Miping the probabilistic integer programming problem
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MIPing the Probabilistic Integer Programming Problem. Anureet Saxena ACO PhD Student, Tepper School of Business, Carnegie Mellon University. (Joint Work with Vineet Goyal and Miguel Lejuene). Why Probabilistic Programming?. Fixed Cost. Transportation Cost. Demand Constraints.

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MIPing the Probabilistic Integer Programming Problem

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MIPing the Probabilistic Integer Programming Problem

Anureet Saxena

ACO PhD Student,

Tepper School of Business,

Carnegie Mellon University.

(Joint Work with Vineet Goyal and Miguel Lejuene)


Why Probabilistic Programming?

Fixed Cost

Transportation Cost

Demand Constraints

Capacity Constraints

Set of Customers

Set of Facilities


Why Probabilistic Programming?

Fixed Cost

Transportation Cost

Demand Constraints

Capacity Constraints

  • Uncertain Future

  • Population Shift

  • Evolution of Market Trends

  • Ford opens a manufacturing unit

  • Google closes its R&D center

Set of Customers

Set of Facilities


Why Probabilistic Programming?

A random 0/1 vector which incorporates the uncertain future into the optimization model


Why Probabilistic Programming?

Reliability Level

Probabilistic Constraint


Probabilistic MIP Model

Random 0/1 Vector

(Joint Distribution)

Reliability

Level

Probabilistic

Deterministic


Why Probabilistic Programming?

  • Facility Location

    • Strategic Planning

    • Population shift

    • Evolution of market trends

    • Demographic Changes

  • Contingency Service

    • Minimum Reliability Principle

  • Production Design and Manufacturing

    • Uncertain Demand

    • Lot Sizing and Inventory Problems

Must Read!

Strategic facility location

by Owen and Daskin


A Simple Algorithm

Random 0/1 Vector

(Joint Distribution)

Reliability

Level

  • Enumerate all possible 0/1 realizations of .

  • For each 0/1 realization whose cdf is greater than or equal to p, solve the deterministic problem


Prekopa, Beraldi, Ruszczynski Approach


Prekopa, Beraldi, Ruszczynski Approach

111

110

101

011

100

010

001

000


Prekopa, Beraldi, Ruszczynski Approach

p-efficient frontier


2-Phase Algorithm

Enumeration of p-efficient points

Solving a Deterministic Problem for each

p-efficient point


2-Phase Algorithm

Enumeration of p-efficient points

Independent

Solving a Deterministic Problem for each

p-efficient point


Beraldi & Ruszczynski Approach

Explosive Growth

In computation

time

scp41

scp42


2-Phase Algorithm

Pitfall

Enumeration of p-efficient points

Solving a Deterministic Problem for each

p-efficient point


Our Approach

Integrate the 2-phases

Enumeration of p-efficient points

Solving a Deterministic Problem for each

p-efficient point


Our Approach

Integrate the 2-phases

Enumeration of p-efficient points

Independent

Solving a Deterministic Problem for each

p-efficient point


Our Model

Log of cumulative probability of block t

Non-Linear

MIPing


Our Model

Log of cumulative probability of block t


Our Model

Log of cumulative probability of block t


Beraldi & Ruszczynski Approach: Comparison

All instances solved in less than 1sec by CPLEX 9.0. CPLEX enumerated less than 50 nodes solving most instances at the root node

scp41

scp42


Key Observations

  • Models any arbitrary distribution

  • Exponential number of constraints for each block

  • Linear in the input size for generic distribution

  • Encodes the enumeration phase as a Mixed Integer Program

  • Allows us to exploit state-of-art MIP solvers to perform intelligent enumeration.


Key Observations

  • Models any arbitrary distribution

  • Exponential number of constraints for each block

  • Linear in the input size for generic distribution

  • Encodes the enumeration phase as a Mixed Integer Program

  • Allows us to exploit state-of-art MIP solvers to perform intelligent enumeration.

Research Question

The model has an exponential number of constraints for each block. Is there a way to reduce the number of constraints?


The Answer is Yes


p-Inefficient Frontier


Refined Formulation

Add t constraints only for lattice points above the frontier

Set-Covering Constraint for maximally p-inefficient points


Refined Formulation

Block Size10


A Tough Instance - p31

  • SSCFLP instance from the Holmberg test-bed

  • 30 facilities and 150 customers

  • Deterministic instance can be solved in 80 sec.

  • Probabilistic instance has 15 blocks of size 10 each

  • CPLEX was unable to solve the probabilistic instance within 2 hours!!


A Tough Instance - p31


A Tough Instance - p31

Research Question

Why is this instance so difficult to solve?


Answer

Big-M Constraints


Polarity Cuts

Facets of P can strengthen the model

Big-M Constraints model P


Polarity Cuts

  • We know all the extreme points and extreme rays of P

  • Compact description of polar

  • Facets of P can be found by solving the linear program derived from the polar

  • The linear program has lot more rows than columns – dual simplex algorithm.


A Tough Instance - p31

  • Tough Instance Solved

  • % Gap closed at Root Node 67.84%

  • Time Spent in Strengthening 0.83 sec

  • Time Spent in Solving Separation LP 0.30 sec

  • Time Taken by CPLEX 9.0 after Strengthening 51.65 sec

  • No. of Branch-and-Bound enumerated by CPLEX 9.0 2300

  • Total time taken to solve the instance to optimality 53.04 sec


Computational Results

  • Implementation

    • COIN-OR Modules

    • CPLEX 9.0

  • Selection Criterion

    • ORLIB & Holmberg Instances

    • Instances which can be solved in 1hr

  • Computational Power

    • P4 Processor

    • 2GB RAM

  • Library of Instances – PCPLIB


Test Bed

  • 2 Distributions – as in BR [2002]

  • 4 Reliability levels – 0.80, 0.85, 0.90, 0.95

  • 2 Block Sizes – 5, 10

  • Total Number of Instances per Deterministic Instance = 16


Computational Results


Computational Results


Impact of Polarity Cuts


Value of Information


Value of Information

Empirical Observation

Probabilistic versions of simple and moderately difficult mixed integer programs can themselves be formulated as MIPs which can be solved in reasonable amount of time.


Structured Distributions

Research Question

Is it possible to exploit structure of distributions to design models which are polynomial in the input size?


Stationary Distributions

Definition

A distribution function F is said to be stationary if F(z) depends only on the number of ones in z.

Principle of Indistinguishability.


Stationary Distributions

111

110

101

011

100

010

001

000


Stationary Distributions

Can be converted to a MIP with linear number of additional variables and constraints!!


Stationary Distributions

A model with linear number of variables and constraints!!


Stationary Distributions

  • 8 Block Sizes: 5, 10, 20, 50, m/4, m/3, m/2, m

  • 4 Threshold Probabilities: 0.80, 0.85, 0.90, 0.95

  • Number of Instances per deterministic instance= 32


Stationary Distributions

Research Question

What is that unique property of stationary distributions which allowed us to design a linear sized model?

Disjunctive Shattering Property

The lattice of a stationary distribution can be partitioned into polynomial number of pieces each of which has a polynomial sized description.


Stationary Distributions

111

110

101

011

100

010

001

000


Summary

Super Linear

Speedup

MIP Model

BR Algorithm

p-Inefficiency

Refinement

Polarity Cuts

Strengthening

Stationary

Distributions

Computational

Results


Our Contribution


Thank you for your attention


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