Chapter 4 ITIS

7/27/2019 Chapter 4 ITIS

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Chapter 4 -

Optimization andMathematical

Programming

2002 South-Western/Thomson Learning


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10/10/201310/10/2013 DIPANJAN CHATTERJEE ITIS 1P97FALL 2010

Startron Data

Computer Model

Desktop Server Notebook

Prof i t per unit $75 $145 $125

Assembly

t ime (hrs ) 0.5 0.75 1.5 12

Software

ins tal lat ion (hrs ) 0.25 0.4 0.3 3

Test ing (hrs ) 1 1.5 1 20

Packaging (hrs) 0.1 0.1 0.2 2

Number of

Workers

ANDeach type should be at least 20% of total production


3/40


Startrons Production

Planning Problem

Object ive:Maxim ize prof i t =75DT +145SV +125NB

Subject to:

0.5DT +0.75SV +1.5NB 96 (avai lable assembly hou rs)0.25DT +0.4SV +0.3NB 24 (avai lable so ftware instal lat ion hou rs)1DT +1.5SV +1NB 160 (avai lable test ing hou rs)0.1DT +0.15SV +0.2NB 16 (avai lable packaging hou rs)0.8DT- 0.2SV- 0.2NB 0 (each typ e sho uld accoun t for at

least 20% of the total produ ct ion )

DT 0; SV 0; NB 0 (Non negat ivi ty cons traint)


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The Modeling Process

for Optimization Studies

Optimizat ion:the name of a fami ly of tools

designed to help solve prob lems invo lv ing

l im i ted resources d ist r ibuted among var iousact iv i t ies to opt im ize a measu rable goal

A l im i ted quant i ty of resou rces is avai lable

The resources are used in the product ion o f

produc ts or serv ices

There are two or m ore possib le solut ions

Each act ivi ty y ields a return related to the goal

The al location o f resources is cons trained


5/40


Linear Programming

Output-m ix prob lems:how much o f wh ich

ou tputs (product or serv ice m ix) should be

planned to best ob tain the ob ject ive (i .e.,maximum prof i t ) so (con strained) resou rces

can be approp riately al loc ated

Blending prob lems:what is the best b lend

of available (cons trained) components that

w i l l ach ieve the ob ject ive (i .e., low cost)

whi le meet ing speci f icat ion


6/40


Formulating the Linear

Programming Model

Al l l inear programm ing(LP) models havethese s ix components :

Decis ion Variables

Object ive Func t ion

Prof i t or Cost Coeff ic ients

Constraints Cons traint Coeff ic ients

Righ t-Hand -Side Constants


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The Blending Problem:

Minimization Problem

A paint can be prod uced us ing two ingredients

Alpha and Beta

Both ingredients contribute towards the paintsquality, determined by its granularityand density

One canno t com prom ise on the paint qual ity as

determ ined by the minim um levels of densi ty and

granular i ty

You w i l l need to m inimize the cost of ing redients

whi le at the same time achieving th e desired quali ty


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The Blending Problem

Decisio n Variablesx1 = Quanti ty of Alpha to be inclu ded, in oun ces,

x2 = Quanti ty of Beta to be includ ed, in ounces

Constra ints:

Each d rum mus t have a granular i ty rat ing of 300

One ounce of x1inc reases the granulari ty by 1 unitOne ounce of x2increases the granular i ty by 1 uni t

Each d rum must h ave a dens ity rat ing o f 250 un itsOne ounce of x1increases the densi ty b y 3 un i ts

Oneounce of x2incr eases the density b y 0 units (no effect on

densi ty)

Costs

x1 = $45.00 per ounc ex2 = $12.00 per ounc e


9/40


The Blending Problem: Diskote

Minim ize z =45x1 +12x2

Subject to:

1x1 + 1x2 300 (granu lar i ty cons traint)3x1 + 0x2 250 (dens i ty cons traint)x1, x2 0 (nonnegat iv i ty cons traint)


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Assumptions of LP

Certainty

Linear Object ive Funct ion

Linear Cons traints : Addi t iv i ty

Independence

Proport ional i ty

Nonnegat iv i ty

Divis ib i l i ty


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10/10/201310/10/2013 DIPANJAN CHATTERJEE ITIS 1P97

FALL 2010

The Graphical Method

Maxim ize z =300x1 +250x2

Subject to:

2x1

+ 1x2 40 (labor con straint)

1x1 + 3x2 45 (machine t ime con straint)1x1 + 0x2 12 (market ing constraint)x1, x2 0 (non negat ivi ty constraint)

The Outpu t-Mix Problem


12/40


FALL 2010



Subject to:

2x1




x1

Product ion of leather chairs

| | | |10 20 30 40 45

x2

Pro

duc

tion

offabr

ic

cha

irs 40

30

20

10

0

15

A

B

C

DE

F

G

O

Feasible A rea


13/40


FALL 2010


Maxim ize z = 300x1 + 250x2

Subject to:

2x1

+ 1x2 40 (labor constraint)

1x1 + 3x2 45 (machine t ime cons traint)1x1 + 0x2 12 (market ing c ons traint)x1, x2 0 (non negat ivi ty constraint)


x1


| | | |10 20 30 40 45

x2

Pro

duc

tion

offabr

ic

cha

irs 40

30

20

10

0

15

A

B

C

DE

F

G

O

Feasible A rea

Solut ion Total Prof i t

Poin t (Corn er) Coo rdin ates 300x1 +250x2 = z

O (0, 0) 300 (0) + 250 (0) = 0

C (0, 15) 300 (0) + 250 (15) = $3,750

E (12, 0) 300 (12) + 250 (0) = $3,600

G (12, 11) 300 (12) + 250 (11) = $6,350


14/40


FALL 2010


Maxim ize z = 300x1 + 250x2

Subject to:

2x1

+ 1x2 40 (labor constraint)

1x1 + 3x2 45 (machine t ime cons traint)1x1 + 0x2 12 (market ing c ons traint)x1, x2 0 (non negat ivi ty constraint)


x1


| | | |10 20 30 40 45

x2

Pro

duc

tion

offabr

ic

cha

irs 40

30

20

10

0

15

A

B

C

DE

F

G

O

Feasible A rea

Solut ion Total Prof i t

Poin t (Corn er) Coo rdin ates 300x1 + 250x2 = z

O (0, 0) 300 (0) + 250 (0) = 0

C (0, 15) 300 (0) + 250 (15) = $3,750

E (12, 0) 300 (12) + 250 (0) = $3,600

G (12, 11) 300 (12) + 250 (11) = $6,350

Maximum


15/40


FALL 2010



Subject to:

2x1





16/40


FALL 2010



Subject to:

2x1




x1


| | |5 10 15

x2

Pro

duc

tion

offabr

ic

cha

irs

15

10

5

0

12

C

E

G

O


17/40


FALL 2010



Subject to:

2x1




x1


| | |5 10 15

x2

Pro

duc

tion

offabr

ic

cha

irs

15

10

5

0

12

C

E

G

O

M

N

z = 300x1 + 250x2 = 1,500


18/40


FALL 2010



Subject to:

2x1




x1


| | |5 10 15

x2

Pro

duc

tion

offabr

ic

cha

irs

15

10

5

0

12

C

E

G

O

M

N

z = 300x1 + 250x2 = 1,500

G

V

W

z = 6,350


19/40


FALL 2010

Slack and Surplus Variables


Subject to:

2x1 + 1x2 + s1 =40 (labor con straint)

1x1 + 3x2 + s2 =45 (machine t ime con straint)1x1 + 0x2 + s3 =12 (market ing constraint)

x1, x2 0 (non negat ivi ty con straint)

The Outpu t-Mix Prob lem


20/40


FALL 2010



Subject to:





Slack variables have been added to each cons traint

At the opt imal solu t ion o f x1 =12, x2 =11, the slackvariables have the fol low ing values:

s1 =5 s2 =0 s3 =0


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FALL 2010



Subject to:





Slack variables have been added to each cons traint

At the opt imal solu t ion o f x1 =12, x2 =11, the slackvariables have the fol low ing values:

s1 =5 s2 =0 s3 =0

There are 5 unused hours

of labo r capaci ty

The avai lable machinet ime is ful ly u t i l ized

The market ing constra int

is exact ly m et

Analysis


22/40


FALL 2010

Sensitivity Analysis:

Objective Function

When does a pr ice (pro f i t )

decrease of an output jus t i fyd iscont inu ing or reduc ing the

quant i ty?

How much o f a pr ice (pro f i t)

inc rease jus t i f ies increasing the

quant i ty in the op t imal so lut ion?


23/40


FALL 2010

Changes in the Coefficients


Subject to:

2x1

+ 1x2

40 (labor con straint)1x1 + 3x2 45 (machine t ime con straint)1x1 + 0x2 12 (market ing constraint)x1, x2 0 (non negat ivi ty constraint)



24/40


FALL 2010


Subject to:

2x1

+ 1x2




x1


| | | |20 45

x2

Pro

duc

tion

offabr

ic

cha

irs 40

0

15


25/40


FALL 2010


Subject to:

2x1

+ 1x2




x1


| | | |20 45

x2

Pro

duc

tion

offabr

ic

cha

irs 40

0

15

A

B

C

DE

F

G

O


26/40


FALL 2010


Subject to:

2x1

+ 1x2




x1


| | | |20 45

x2

Pro

duc

tion

offabr

ic

cha

irs 40

0

15

A

B

C

DE

F

G

O

K

L


27/40


FALL 2010


Subject to:

2x1

+ 1x2




x1


| | | |20 45

x2

Pro

duc

tion

offabr

ic

cha

irs 40

0

15

A

B

C

DE

F

G

O

K

L

M

N


28/40


FALL 2010


Subject to:

2x1

+ 1x2




x1


| | | |20 45

x2

Pro

duc

tion

offabric

cha

irs 40

0

15

A

B

C

DE

F

G

O

K

L

M

N

Changes in the coeff ic ients change

the slope of the ob ject ive func t ion

This w i l l change the value of the

solut io n, bu t not n ecessar ily the

opt imal point


29/40


FALL 2010


Subject to:

2x1

+ 1x2




x1


| | | |20 45

x2

Pro

duc

tion

offabric

cha

irs 40

0

15

A

B

C

DE

F

G

O

K

L

M

N

Changes in the coeff ic ients change

the slope of the ob ject ive func t ion

This w i l l change the value of the

solut io n, bu t not n ecessar ily the

opt imal point

Upper and Lower Lim i ts

There are l im its to the extent

coeff ic ients can be changed

before the opt imal so lut ion

poin t changes


30/40


FALL 2010

Changes in Right-Hand Side Values


Subject to:

3x1 + 5x2 152x1 + 1x2 80x1 + 1x2 2x1, x2 0


31/40


FALL 2010


x1| |4 5

x2

9

8

2

0

Exhibit 4.33Maxim ize z =3x1 +4x2

Subject to:

3x1 + 5x2 152x1 + 1x2 80x1 + 1x2 2x1, x2 0


32/40


FALL 2010


x1| |4 5

x2

9

8

2

0

Exhibit 4.33

1

2

3

A B

C

D

F

Isoprof i t

l ine


Subject to:

3x1 + 5x2 152x1 + 1x2 80x1 + 1x2 2x1, x2 0


33/40


FALL 2010


x1| |4 5

x2

9

8

2

0

Exhibit 4.33

1

2

3

A B

C

D

F

Isoprof i t

l ine G

E

S

T


Subject to:

3x1 + 5x2 152x1 + 1x2 80x1 + 1x2 2x1, x2 0


34/40


FALL 2010


Subject to:

3x1 + 5x2 152x1 + 1x2 80x1 + 1x2 2x1, x2 0


x1| |4 5

x2

9

8

2

0

Exhibit 4.33

1

2

3

A B

C

D

F

Isoprof i t

l ine G

E

S

T

Changin g the RHS value shif ts

Cons traint 2 fur ther from the

or ig in .

This resul ts in a new opt imal

so lut ion at G


35/40


FALL 2010


Subject to:

3x1 + 5x2 152x1 + 1x2 80x1 + 1x2 2x1, x2 0


x1| |4 5

x2

9

8

2

0

Exhibit 4.33

1

2

3

A B

C

D

F

Isoprof i t

l ine G

E

S

T

Changin g the RHS value shif ts

Cons traint 2 fur ther from the

or ig in .

This resul ts in a new opt imal

so lut ion at G

Upper and Lower Lim i ts

There are l im its to the extent

RHS values can be changed

before other cons traints

become l imi t ing factors


36/40


FALL 2010

Integer Programming

A form of programm ing where some

or al l of the decis ion var iables mus t

be in tegers

It al low s us to solve prob lems

where indiv is ib i li ty is p resent

Prov ides solut ion opt ions toprob lems not o therwise solvable


37/40


FALL 2010

Integer Programming

Dif fers from no rmal LP in several ways :

Ind iv is ibi l i ty resul ts in add it ional

constra ints

Solut ion is general ly infer ior to theop t imal LP so lut ion

Complex so lut ion techn iques Gomory Cu tt ing Plane Method

Complete enumeration

Branch and bound

Typ ical ly a f in i te number of so lut ions


38/40


FALL 2010

Integer Programming

Integer Prog ramm ing Models:

A ll-In teger Model

All decis ion var iables mus t be integers

Mixed In teger Model Only som e of the decis ion var iables are

integers Zero -One In teger Model

Useful for modeling yes/no decisions


39/40


FALL 2010

Southern General Hospital

Decis ion variables:

x1 = Number of beds to add to the materni ty ward

x2 = Number of beds to add to the cardiac ward

Constra ints:

Only 10 beds to tal can be added

$210,000 to tal budget At least two of the beds must be in materni ty


40/40


FALL 2010

Southern General Hospital


Subject to :

1x1 + 1x2 10 (total bed l im itat ion )$20,000x1 + $24,600x2 $210,000 (budget con stra int)1x1 2 (at least two beds in

materni ty ward)

x1, x2 0 (nonnegativi tyconstra int)

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Chapter 4 ITIS

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