Stochastic Constraint Programming: A Scenario-Based Approach

arXiv:0903.1150v1 [cs.AI] 6 Mar 2009 Submission to CONSTRAINTS Stochastic Constraint Programming: A Scenario-Based Approach Abstract To model combinatorial decision problems involving uncertainty and probability, we introduce scenario based stochastic constraint programming. Stochastic constraint programs contain both decision variables, which we can set, and stochastic variables, which follow a discrete probability distribution. We provide a semantics for stochastic constraint programs based on scenario trees. Using this semantics, we can compile stochastic constraint programs down into conventional (non- stochastic) constraint programs. This allows us to exploit the full power of existing constraint solvers. We have implemented this framework for decision making under uncertainty in stochastic OPL, a language which is based on the OPL constraint modelling language [Hentenryck et al., 1999]. To illustrate the potential of this framework, we model a wide range of problems in areas as diverse as portfolio diversification, agricultural planning and production/inventory management. Keywords: constraint programming, constraint satisfaction, reasoning under uncertainty S. Armagan Tarim (primary contact author) Cork Constraint Computation Centre, Department of Computer Science, University College Cork, Cork, Ireland Tel: +353-21-4255411 at@4c.ucc.ie, http://www.cs.york.ac.uk/∼at/ Suresh Manandhar Artificial Intelligence Group, Department of Computer Science, University of York, York, UK Tel: +44-1904-432746 suresh@cs.york.ac.uk, http://www.cs.york.ac.uk/∼suresh/ Toby Walsh National ICT Australia and School of CSE University of New South Wales, Sydney, Australia Tel:+61-2-93857343 tw@cse.unsw.edu.au, http://4c.ucc.ie/∼tw Stochastic Constraint Programming: A Scenario-Based Approach S. Armagan Tarim, Cork Constraint Computation Centre, University College Cork, Ireland Suresh Manandhar, Department of Computer Science, University of York, United Kingdom Toby Walsh, National ICT Australia and School of Computer Science and Engineering, University of New South Wales, Australia Abstract To model combinatorial decision problems involving uncertainty and probability, we introduce scenario based stochastic constraint programming. Stochastic constraint programs contain both decision variables, which we can set, and stochastic variables, which follow a discrete probability distribution. We provide a semantics for stochastic constraint programs based on scenario trees. Using this semantics, we can compile stochastic constraint programs down into conventional (non- stochastic) constraint programs. This allows us to exploit the full power of existing constraint solvers. We have implemented this framework for decision making under uncertainty in stochastic OPL, a language which is based on the OPL constraint modelling language [Hentenryck et al., 1999]. To illustrate the potential of this framework, we model a wide range of problems in areas as diverse as portfolio diversification, agricultural planning and production/inventory management. Keywords: constraint programming, constraint satisfaction, reasoning under uncertainty Content areas: constraint programming, constraint satisfaction, reasoning under uncertainty 1 Introduction Many decision problems contain uncertainty. Data about events in the past may not be known exactly due to errors in measuring or difficulties in sampling, whilst data about events in the future may simply not be known with certainty. For example, when scheduling power stations, we need to cope with uncertainty in future energy demands. As a second example, nurse rostering in an accident and emergency department requires us to anticipate variability in workload. As a final example, when constructing a balanced bond portfolio, we must deal with uncertainty in the future price of bonds. To deal with such situations, [27] proposed an extension of constraint programming, called stochastic constraint programming, in which we distinguish between decision variables, which we are free to set, and stochastic (or observed) variables, which follow some probability distribution. A semantics for stochastic constraint programs based on policies was proposed and backtracking and forward checking algorithms to solve such stochastic constraint programs were presented. 1 In this paper, we extend this framework to make it more useful practically. In particular, we permit multiple chance constraints and a range of different objectives. As each such extension requires large changes to the backtracking and forward checking algorithms, we propose instead a scenario based view of stochastic constraint programs. One of the major advantages of this approach is that stochastic constraint programs can then be compiled down into conventional (non-stochastic) constraint pro- grams. This compilation allows us to use existing constraint solvers without any modification, as well as call upon the power of hybrid solvers which combine constraint

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