The Pooling Problem Charles Audet Jack Brimberg Pierre Hansen Sébastien Le Digabel Nenad...

The Pooling Problem

Charles Audet

Jack Brimberg

Pierre Hansen

Sébastien Le Digabel

Nenad Mladenović

Submitted to Management Science

Outline

• The Pooling Problem

• Model Formulations– Flow model– Proportion model

• Exact method

• Heuristic methods

• Conclusion

The Pooling Problem

• Petroleum industry

• Products with different attribute qualities are mixed in a series of pools

• End pools (blends) : constraints on the attribute qualities

• Bilinear problem

Attribute Qualities

• Sulfur composition, density, octane number…• Attribute qualities blend linearly :

F2

F1

P1

10

24

q1 = 12

q2 = 4

35.62410

24*412*103

q

The Blending Problem

Fi

BnB

F1

FnF

Bk

B1

feeds blends

...

...

...

...

linear problem

feeds blendspools

PjFi

BnB

F1

FnF

Bk

B1

PnP

P1

...

...

...

...

...

...

The Pooling Problem

bilinear problem

Flow Model

iF feed kB blendjP pool

ijw

: arcs theofcapacity / variablesflow

FPijl

FPiju

ikxFBikl

FBiku

jkyPBjkl

PBjku

0 ,, yxw

PjFi

BnB

F1

FnF

Bk

B1

PnP

P1

...

...

...

...

...

...

x

y

w

Flow Model

yxwt ,,,max

W(i,j)ij

Fi wp

Xki

ikFi

Bk xpp

),(

Ykj

jkBk yp

),(


)( iXk

ikx )( iWj

ijwFil

Fiu:supply

iF

Flow Model


1

)( jWiijw

: pooleach for variables and between link jPyw

jP )( jYk

jky 0

Flow Model


1

)( kYjjky

1)( kXiikxB

klBku:demand kB

)( jYk

jkyPjl

Pju:capacity pool jP

Flow Model

ajt

1)( jWi

ijai ws

)( jYk

jky

PjFi2

Bk2

Fi1

Fi3

Bk1

attribute a of pool Pj :

iai Fas feed of attribute:

Flow Model


)(

.jYk

jkaj yt

: pool of attribute jPa

jP

a

1)(

.jWi

ijai ws 0

bilinear term

Flow Model


: blend of attribute oft requiremen kBa

kB

a

1

)(

.kYj

jkaj yt

1)(

.k

Xiik

ai xs

1

)( kYjjky

1)( k

Xiikx

akl

1

)( kYjjky

1)( k

Xiikx

a

ku

Proportion Model

)(

.jYk

jkijij yqw

1

)( jWiij

ai qs

link with flow model

attribute a of pool Pj

ijqproportion of flow entering Pj from Fi


Proportion Model


)(

.jYk

jkij yq

: arcs theofcapacity / variablesflow

FPijl

FPiju

ikxFBikl

FBiku

jkyPBjkl

PBjku

0 ,, yxq

11)(

jWhhjq

Proportion Model

qyx ,,max

W(i,j) Ykjkij

Fi

j

yqp)(

.

Xki

ikFi

Bk xpp

),(

Ykj

jkBk yp

),(


bilinear terms

Proportion Model


)( iXk

ikx )( )(

.i jWj Yk

jkij yqFil

Fiu

:supply

iF

1

)( kYjjky

1)( kXiikxB

klBku

:demand

kB

)( jYk

jkyPjl

Pju

:capacity pool

jP

Proportion Model


: blend of attribute oft requiremen kBa

kB

a

1)(

1)(k jYj

jkWi

ijai yqs

1)( k

Xiik

ai xs

1

)( kYjjky

1)( k

Xiikx

akl

1

)( kYjjky

1)( k

Xiikx

a

ku

Numerical examples

Examples nF nP nB nAAST1 5 2 4 4AST2 5 2 4 6AST3 8 3 4 6AST4 8 2 5 4BT4 4 1 2 1BT5 5 3 5 2F2 6 2 4 1H1 3 1 2 1RT1 3 2 3 4RT2 3 2 3 4

Linear Bilinear BilinearVariables Variables Terms L ≤ Q ≤ Q =

AST1 3 16 32 10 32 8BT4 4 3 2 6 4 1F2 10 10 8 12 8 2H1 3 (2-1+2) 3 (1+2) 2 (1*2) 6 4 1RT2 8 14 24 14 21 8

AST1 0 11 12 7 37 0BT4 2 4 4 5 7 0F2 8 10 8 8 12 0H1 2 3 (2-1+2) 2 (2*(2-1)) 4 6 0RT2 4 10 12 11 24 0

Constraints

Flow Model

Proportion Model

Examples

Characteristics of examples

Bilinear Terms :• flow : t.y• prop. : q.y

Linear Terms :• flow : w, x• prop. : x

F1

P1F2

F3

B1

B2

H1 w1/q1

impl.

x1

y1

y2

t1

x2

QP: exact method

• Program: QP

• Algorithm:

– Relaxation-Linearization Technique (RLT)– Branch-And-Cut– Sherali et al., Audet et al.

Results of exact method

best timesNodes CPU time (s) Nodes CPU time (s) from litterature

AST1 4145 7,786.04 245 9.06 425.00AST2 … … 267 9.67 1,115.00AST3 … … 537 68.50 19,314.00AST4 723 953.90 693 177.98 182.00BT4 9 0.22 43 1.03 0.11BT5 97 665.63 39 31.10 1.12F2 15 0.67 1 0.40 0.10H1 9 0.26 9 0.22 0.09RT1 179 31.81 7 0.60 …RT2 489 204.99 59 1.96 …

ExamplesFlow Model Proportion Model

ALT heuristic

• Bilinear problem in (x,y,z)

• y variables fixed: linear problem LPy(x,z)

• z variables fixed: linear problem LPz(x,y)

• MALT: ALT with multistart

zy

c

zz

c

yy

c

vv

yy

yxLPv

zz

zxLPv

yy

c

c

while

),(

),(

do0

VNS heuristic

) ( while

1 k

max

0

kk

ss

sks of odneighborho- in thepoint a : shaking 1 solution initial as with ALT :search local 12 ss

1

else

1

n better tha if

:not or move

2

12

kk

k

ss

ss

Heuristics Resultsexact MALT VNS MALT VNS MALT VNS

AST1 549.80 532.90 545.27 2.45 2.81 3.07 0.82AST2 549.80 535.62 543.91 5.21 5.68 2.58 1.07AST3 561.05 397.44 412.15 4.96 5.34 29.09 26.47AST4 877.65 876.21 876.21 0.77 1.01 0.16 0.16BT4 45.00 45.00 45.00 0.01 0.01 0.00 0.00BT5 350.00 324.08 350.00 0.09 1.11 7.41 0.00F2 110.00 107.87 110.00 0.44 0.57 1.94 0.00H1 40.00 40.00 40.00 0.01 0.01 0.00 0.00RT1 4,136.22 4,136.22 4,136.22 0.04 0.04 0.00 0.00RT2 4,391.83 4,330.78 4,391.83 0.47 0.60 1.39 0.00

AST1 549.80 532.90 533.78 2.38 2.61 3.07 2.91AST2 549.80 535.62 542.54 4.97 5.37 2.58 1.32AST3 561.05 397.44 558.84 4.98 5.93 29.09 0.30AST4 877.65 876.21 876.21 1.21 1.55 0.16 0.16BT4 45.00 45.00 45.00 0.02 0.02 0.00 0.00BT5 350.00 323.12 350.00 0.16 1.53 7.68 0.00F2 110.00 110.00 110.00 0.49 0.49 0.00 0.00H1 40.00 40.00 40.00 0.01 0.01 0.00 0.00RT1 4,136.22 4,136.22 4,136.22 0.03 0.03 0.00 0.00RT2 4,391.83 4,330.77 4,391.82 0.58 0.72 1.39 0.00

Flow Model

Proportion Model

ExamplesSolution CPU time (s) Error (%)

Conclusion• Proportion and Flow models

• Proportion better than flow for both exact and heuristic methods

• Elimination of variables

• Interconnected pools: Generalized Pooling Problem

The Pooling Problem Charles Audet Jack Brimberg Pierre Hansen Sébastien Le Digabel Nenad...

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