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?
Transportation Cost
Fixed Cost
Demand Constraints
Capacity Constraints
Set of Customers
Set of Facilities
Why Probabilistic Programming?
Transportation Cost
Fixed Cost
Demand Constraints
Capacity Constraints
Uncertain Future
• Population Shift
• Evolution of Market Trends
of Customers
• Ford opens aSet
manufacturing
unit
• Google closes its R&D center
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)
Deterministic
Reliability
Level
Probabilistic
Why Probabilistic Programming?
• Facility Location
–
–
–
–
Strategic Planning
Population shift
Evolution of market trends
Demographic Changes
Must Read!
Strategic facility location
by Owen and Daskin
• Contingency Service
– Minimum Reliability Principle
• Production Design and Manufacturing
– Uncertain Demand
– Lot Sizing and Inventory Problems
A Simple Algorithm
Random 0/1 Vector
(Joint Distribution)
1.
2.
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
Researchphase
Question
Encodes the enumeration
as a Mixed Integer
Program
The model has an exponential number of
• Allows us
to exploitforstate-of-art
solvers
to perform
constraints
each block.MIP
Is there
a way
intelligent
to enumeration.
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 pinefficient 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
Big-M Constraints
model P
Facets of P can
strengthen the model
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
Problem Set
OrLib Set Covering
OrLib Warehouse Location (Cap)
OrLib p-Median (Cap)
Holmberg Facility Location (Cap)
Number of Instances
60
37
20
70
# Rows
50-500
66-100
101-201
60-230
# Columns
500-5000
816-2550
2550-10100
510-6030
• 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
Deterministic
Problem
Set Covering
CWLP
Cap k-Median
SSCFLP
Number of Probabilistic
Instances
1440
888
480
1680
Number of Unsolved
Instances
37
0
0
22
% Relative Gap
(Unsolved Instances)
11.69
0.45
Computational Results
Deterministic
Problem
Set Covering
CWLP
Cap k-Median
SSCFLP
Solution
Number of BranchTime (sec)
and-Bound Nodes
160.81
7440
0.31
30
43.79
1464
31.27
2248
Impact of Polarity Cuts
Polarity Cuts' Strengthening
Deterministic
Problem
% Duality Gap
% Time Spent
Closed
Set Covering
23.74
0.22
CWLP
11.44
9.43
0.00
0.21
18.45
0.29
Cap k-Median
SSCFLP
Value of Information
Deterministic
Problem
Set Covering
CWLP
Cap k-Median
SSCFLP
Value of Information
(%)
5.75
15.05
9.54
4.60
Value of Information
Value of Information
(%)
Set Covering
5.75
CWLP
15.05
Cap k-Median
9.54
Empirical Observation
SSCFLP
4.60
Deterministic Problem
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
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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
Deterministic
Problem
Number of
Probabilistic
Instances
Number of
Unsolved
Instances
% Relative
Solution Time
Gap (Unsolved
(sec)
Instances)
Value of
Information
Set Covering
1920
127
21.34
112.98
10.42
CWLP
1184
0
-
0.09
25.15
Cap k-Median
SSCFLP
640
2240
0
17
0.45
2.90
9.36
15.25
8.51
• 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
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011
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Summary
BR Algorithm
Stationary
Distributions
MIP Model
Super Linear
Speedup
p-Inefficiency
Refinement
Polarity Cuts
Strengthening
Computational
Results
Our Contribution
Thank you for your attention