decomposition method for the multiperiod blending...

Decomposition method for the Multiperiod Blending Problem

Irene Lotero, Francisco Trespalacios and Ignacio Grossmann September 17-18, 2014

Center for Advanced Process Decision-making Department of Chemical Engineering Carnegie Mellon University

Pittsburgh, PA 15213

2

Outline

• Problem description • Applications • Mathematical Formulation

Background

• State-of-the-art commercial MINLP solvers struggle • Generate “good” solutions fast Motivation

Approach • Alternative mathematical formulation based on GDP • Decomposition algorithm

Supply Tanks (s) Blending Tanks (b) Demand Tanks (d)

Multiperiod blending problem is defined over supply, blending and demand tanks

Assumptions: • Supply concentrations are constant • Perfect mixing • No simultaneous input/output to blending tanks

Given: • Topology, initial conditions and flow profit/costs • Supply tank flow and concentration • Demand tank flow and concentration limits

Determine • Flows between which tanks in which time periods • Inventories/concentrations for tanks in each period • Maximum total profit of blending operation

Fs1,t Cs1

Fs3,t Cs3

Fs2,t Cs2

Fd1,t CL

d1-CUd1

Fd2,t CL

d2-CUd2

Fd3,t CL

d3-CUd3

Fs,b,t

Fb,d,t

Ib,t

Is,t

Id,t

Fs,d,t

3

ys,b,t

yb,d,t

Cq,s,t

General model for many applications

4

• Gasoline and crude oil blending • Raw material feed scheduling • Storage of intermediate streams • Water treatment • Emissions regulation

Supply Tanks (s) Blending Tanks (b) Demand Tanks (d)

Fs1,t Cs1

Fs3,t Cs3

Fs2,t Cs2

Fd1,t CL

d1-CUd1

Fd2,t CL

d2-CUd2

Fd3,t CL

d3-CUd3

Fs,b,t

Fb,d,t

Ib,t

Is,t

Id,t

Fs,d,t

Applications Nomenclature

ys,b,t

yb,d,t

Cq,s,t

MINLP formulation contains bilinear terms

[ [ ] ] ⋁

Type

max profit - network flow cots

linear

mixed-integer linear


linear

nonlinear (nonconvex)

integer linear

linear

s.t for flows into blending tanks: [flow within bounds] ⋁ [flow = 0] for flows into demand tanks: flow within bounds flow = 0 concentrations within demand spec. “no bounds” on concentration

total inventory mass balance in tanks inventory mass balance by component in blending tanks no simultaneous in/out flow variable bounds

s.t

Contains bilinear terms, for each contaminant, each blending tank

and each time period

5

6

Outline


Background



Commercial solvers not able to find feasible solutions

Commercial Solvers Performance Objectives

Increase size of the problems • Current problems: 4 time periods 8 qualities 4 blending tanks • Target problems: 12 time periods 10 qualities 16 blending tanks

• After 5 minutes of computational time,

Average normalized best feasible solution

Optimal 1.0

GloMIQO 0.57

BARON 0.19

SCIP 0.35

• After 30 minutes of computational time,

Found Feasible Solution Large Instances

GloMIQO 54%

BARON 0%

SCIP 15%

• 36 randomly generated instances

7

Generate “good solutions” fast • Guaranteeing global optimality not a priority • Obtain good feasible solutions

8

Outline


Background



1. Raman R. and Grossmann I.E., “Modelling and Computational Techniques for Logic-Based Integer Programming”, Computers and Chemical Engineering, 18, 563, 1994.

GDP is a higher level of representation for MILP/MINLP Optimization problem with algebraic expressions, disjunctions & logic propositions

General form of GDP1 Illustration – Process network

9

Objective Function

Global Constraints

Disjunctions

Logic Propositions

Objective function

Global Constraints

Logic

R1

R2 S2

F1

F3

F2

F4

F6

F7

F5

S1

Disjunctions

1. The new binary variable can in fact be continuous due to the problem structure. Furthermore, using it as binary variable and giving it priority over the network variables can increase the solution time of a branch-and-bound method

Remark: Possible to reduce bilinear terms using a disjunction

10

If some of the tanks are fixed as “split tanks”, the resulting MINLP becomes easier to solve

Blending tank Split tank

⋁

Requires 0-1 variable, but it does not increase combinatorial

complexity (it might even reduce it!)1

Blend + Split tank

Tank

Mass balance

Comp. mass

balance Fewer or none bilinear terms

Original MINLP formulation Type

linear



linear


integer linear

linear

s.t

11

Alternative MINLP formulation based on GDP Type

linear



linear


integer linear

linear

s.t

12

Implies decomposition

Decomposition algorithm seeks to find feasible solutions in short periods of time

13

Algorithm Decisions in the algorithm

Master Problem (MILP) Provides Upper Bound (UB)

Add

cut

s: o

ptim

ality

or f

easi

bilit

y

Stop

No

Yes

Master MILP (MINLP relaxation) • Multiparametric disaggregation[1]

Subproblem (MINLP) Provides Lower Bound (LB)

Fix split tanks

If feasible, it provides feasible solution for the original MINLP

[2] Kolodziej, S.P., Castro, P.M., Grossmann, I.E. Global optimization of bilinear programs with a multiparametric disaggregation technique. Journal of Global Optimization (2013)

Solving MINLP • Smaller nonconvex MINLP • Global solution through specialized technique,

e.g. multiparametric disaggregation

Other considerations • “no-good” are included in the subproblem,

eliminating regions already evaluated in previous subproblems

Multiparametric disaggregation provides a lower bound

Discretization Technique Notes and examples

Multiperiod blending problem

We can use other bases for more/fewer binary variables

• Base two takes fewer binary variables

Precision

Range 0 – 15 0 – 9

0 – 12.7 0 – 9.9

0 – 10.23 0 – 9.99

Increment 1 0.1 0.01

Binary Variables

Base 2 8 14 20

Base 10 10 20 30

Binary variables

Scales up better If approx. MILP is feasible,

provides solution for original MINLP 14

Slack variables are introduced for upper bound

Discretization Technique + Slack Notes

Is a relaxation problem because it includes at least the entire feasible region of the problem

The relaxed MILP provides a valid upper bound for original MINLP

15

Subproblem algorithm solves the discretized and relaxed MILPs successively increasing the precision

16

Decomposition Algorithm

Master Problem (MILP) Provides Upper Bound (UB)

Add

cut

s: o

ptim

ality

or f

easi

bilit

y

Stop?

Stop

No

Yes

Subproblem (MINLP) Provides Lower Bound (LB)

Fix split tanks

Subproblem Algorithm[2]

[2] Kolodziej, S.P., Grossmann, I.E., Furman, K.C. and Sawaya, N.W. A discretization-based approach for the optimization of the multiperiod blend scheduling problem, Computers & Chemical Engineering, (2013).

GLOMIQO BARON SCIP

• 18 randomly generated instances • 3 – 4 time periods, 2 – 10 qualities, 4 – 12 blending tanks

0 – 300 s

Results Small Instances

17

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 60 120 180 240 300

Aver

age

norm

aliz

ed

best

feas

ible

sol

utio

n

TIME (s)

GLOMIQO BARON SCIP DECOMPOSITION


0 – 1800 s 0 – 300 s

Results Small Instances

17

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 60 120 180 240 300

Aver

age

norm

aliz

ed

best

feas

ible

sol

utio

n

TIME (s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 600 1200 1800

Ave

rage

nor

mal

ized

be

st fe

asib

le s

olut

ion

TIME (s)


0 – 300 s

Results Large Instances

18


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 50 100 150 200 250 300

Ave

rage

nor

mal

ized

be

st fe

asib

le s

olut

ion

TIME (s)


0 – 1800 s 0 – 300 s


18


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 50 100 150 200 250 300

Ave

rage

nor

mal

ized

be

st fe

asib

le s

olut

ion

TIME (s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 600 1200 1800

Ave

rage

nor

mal

ized

be

st fe

asib

le s

olut

ion

TIME (s)


0 – 1800 s 0 – 300 s


18


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 50 100 150 200 250 300

Ave

rage

nor

mal

ized

be

st fe

asib

le s

olut

ion

TIME (s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 600 1200 1800

Ave

rage

nor

mal

ized

be

st fe

asib

le s

olut

ion

TIME (s)

After 10 minutes the average normalized best feasible solution is twice the average normalized

best feasible solution of the following best commercial solver


0 – 300 s

Results All Instances

19

GLOMIQO BARON SCIP

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 50 100 150 200 250 300

Aver

age

norm

alize

d be

st fe

asib

le so

lutio

n

TIME (s)


0 – 1800 s 0 – 300 s


19


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 50 100 150 200 250 300

Aver

age

norm

alize

d be

st fe

asib

le so

lutio

n

TIME (s)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 600 1200 1800

Aver

age

norm

alize

d be

st fe

asib

le so

lutio

n

TIME (s)


0 – 1800 s 0 – 300 s


19


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 50 100 150 200 250 300

Aver

age

norm

alize

d be

st fe

asib

le so

lutio

n

TIME (s)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 600 1200 1800

Aver

age

norm

alize

d be

st fe

asib

le so

lutio

n

TIME (s)

After 5 minutes the average normalized best feasible solution is >0.7

The algorithm performs better than state-of-the-art MINLP commercial solvers

Good MILP approximation and decomposition allow finding good solutions in first iteration

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Instance

1. Normalized to best known solution

Incr

ease

siz

e

Group 1

Group 2

Group 3

Group 4

Group 5

Group 6

Instance

0-1 vars

Bilinear terms

114

235

539

988

1524

2736

195

428

1071

1335

1940

3200

First MINLP

0-1 vars

Bilinear terms

30

63

152

223

305

494

72

132

516

580

446

1089

UB

First iteration (normalized1)

LB Time (s)

1.07

1.04

1.10

1.06

1.06

1.08

0.94

0.62

0.69

0.80

0.50

0.48

14.5

60.9

95.1

79.7

95.4

213.7

70%-80% fewer 0-1 variables

~55% fewer bilinear terms



Very good solution after 1st iteration

Good solution after 1st iteration

Not as good but at least feasible

solution found after 1st iteration

Feasible solution found in less than

a minute

Feasible solution found in less than a

two minutes

Almost 5 minutes for large instances

To wrap up

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Propose an alternative formulation based on GDP • Blending/Splitting tanks

Decomposition algorithm that simplifies the search for feasible solutions • Solving smaller MINLPs with fewer 0-1 variables and bilinear terms • “Guided” by an MILP relaxation of the problem

Tested the approach in randomly generated instances • Increase problem sizes to match industrial applications

Generate “good” solutions fast

Thank you

Dimitri J. Papageorgiou and Myun-Seok Cheon from ExxonMobil

Acknowledgments

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decomposition method for the multiperiod blending...

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