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Models for Multiple Criteria Decision Analysis (MCDA) often separate per-criterion attractiveness evaluation from weighted aggregation of these evaluations across the different criteria. In simulation-based MCDA methods, such as Stochastic Multicriteria Acceptability Analysis, uncertainty in the weights is modeled through a uniform distribution on the feasible weight space defined by a set of linear constraints. Efficient sampling methods have been proposed for special cases, such as the unconstrained weight space or complete ordering of the weights. However, no efficient methods are available for other constraints such as imprecise trade-off ratios, and specialized sampling methods do not allow for flexibility in combining the different constraint types. In this paper, we explore how the Hit-And-Run sampler can be applied as a general approach for sampling from the convex weight space that results from an arbitrary …