On the Use of Integer Programming Models in AI Planning

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On the Use of Integer Programming Models in AI
Planning

• by
• Thomas Vossen, Michael Ball,
• Amnon Lotem, Dana Nau

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Overview

• Borrows Integer Programming (IP) from Operations
  Research (OR)
• Applies IP to AI Planning

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What is Integer Programming?

• An extension of linear programming where some
  variables are constrained to integer values.
• So whats linear programming?

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LP Definition and Introduction to Graphical
SolutionActive Learning Module 2

• J. René Villalobos and Gary L. Hogg
• Arizona State University
• Paul M. Griffin
• Georgia Institute of Technology

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The Windsor Glass Company Problem (Hillier and
Liberman)

• The Windsor Glass Company is planning to launch
  two new products. Product 1 is an 8-foot glass
  door with aluminum framing and Product 2 a 4x6
  foot double-hung wood-framed window
• Aluminum frames are made in Plant 1, wood frames
  are made in Plant 2, and Plant 3 produces the
  glass and assembles the products. Product 1
  requires some of the production capacity in
  Plants 1 and 3, but none in Plant 2. Product 2
  needs only Plants 2 and 3. The marketing
  division has concluded that the company could
  sell as much of either product as could be
  processed by these plants. The management of the
  company wants to determine what mixture of both
  products would be the most profitable. The
  following table provides the information
  available.

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The Windsor Glass Company Problem Formulation
(Hillier and Liberman)
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Why use IP

• IP formulations quite naturally allow the
  incorporation of numeric constraints and
  objectives into planning domains.
• Can apply work from OR to AI Planning.

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Two Formulations

• SATPLAN-based formulation, based on Kautz and
  Selman
• State-change formulation

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Sets used in the formulations
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More Sets
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SATPLAN - Variables
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SATPLAN - Constraints
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SATPLAN - Constraints contd
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SATPLAN - Constraints contd
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SATPLAN - Constraints contd
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SATPLAN - Objective function

• In IP the objective function gives the goal of
  the search
• Here it was set to minimize the number of actions
  in the plan

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State-change formulation

• Fluent variables are replaced by state change
  variables
• Propagation through no-ops is restricted to cut
  down on equivalent solutions

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More variables
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More variables contd
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State Change - Constraints
These constraints represent mutual exclusion.
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State Change Constraints contd
This is equivalent to the backward chaining
constraint of the SATPLAN formulation
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State Change Constraints contd
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Problem

• Performance critically depends on how problems
  are formulated as Integer Programs

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Conclusions

• IP has the potential to do efficient planning
• More work is needed to incorporate numeric
  constraints and to develop better encodings