A Path Formulation for the Generalized Pooling Problemmohammeda/publications/stockholm.pdf · Branch{And{Bound Column Generation Conclusion A Path Formulation for the Generalized

Introduction to the pooling problemFormulations for general networks

Branch–And–BoundColumn Generation

Conclusion

A Path Formulation for the GeneralizedPooling Problem

Mohammed Alfaki

3rd Nordic Optimization Symposium, Stockholm, KTH

March 13, 2009

Mohammed Alfaki A Path Formulation for the Generalized Pooling Problem



Conclusion

Outline

1 Introduction to the pooling problemWhat is the pooling problem?State of the art in global methodsGeneralization and important applications

2 Formulations for general networksArc and path–oriented formulations

3 Branch–And–BoundLinear relaxation

4 Column GenerationColumn Generation–General ideaHow it works here?

5 Conclusion




Conclusion

What is the pooling problem?State of the art in global methodsGeneralization and applications

What is the pooling problem?

Example

Blending problem:




Conclusion


What is the pooling problem?

Example

Pooling problem:




Conclusion


NP–hardness

Definitions

Mixing raw materials in pools

Two stage blending problem

The model is non–linear

Many local optima

Hard to find global optimum

Theorem

The pooling problem is NP-hardeven in the case of single-layer ofpools.Proof: A poly reduction from the3-dimensinal matching problem tothe single-layered pooling problem.




Conclusion


State of the art in global methods

The mathematical models for the pooling problem based onnon–linear formulation

Nonlinearity appears in two type of constraints1 Quality balance at pools2 Quality bound at the terminals (products)

Two formulations exist:1 p–formulation (Haverly 1978): consists of flow and quality

variables2 q–formulation (Ben–Tal et al. 1994): replacing the quality

variables with variable representing flow proportions.

q–formulation is tighter than p–formulation

Tawarmalani & Sahinidis 2005 gave the pq–formulation,which is even tighter




Conclusion


Generalization and important applications

The pq–formulation is not applicablewhen we have multi–layered of pools

Important application do exist fo themulti–layered pooling problem:

1 Pipeline transportation of dry gas2 Multi–period inventory model




Conclusion

Arc and path–oriented formulations

Arc–oriented formulation

G = (N, A) is DAGParameters:

1 qi , i ∈ S ∪ T , k ∈ K2 ui node capacity of i ∈ N3 cij unit cost of (i , j) ∈ A

Variables:1 wi S quality leaving i ∈ Q2 vij the flow along (i , j) ∈ A

Minimize the cost

Sulfur Quality balance

wi =∑j∈N−i

wjvji

∑j∈N−ivji

, i ∈ Q




Conclusion

Arc and path–oriented formulations

Path–oriented formulation

Variables:1 vij the flow along (i , j) ∈ A2 v s

ij flow originating from s ∈ S3 xp flow along path p ∈ P4 y s

i proportion of flow at pool icoming from s ∈ S

Minimize the cost

Propostion variables:

y si =

v sij

vij, s ∈ S , i ∈ Q, (i , j) ∈ A

legal flow:

v sij − ∑

p∈Psij

xp = 0, s ∈ S , (i , j) ∈ A

P: set of directed paths from some source to some sink

Ps ⊆ P: set of paths start with source s ∈ S

Pij ⊆ P: set of paths intersecting (i , j) ∈ A

Psij = Ps ∩ Pij




Conclusion

Linear relaxation

Linear relaxation

The idea based on B&B algorithms forinteger programming

Subproblem: linear relaxation

Solve the relaxed problem at eachnode to get upper or lower bounds

Prune: by bound, infeasibility,optimality

Stop when the tree is empty




Conclusion

Linear relaxation

Convex and concave envelopes

Sutting ζ = yv the convex andconcave envelopes on rectangleC = [y , y ]× [v , v ] are:

max{yv + yv − yv , yv + yv − y v}

min{yv + yv − yv , yv + yv − yv}

Replacing the bilinear constraintswith ζs

ij − v sij = 0

Adding the above bounds to givelinear relaxation problem R




Conclusion

Column Generation–General ideaHow it works here?

Column Generation–General idea

Column generation helpful in problems with manyvariables/columns but relatively few constraints

Exclude some of the variables

The master problem: original problem with only a subset ofvariables

The subproblem: to identify whether there is excluded columnthat reduces the objective function value




Conclusion


How it works here?

The master problem R ′ will be the problem R with onlyP ′ ⊆ P

The idea is to solve R ′, and whenever we detect that the costcan be reduced by sending flow along any path p ∈ P \ P ′, weaugment P ′ by p

We are looking for a path that reduces the cost

The subproblem will be the Shortest–Path problem, which issolvable in polynomial time, since G is DAG




Conclusion


Algorithm for solving R

At each node in the B&B tree we will solve the following:

repeatsolve the master problem R ′

for (∀ s ∈ S) dodefine arc lengthsp ← shortest path from s to some t ∈ T in Gif (length(p) < 0) then

P ′ ← P ′ ∪ {p}end

enduntil (no path of length < 0 found)




Conclusion

Conclusion

This is work in progress.

The pooling problem is NP–hard

The path–oriented formulation is equivalent to thepq–formulation, when we have one layer of pools

Generalization of the strongest known formulation forone–layer instances

Outlook

Experimental evaluation


Documents

A Path Formulation for the Generalized Pooling Problemmohammeda/publications/stockholm.pdf · Branch{And{Bound Column Generation Conclusion A Path Formulation for the Generalized