fitting problem, presented in the first section, is ane)ample of an unconstrained optimization problem.

• The obCe9tive fun9tion must be nonlinear "because the

minimum of an unconstrained linear obective function isobviously infinity&.

• /'ob&em( wit$ (im%&e boun)( on optimization variables canoften be solved first as un9onstrained.

• -fter e)amining different options, one can pic/ a solution• that satisfies the bounds on the variables.


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Kinear Prorammin KP/ Problems

• If the obCe9tive fun9tion and all the 9onstraints are linear fun9tions of optimization variables, the problem is called a&inea' %'o-'ammin- %'ob&em.

• The tire plant management problem presented here is an

e)ample of a linear optimization problem.• -n efficient and robust algorithm, called the imple/ method ,

is available for solving these problems.


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Lua"ratic Prorammin LP/ Problems

• If the obCe9tive fun9tion is a >uadrati9 fun9tion and all9onstraint fun9tions are linear fun9tions of optimizationvariables the problem is called a ua)'atic %'o-'ammin-

%'ob&em*

• The portfolio management problem is an e)ample of a!uadratic optimization problem.

• It is possible to solve HP problems using e)tensions of themethods for P problems.


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#onlinear Programming $#%P& Problems

• The general constrained optimization problems- in whi9h oneor more fun9tions are nonlinear- are called non&inea'%'o-'ammin- problems.

• The bui&)in- )e(i-n %'ob&em, presented in here, is ane)ample of a general nonlinear optimization problem.


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%ypes of optimization metho"s

! Single4variable optimization algorithms! -ulti4variable optimization algorithms

! onstrained optimization algorithms

! Specialized optimization algorithms

! ontraditional optimization algorithms


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!ingle'variable optimization algoritms

• (ecause of their simplicity, single!variable optimizationtechniques will be discussed first.

• These algorithms provide a good understanding of theproperties of the minimum and maimum points in a function

and how optimization algorithms wor* iteratively to find theoptimum point in a problem.

• The algorithms are classified into two categories

• Direct methods and

•

gradient-based methods.• 0irect methods do not use any derivative information of the

objective function3 only objective function values are used to

guide the search process.


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Multi-variable optimization algorithms! ( number of alorithms for unconstraine"&

multi#ariable optimization problems $ill be

"iscusse".

! %hese alorithms "emonstrate ho$ the searchfor the optimum point proresses in multiple

"imensions. Depen"in on $hether the

ra"ient information is use" or not use"& these

alorithms are also classi)e" into "irect an"ra"ient,base" techni3ues.


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Constrained optimization algorithms! 6onstraine" optimization alorithms use the sinle

#ariable an" multi#ariable optimization alorithms

repeate"ly an" simultaneously maintain the search

eort insi"e the feasible search reion.! 2ince these alorithms are mostly use" in

enineerin optimization problems& the "iscussion

of these alorithms $ill be "one in "etail.


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Specialized optimization algorithms

! %here e0ist a number of structure" alorithms&$hich are i"eal for only a certain class ofoptimization problems.

! %$o of these alorithms , inteer proramminan" eometric prorammin,are often use" inenineerin "esin problems.

! 4nteer prorammin metho"s can sol#eoptimization problems $ith inteer "esin

#ariables. eometric prorammin metho"s sol#eoptimization problems $ith ob'ecti#e functionsan" constraints $ritten in a special form.


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Nontraditional optimizationalgorithms

! %here e0ist a number of other search an"optimization alorithms $hich are

comparati#ely ne$ an" are becomin popularin enineerin "esin optimization problems inthe recent past.

! %$o such alorithms: enetic alorithms an"

simulate" annealin, $ill be "iscusse".

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lec 9 OT problem formulation.ppt