Integrated Bayesian Networks and Linear Programming for Decision Optimization

Abstract

This paper develops a general BN → LP framework for decision optimization under complex, structured uncertainty. A Bayesian network encodes causal dependencies among drivers and yields posterior joint probabilities; a linear program then reads expected coefficients directly from BN marginals to optimize the objective under operational constraints with explicit risk control via chance constraints or small ambiguity sets centered at the BN posterior. This mapping avoids explicit scenario enumeration and separates feasibility from credibility, so extreme but implausible cases are down-weighted rather than dictating decisions. A farm-planning case with interacting factors (weather → disease → yield; demand ↔ price; input costs) demonstrates practical feasibility. Under matched risk control, the BN → LP approach maintains the target violation rate while avoiding the over-conservatism of flat robust optimization and the optimism of independence-based stochastic programming; it also circumvents the inner minimax machinery typical of distributionally robust optimization. Tractability is governed by BN inference over the decision-relevant ancestor subgraph; empirical scaling shows that Markov-blanket pruning, mutual-information screening of weak parents, and structured/low-rank CPDs yield orders-of-magnitude savings with negligible impact on the objective. A standardized, data-and-expert construction (Dirichlet smoothing) and a systematic sensitivity analysis identifies high-leverage parameters, while a receding-horizon DBN → LP extension supports online updates. The method brings the largest benefits when uncertainty is high-dimensional and coupled, and it converges to classical allocations when drivers are few and essentially independent.