## Abstract

The mixing set with a knapsack constraint arises in deterministic equivalent of chance-constrained programming problems with finite discrete distributions. We first consider the case that the chance-constrained program has equal probabilities for each scenario. We study the resulting mixing set with a cardinality constraint and propose facet-defining inequalities that subsume known explicit inequalities for this set. We extend these inequalities to obtain valid inequalities for the mixing set with a knapsack constraint. In addition, we propose a compact extended reformulation (with polynomial number of variables and constraints) that characterizes a linear programming equivalent of a single chance constraint with equal scenario probabilities. We introduce a *blending* procedure to find valid inequalities for intersection of multiple mixing sets. We propose a polynomial-size extended formulation for the intersection of multiple mixing sets with a knapsack constraint that is stronger than the original mixing formulation. We also give a compact extended linear program for the intersection of multiple mixing sets and a cardinality constraint for a special case. We illustrate the effectiveness of the proposed inequalities in our computational experiments with probabilistic lot-sizing problems.

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This study was supported, in part, by NSF-CMMI Grant 0917952.

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Küçükyavuz, S. On mixing sets arising in chance-constrained programming.
*Math. Program.* **132, **31–56 (2012). https://doi.org/10.1007/s10107-010-0385-3

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DOI: https://doi.org/10.1007/s10107-010-0385-3

### Keywords

- Mixed-integer programming
- Facets
- Compact extended formulations
- Chance constraints
- Lot-sizing
- Computation

### Mathematics Subject Classification (2000)

- 90C11
- 90C57