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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)

Fixed Cost

Transportation Cost

Demand Constraints

Capacity Constraints

Set of Customers

Set of Facilities

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

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

Reliability Level

Probabilistic Constraint

Random 0/1 Vector

(Joint Distribution)

Reliability

Level

Probabilistic

Deterministic

- 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

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

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p-efficient frontier

Enumeration of p-efficient points

Solving a Deterministic Problem for each

p-efficient point

Enumeration of p-efficient points

Independent

Solving a Deterministic Problem for each

p-efficient point

Explosive Growth

In computation

time

scp41

scp42

Pitfall

Enumeration of p-efficient points

Solving a Deterministic Problem for each

p-efficient point

Integrate the 2-phases

Enumeration of p-efficient points

Solving a Deterministic Problem for each

p-efficient point

Integrate the 2-phases

Enumeration of p-efficient points

Independent

Solving a Deterministic Problem for each

p-efficient point

Log of cumulative probability of block t

Non-Linear

MIPing

Log of cumulative probability of block t

Log of cumulative probability of block t

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

- 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.

- 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?

Add t constraints only for lattice points above the frontier

Set-Covering Constraint for maximally p-inefficient points

Block Size10

- 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!!

Research Question

Why is this instance so difficult to solve?

Big-M Constraints

Facets of P can strengthen the model

Big-M Constraints model P

- 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.

- 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

- 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

- 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

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.

Research Question

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

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.

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

A model with linear number of variables and constraints!!

- 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

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.

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Super Linear

Speedup

MIP Model

BR Algorithm

p-Inefficiency

Refinement

Polarity Cuts

Strengthening

Stationary

Distributions

Computational

Results

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