formulation exercises version 7
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8/9/2019 Formulation Exercises Version 7
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Formulation Exercises in LP
8/9/2019 Formulation Exercises Version 7
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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*
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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
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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:
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LP Formulation
maximize Z:701x3 ; 91x4
sub(ect to 3x3 ; 4x4≤ 01
0x4 ; 8x4≤ 341
x32 x4≥ 1
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>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
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>rap-ical olution o! a $aximization $o+el )4 o! 34*
Labor Constraint
maximize Z:701x3 ; 91x4
sub(ect to
3x3 ; 4x4≤ 01 -ours o! labor
0x4 ; 8x4≤ 341 poun+s o! clay
x32 x4≥ 1
Figure 2.2 >rap- o! t-e labor constraint line
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>rap-ical olution o! a $aximization $o+el )8 o! 34*
Labor Constraint Area
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 2.3 ?-e labor constraint area
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>rap-ical olution o! a $aximization $o+el )0 o! 34*
Clay Constraint Area
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 2.4 ?-e constraint area !or clay
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>rap-ical olution o! a $aximization $o+el )9 o! 34*
Bot- Constraints
maximize Z:701x3 ; 91x4
sub(ect to
3x3 ; 4x4≤ 01 -ours o! labor
0x4 ; 8x4≤ 341 poun+s o! clay
x32 x4≥ 1
Figure 2.5 >rap- o! bot- mo+el Constraints
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>rap-ical olution o! a $aximization $o+el )@ o! 34*
Feasible olution Area
maximize Z:701x3 ; 91x4
sub(ect to
3x3 ; 4x4≤ 01 -ours o! labor
0x4 ; 8x4≤ 341 poun+s o! clay
x32 x4≥ 1
Figure 2.6 ?-e !easible solution area constraints
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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
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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
$o+el constraints/
4x3 ; 0x4 ≥ 3@ lb )nitro"en constraint*
0x3 ; 8x4 ≥ 40 lb )p-osp-ate constraint*
x32 x4 ≥ 1 )non ne"ativity constraint*
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$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
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Feasible olution Area
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 2.15 Feasible solution area
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Dptimal olution Point
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 2.16 ?-e optimal solution point
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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.
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'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
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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
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Formulation
$ax Z:319x8; 334y4
ub(ecte+ to
x3 ; y3 :311111
#319x3 ; x4 ; y4 :1
#334y3 #319x4 ; x8; y8 : 1
x32 y3 2 x42 y4 2 x82 y8 G: 1
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Dptimal Investment Plan
tart
3st Year 4n+ Year 8r+ Year En+ o! 8r+ Year
x3 :311111
y3 :1
x4 :1
y4 : 319111
x8 :1
y8 :1$oney %eceive+
"#6$$ x3 :1
y3 :311111
x4 :1
y4 : 1
x8 :334111
y8 :1
%lan
%lan 2
OR
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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.
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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
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'ecision ariables
• xi: umber o! people &-o be"in &or on
i t- +ay
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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
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Dptimal olution
x3 :8
x0 :4
x4 2x8 2 x9 2 x@ 2 xI :1
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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
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'ecision ariables
• xi: umber o! people &-o be"in &or on
i t- +ay
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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
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Dptimal olution )-i!t#3*
x3 :
x8 :8
x9 :0 x@ :
xI :3
x4 2x0 :1
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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
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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,(,#
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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*
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$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
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Formulation
Z G: a3
Z G: a4
Z G: a8
/
Z G: an
$in Z
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%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
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'ecision ariables
• pi: uantity o! rice purc-ase+ at t-e
be"innin" o! mont- i
• si: uantity o! rice sol+ +urin" mont- i
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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
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$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
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$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
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's
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
xAB
xA'
xBE
xAC
xBC 2xCBx
CF
xEF
x'FxC' 2x'C
xI
xDQ?
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Formulation
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
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
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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
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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 *
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Pro(ect $ana"ement
A B
C
'
E
F
>
, I
9
9
0
9
43
4
4
8
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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
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Ballmart )A*
1
10
3
8
9
6
5
4
2
7
D
F
B
C
E
A
Outlets
Factories
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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$$$
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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
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'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