Stochastic Constraint Programming: A Scenario-Based Approach

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 [Van 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.

[1]  Jitka Dupacová,et al.  Scenario reduction in stochastic programming , 2003, Math. Program..

[2]  R. M. Oliver,et al.  Influence diagrams, belief nets and decision analysis , 1992 .

[3]  Les G. Proll,et al.  Integer Linear Programming and Constraint Programming Approaches to a Template Design Problem , 1998, INFORMS J. Comput..

[4]  Michael L. Littman,et al.  Constraint Satisfaction with Probabilistic Preferences on Variable Values , 1999, AAAI 1999.

[5]  Thomas Schiex,et al.  A constraint satisfaction framework for decision under uncertainty , 1995, UAI.

[6]  Jérôme Lang,et al.  Uncertainty in Constraint Satisfaction Problems: a Probalistic Approach , 1993, ECSQARU.

[7]  M. Allais.  Le comportement de l'homme rationnel devant le risque : critique des postulats et axiomes de l'ecole americaine , 1953 .

[8]  Ş. Tarim,et al.  The stochastic dynamic production/inventory lot-sizing problem with service-level constraints , 2004 .

[9]  Peter Kall,et al.  Stochastic Programming , 1995 .

[10]  R. M. Dudley,et al.  Influence Diagrams, Belief Nets and Decision Analysis. , 1991 .

[11]  Michael L. Littman,et al.  The Computational Complexity of Probabilistic Planning , 1998, J. Artif. Intell. Res..

[12]  Robert J. Vanderbei,et al.  Robust Optimization of Large-Scale Systems , 1995, Oper. Res..

[13]  Martin L. Puterman,et al.  Markov Decision Processes: Discrete Stochastic Dynamic Programming , 1994 .

[14]  J. Dupacová,et al.  Scenario reduction in stochastic programming: An approach using probability metrics , 2000 .

[15]  H. Konno,et al.  Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market , 1991 .

[16]  Thomas Schiex,et al.  Semiring-Based CSPs and Valued CSPs: Basic Properties and Comparison , 1995, Over-Constrained Systems.

[17]  Toby Walsh,et al.  Stochastic Constraint Programming , 2002, ECAI.

[18]  S. Rachev,et al.  Mass transportation problems , 1998 .

[19]  Carmen Gervet,et al.  Certainty Closure: A Framework for Reliable Constraint Reasoning with Uncertainty , 2003, CP.

[20]  Kenneth N. Brown,et al.  Branching Constraint Satisfaction Problems for Solutions Robust under Likely Changes , 2000, CP.

[21]  Toniann Pitassi,et al.  Stochastic Boolean Satisfiability , 2001, Journal of Automated Reasoning.

[22]  Richard J. Beckman,et al.  A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output From a Computer Code , 2000, Technometrics.

[23]  A Gerodimos,et al.  Robust Discrete Optimization and its Applications , 1996, J. Oper. Res. Soc..

[24]  Thomas Schiex,et al.  Semi-ring based CSPs and valued CSPs: Basic properties and com-parison , 1996 .

[25]  Hiroshi Konno,et al.  PIECEWISE LINEAR RISK FUNCTION AND PORTFOLIO OPTIMIZATION , 1990 .

[26]  Thomas Schiex,et al.  Mixed Constraint Satisfaction: A Framework for Decision Problems under Incomplete Knowledge , 1996, AAAI/IAAI, Vol. 1.

[27]  Pedro Barahona,et al.  PSICO: Solving Protein Structures with Constraint Programming and Optimization , 2002, Constraints.

[28]  Michael L. Littman,et al.  Solving Crossword Puzzles as Probabilistic Constraint Satisfaction , 1999, AAAI/IAAI.

[29]  Mtw,et al.  Mass Transportation Problems: Vol. I: Theory@@@Mass Transportation Problems: Vol. II: Applications , 1999 .

[30]  Richard J. Wallace,et al.  Partial Constraint Satisfaction , 1989, IJCAI.

[31]  Pascal Van Hentenryck,et al.  Constraint Programming in OPL , 1999, PPDP.