formulation exercises version 7

8/9/2019 Formulation Exercises Version 7

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Formulation Exercises in LP



Exercises• ABC Pottery Company

• XYZ Fertilizer Company

• Investment Problem

• Investment Problem II

• ta!!in" Problem

• ta!!in" Problem#II

• $AX Function

• %etailer Problem

• Flo& $aximization

• Flo& $aximization#II

• Port!olio 'esi"n• Pro(ect $ana"ement

• Ballmart)A*



ABC Pottery Company

• Pro+uct mix problem # ABC Pottery Company

• ,o& many bo&ls an+ mu"s s-oul+ be pro+uce+ to maximize pro!its"iven labor an+ materials constraints.

• %esource Availability/ 01 -rs o! labor per +ay2 341 lbs o! clay

• Pro+uct resource re5uirements an+ unit pro!it/

%esource %e5uirements

Pro+uct Labor

)-r6unit*

Clay

)lb6unit*

Pro!it

)76unit*Bo&l 3 0 01

$u" 4 8 91



Problem 'e!inition

Resource 01 -rs o! labor per +ayAvailability: 341 lbs o! clay

Decision x3 : number o! bo&ls to pro+uce per +ay

Variables: x4 : number o! mu"s to pro+uce per +ay

Objective $aximize Z : 701x3 ; 791x4

Function: <-ere Z : pro!it per +ay

Resource 3x3 ; 4x4 ≤ 01 -ours o! labor

Constraints: 0x3 ; 8x4 ≤ 341 poun+s o! clay

Non-Negativity x3≥ 1= x4≥ 1

Constraints:



LP Formulation

maximize Z:701x3 ; 91x4

sub(ect to 3x3 ; 4x4≤ 01

0x4 ; 8x4≤ 341

x32 x4≥ 1



>rap-ical olution o! a $aximization $o+el )3 o! 34*

Coor+inate Axes

maximize Z:701x

3 ; 91x

4

sub(ect to

3x3 ; 4x4≤ 01 -ours o! labor

0x4 ; 8x4≤ 341 poun+s o! clay

x32 x4≥ 1

Figure 2.1 Coor+inates !or "rap-ical analysis




Labor Constraint


sub(ect to



x32 x4≥ 1

Figure 2.2 >rap- o! t-e labor constraint line




Labor Constraint Area

maximize Z:701x

3 ; 91x

4

sub(ect to



x32 x4≥ 1

Figure 2.3 ?-e labor constraint area




Clay Constraint Area

maximize Z:701x

3 ; 91x

4 sub(ect to



x32 x4≥1

Figure 2.4 ?-e constraint area !or clay




Bot- Constraints


sub(ect to



x32 x4≥ 1

Figure 2.5 >rap- o! bot- mo+el Constraints



>rap-ical olution o! a $aximization $o+el )@ o! 34*

Feasible olution Area


sub(ect to



x32 x4≥ 1

Figure 2.6 ?-e !easible solution area constraints



XYZ Fertilizer Company

• ?&o bran+s o! !ertilizer available # uper#"ro2 Crop#5uic• Fiel+ re5uires at least 3@ poun+s o! nitro"en an+ 40 poun+s o! p-osp-ate

• uper#"ro costs 7@ per ba"2 Crop#5uic 78 per ba"

• Problem / ,o& muc- o! eac- bran+ to purc-ase to minimize total cost o!

!ertilizer "iven !ollo&in" +ata .

C-emical Contribution

Bran+

itro"en

)lb6ba"*

P-osp-ate

)lb6ba"*

uper#"ro 4 0

Crop#5uicA 0 8



Problem 'e!inition

'ecision varIables

x3 : ba"s o! uper#"ro

x4 : ba"s o! Crop#5uic

?-e ob(ective !unction/

minimize Z : 7@x3

; 8x4

&-ere 7@x3 : cost o! ba"s o! uper#"ro

8x4 : cost o! ba"s o! Crop#5uic

$o+el constraints/

4x3 ; 0x4 ≥ 3@ lb )nitro"en constraint*

0x3 ; 8x4 ≥ 40 lb )p-osp-ate constraint*

x32 x4 ≥ 1 )non ne"ativity constraint*



$o+el Formulation an+ >rap-

Complete mo+el !ormulation/

minimize Z : 7@x3

; 8x4

sub(ect to

4x3 ; 0x4 ≥ 3@ lb o! nitro"en

0x3 ; 8x4 ≥ 40 lb o! p-osp-ate

x32 x4 ≥ 1

Figure 2.! Constraint lines !or !ertilizer mo+el



Feasible olution Area

minimize Z : 7@x3 ; 8x4

sub(ect to

4x3 ; 0x4 ≥ 3@ lb o! nitro"en


x32 x4 ≥ 1

Figure 2.15 Feasible solution area



Dptimal olution Point

minimize Z : 7@x3 ; 8x4

sub(ect to 4x3 ; 0x4 ≥ 3@ lb o! nitro"en


x32 x4 ≥ 1

Figure 2.16 ?-e optimal solution point



Investment Problem

$r Praban+- -i-ar is looin" at a 8 year plannin"

-orizon !or investin" %s311111 ,e -as t&o options/

• a* Invest in a savin"s account at 9 rate o! interest

&it- maturity perio+ o! 3 year

• b* Invest in a security Y at 34 rate o! interest &it-

maturity perio+ o! 4 years

Assume t-at t-e &it-+ra&als can only be ma+e at t-e en+

o! maturity perio+ <-at s-oul+ be -is investment strate"y

t-at &oul+ maximize t-e total return at t-e en+ o! 8 years.



'ecision ariables

• xi: $oney investe+ in t-e savin"s account

at t-e be"innin" o! i t- year

• yi: $oney investe+ in t-e security Y at t-e be"innin" o! i t- year



Constraints

tart

3st Year 4n+ Year 8r+ Year En+ o! 8r+ Year

x3

y3

x4

y4

x8

y8

x3

; y3

:311111 x4

; y4

:319x3

x8

; y8

:319x4

; 334y3

$oney %eceive+

:319x8; 334y4



Formulation

$ax Z:319x8; 334y4

ub(ecte+ to

x3 ; y3 :311111

#319x3 ; x4 ; y4 :1

#334y3 #319x4 ; x8; y8 : 1

x32 y3 2 x42 y4 2 x82 y8 G: 1



Dptimal Investment Plan

tart

3st Year 4n+ Year 8r+ Year En+ o! 8r+ Year

x3 :311111

y3 :1

x4 :1

y4 : 319111

x8 :1

y8 :1$oney %eceive+

"#6$$ x3 :1

y3 :311111

x4 :1

y4 : 1

x8 :334111

y8 :1

%lan

%lan 2

OR



Investment Problem #II

$r Praban+- -i-ar is looin" at a 9 year plannin" -orizon !or investin"%s3111111 ,e -as !our options/

• Invest in a savin"s account X at 9 rate o! interest &it- maturity perio+ o!3 year

• Invest in a security Y at 34 rate o! interest &it- maturity perio+ o! 4 years

• Invest in a security Z at 3H rate o! interest &it- maturity perio+ o! 8 years

• Invest in a security < at 40 rate o! interest &it- maturity perio+ o! 0 years

Assume t-at t-e &it-+ra&als can only be ma+e at t-e en+ o! maturity perio+ <-at s-oul+ be -is investment strate"y t-at &oul+ maximize t-etotal return at t-e en+ o! 9 years.



ta!!in" Problem

'aily re5uirements in t-e ,% +epartment o! a company are as !ollo&s/

'ay $on ?ue <e+ ?-u Fri at un

%e5uirement 9 8 4 0 9 4 4

A person &ors !or 9 consecutive +ays an+ taes 4 +ays o!! Assumin"

t-at a cyclical sc-e+ule is to be ma+e !in+ t-e minimum number o! people

t-at &ill meet t-e +aily re5uirements



'ecision ariables

• xi: umber o! people &-o be"in &or on

i t- +ay



Formulation

$in Z: x3 ; x4 ; x8 ; x0 ; x9 ; x@ ; x

ub(ecte+ to

x3 ; x0 ; x9 ; x@ ; x G:9x3 ; x4 ; x9 ; x@ ; x G:8

x3 ; x4 ; x8 ; x@ ; x G:4

x3 ; x4 ; x8 ; x0 ; x G:0

x3 ; x4 ; x8 ; x0 ; x9 G:9 x4 ; x8 ; x0 ; x9 ; x@ G:4

x8 ; x0 ; x9 ; x@ ; x G:4

x3 2 x4 2 x8 2x0 2x9 2 x@ 2x G: 1



Dptimal olution

x3 :8

x0 :4

x4 2x8 2 x9 2 x@ 2 xI :1



ta!!in" Problem #II

'aily re5uirements in t-e ,% +epartment o! a company are as !ollo&s/

A person &ors !or 9 consecutive +ays an+ taes 4 +ays o!! Assumin"

t-at a cyclical sc-e+ule is to be ma+e !in+ t-e minimum number o! people

t-at &ill meet t-e +aily re5uirements

$on+ay ?ues+ay <e+nes+ay ?-urs+ay Fri+ay atur+ay un+ay

-i!t 3 30 38 39 3@ 3J 3H 33

-i!t 4 44 4H 3J 41 4@ 4H 3H

-i!t 8 33 31 39 34 3J 34 J



'ecision ariables

• xi: umber o! people &-o be"in &or on

i t- +ay



Formulation )-i!t#3*

$in Z: x3 ; x4 ; x8 ; x0 ; x9 ; x@ ; x

ub(ecte+ to

x3 ; x0 ; x9 ; x@ ; x G:30x3 ; x4 ; x9 ; x@ ; x G:38

x3 ; x4 ; x8 ; x@ ; x G:39

x3 ; x4 ; x8 ; x0 ; x G:3@

x3 ; x4 ; x8 ; x0 ; x9 G:3J x4 ; x8 ; x0 ; x9 ; x@ G:3H

x8 ; x0 ; x9 ; x@ ; x G:33

x3 2 x4 2 x8 2x0 2x9 2 x@ 2x G: 1



Dptimal olution )-i!t#3*

x3 :

x8 :8

x9 :0 x@ :

xI :3

x4 2x0 :1



Dptimal olution )All -i!ts*

-i!t 3 -i!t 4 -i!t 8

$on+ay 9 1

?ues+ay 1 @ 0

<e+nes+ay 8 @ 1

?-urs+ay 1 3 1

Fri+ay 0 0

atur+ay 33 8un+ay 3 1 9



X 3 X 3 X 3 X 3 X 3

X 4 X 4 X 4 X 4 X 4

X 8 X 8 X 8 X 8 X 8

X 0 X 0 X 0 X 0 X 0

X 9 X 9 X 9 X 9 X 9

X @ X @ X @ X @ X @

X X X X X

Y 3 Y 3 Y 3 Y 3 Y 3

Y 4 Y 4 Y 4 Y 4 Y 4

Y 8 Y 8 Y 8 Y 8 Y 8

Y 0 Y 0 Y 0 Y 0 Y0

Y 9 Y 9 Y 9 Y 9 Y 9

Y @ Y@ Y @ Y @ Y @

Y Y Y Y Y

Z 3 Z 3 Z 3 Z 3 Z 3

Z 4 Z 4 Z 4 Z 4 Z 4

Z 8 Z 8 Z 8 Z 8 Z 8

Z 0 Z 0 Z 0 Z 0 Z 0

Z 9 Z 9 Z 9 Z 9 Z 9

Z @ Z @ Z @ Z @ Z @

Z Z Z Z Z

22 2' 2( 26 22 # 6%e5

)Z3; Z0 ; Z9 ; Z@; Z *

K : 18 )X3; X0 ; X9 ; X@; X ;

Y3; Y8 ; Y0 ; Y@; Y ;

Z3; Z0 ; Z9 ; Z@; Z *

)X3; X0 ; X9 ; X@; X ;

Y3; Y8 ; Y0 ; Y@; Y ;

Z3; Z0 ; Z9 ; Z@; Z * G: 44

Constraints DAY 1

)i*ilarly +or Day 2,(,#



Db(ective Function

$inimize "

411 )X3; X4 ;X8 ; X0 ; X9 ; X@; X ; Y3; Y4 ;Y8 ; Y0 ; Y9 ; Y@; Y * ;

341 )Z3; Z0 ; Z9 ; Z@; Z * ; 381 )Z3; Z4 ; Z9 ; Z@; Z * ;

301 )Z3; Z4 ; Z8 ; Z@; Z * ; 391 )Z3; Z4 ; Z8 ; Z0; Z * ;

3@1 )Z3; Z4 ; Z8 ; Z0; Z9 * ; 31 )Z4; Z8 ; Z0; Z9; Z@ * ;

3H1 )Z8 ; Z0; Z9; Z@ ; Z*



$ax Function

<rite an LP Formulation !or a $ax Function

s-o&n belo& /

Z:$ax M a3 2 a42 a82 a02 a92 N an O



Formulation

Z G: a3

Z G: a4

Z G: a8

/

Z G: an

$in Z



%etailin" Problem $r X2 &-o operates in In+ore2 nee+s to +etermine -is procurement an+

sales plan !or rice !or t-e next 0 mont-s ,e -as 49 tonnes o! rice to

be"in &it- ,e -as a &are-ouse t-at can stoc maximum 311 tonnes

o! rice ,e -as access to t-e !ollo&in" in!ormation/

$ont- ales Price inIn+ore)%s per

tonne*

Purc-ase price inC-attis"ar- )%s per

tonne*

3 0111 0111

4 9111 8411

8 0111 4J11

0 0111 8111



'ecision ariables

• pi: uantity o! rice purc-ase+ at t-e

be"innin" o! mont- i

• si: uantity o! rice sol+ +urin" mont- i



Formulation

Let bi be t-e be"innin" stoc !or eac- mont- b3 ; p3 # s3 : b4 b3 ; p3K: 311

b4 ; p4 # s4 : b8 b4 ; p4 K: 311 b8 ; p8 # s8 : b0 b8 ; p8 K: 311

b0 ; p0 # s0 : b9 b0 ; p0 K: 311

b3: 49 2 b9:1 2on ne"ativity

$ax Z : 0111s3#0111 p3; 9111s4#8411 p4 ;0111s8#4J11 p8 ;0111s0#8111 p0



$aximize Dut!lo&

Cap:9

Cap:8

Cap:3

Cap:9Cap:9

Cap:8

Cap:

Cap:4

Cap:In!

Dut!lo&

En+tart

A

E

C

'

F

B



$aximize Dut!lo& #II

Cap:9

Cap:8

Cap:3

Cap:9Cap:9

Cap:8

Cap:

Cap:4

Cap:In!

Dut!lo&

En+tart

Leaa"e :31 o! t-eincomin" !lo&

Leaa"e :H o! t-e

incomin" !lo&

Leaa"e :9 o! t-e!lo&

A

E

C

'

F

B



's

Cap:9

Cap:8

Cap:3

Cap:9Cap:9

Cap:8

Cap:

Cap:4

Cap:In!

Dut!lo&

En+tart


Leaa"e :H o! t-e

incomin" !lo&


A

E

C

'

F

B

xAB

xA'

xBE

xAC

xBC 2xCBx

CF

xEF

x'FxC' 2x'C

xI

xDQ?



Formulation

Cap:9

Cap:8

Cap:3

Cap:9Cap:9

Cap:8

Cap:

Cap:4

Cap:In!

Dut!lo&

En+tart


Leaa"e :H o! t-e

incomin" !lo&


A

E

C

'

F

B

xAB

xA'

xBE

xAC

xBC 2xCBx

CF

xEF

x'FxC' 2x'C

xI

xDQ?

xI :xAB ; xAC ;xA'

xAB2 xA' K:9

xAB ;xCB :xBE ; xBC

xBC2 xCB K:4

xBEK: 8

)xA' ;xC'*1J4 :x'C ; x'F

x'F K:9

x'C2 xC' K: 8

xAC ;xBC ; x'C : xCF ;xCB ;xC'

xCFK:

xBE 1J : xEF

xEFK: 3

1J9xCF ; xEF ;x'F: xDQ?

"/a0 0O1



Port!olio 'esi"nA mort"a"e team &ants to !inance %s 31121112111 to various loans Any

uninveste+ money "oes to savin"s account &it- 49 return

Allocate t-e money so as to /

3 $aximize avera"e return per rupee !or one year

4 Avera"e ris s-oul+ not be more t-an 9)exclu+in" t-e savin"s*

8 41 minimum investment in commercial

Loan6Investment %eturn )* %is)*

?ype I $ort"a"es J 8

?ype II $ort"a"es 34 @

Personal 39 H

Commercial H 4

>ovt ecurities @ 3



Formulation

's / xi is t-e amount to be investe+ in it- loan

Z: Jx3 ; 34x4 ; 39x8 ; Hx0 ; @x9 ; 49x@

x3 ; x4 ; x8 ; x0 ; x9 ; x@ ; xIK:311111111

)8x3 ; @x4 ; Hx8 ; 4x0 ; x9*6)x3 ; x4 ; x8 ; x0 ; x9 * K: 9

x0 G: 14)x3 ; x4 ; x8 ; x0 ; x9 *



Pro(ect $ana"ement

A B

C

'

E

F

>

, I

9

9

0

9

43

4

4

8



Formulation

A B

C

'

E

F

>

, I

9

9

0

9

43

4

4

8

?BG: ?A; 9 ?EG: ?C; 8

?EG: ?'; 9

$in ?FII,

?FII,G: ?I; 4

?CG: ?B; 0

?'G: ?B; 0

?FG: ?E; 4

?>G: ?E; 4

?,G: ?F; 9

?,G: ?>; 4

?IG: ?,; 3



Ballmart )A*

1

10

3

8

9

6

5

4

2

7

D

F

B

C

E

A

Outlets

Factories



Factories

Coor3inates

Factor

y 4

%ro3uction

Cost

Rs. 7er

1nit&

%ro3uction

Ca7acity 1nits

7er year&

A 6$.66 '.'8 !.22 5$$$

9 6$.'# 6$.68 (.! 5$$$

C 6#.2! !'.! (.56 5$$$

D 8(.68 85.6! (.#( 5$$$

2#.86 2(.52 !.88 5$$$

F '(.2# #'.8# !.'$ 5$$$



Dutlets Coor3inates

Outlets 4

)ales %rice

Rs. %er

1nit&

De*an3 1nits 7er

year&

3 9@ 044H HHJ 8111

4 H89 1 H41 4111

8 9@0 3H4 J41 3111

0 @814 H3J H91 4111

9 0330 801 J4J 3111

@ 41@ H088 HJJ 3911

0J41 898 H80 911

H 414 830 JJ8 4111

J J4H @JHH H30 3411

31 9H00 3348 H19 4311



'istance $atrix

2 ( ! 5 6 # ' 8 $

A 4H348 44@01 938 488919 3@@38 8H9H39 41300 J4303 4@3JH9 83H1

9 3H8 39J1@ J4048 8984@ 494J 411H04 8HH9 40@31 3844 344@83

C 899 J3@4H 938J8 @18 @94H1 4@J40 409J 431@89 3J3114 4JJ3

D 43193@ 010181 8@H33 91@43 3@4H84 04@43J 4H19H0 @J014H 8HJ093 03H048

9J33@ 301938 09@@@ 4@038H 3881H9 43H089 4J9J 4888 340J1H 9011J

F 31401H 49J43 43JH4H 48H8 J14J4 8833@3 3981@@ 0H81J@ 4H84 4@14HH

formulation exercises version 7

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