Simge Küçükyavuz

On mixing sets arising in chance-constrained programming

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.

Uncapacitated lot sizing with backlogging: the convex hull

An explicit description of the convex hull of solutions to the uncapacitated lot-sizing problem with backlogging, in its natural space of production, setup, inventory and backlogging variables, has been an open question for many years. In this paper, we identify valid inequalities that subsume all previously known valid inequalities for this problem. We show that these inequalities are enough to describe the convex hull of solutions. We give polynomial separation algorithms for some special cases. Finally, we report a summary of computational experiments with our inequalities that illustrates their effectiveness.

Lot Sizing with Inventory Bounds and Fixed Costs: Polyhedral Study and Computation

We investigate the polyhedral structure of the lot-sizing problem with inventory bounds. We consider two models, one with linear cost on inventory, the other with linear and fixed costs on inventory. For both models, we identify facet-defining inequalities that make use of the inventory bounds explicitly and give exact separation algorithms. We also describe a linear programming formulation of the problem when the order and inventory costs satisfy the Wagner-Whitin nonspeculative property. We present computational experiments that show the effectiveness of the results in tightening the linear programming relaxations of the lot-sizing problem with inventory bounds and fixed costs.

Decomposition algorithms with parametric Gomory cuts for two-stage stochastic integer programs

We consider a class of two-stage stochastic integer programs with binary variables in the first stage and general integer variables in the second stage. We develop decomposition algorithms akin to the

Chance-Constrained Binary Packing Problems

L

A Polyhedral Study of Multiechelon Lot Sizing with Intermediate Demands

-shaped or Benders’ methods by utilizing Gomory cuts to obtain iteratively tighter approximations of the second-stage integer programs. We show that the proposed methodology is flexible in that it allows several modes of implementation, all of which lead to finitely convergent algorithms. We illustrate our algorithms using examples from the literature. We report computational results using the stochastic server location problem instances which suggest that our decomposition-based approach scales better with increases in the number of scenarios than a state-of-the art solver which was used to solve the deterministic equivalent formulation.

A polyhedral study of production ramping

We present an O(n^2) dynamic programming algorithm for lot sizing with inventory bounds and fixed costs, where n is the number of time periods. The algorithm utilizes a hierarchy of two layers of value functions and improves the complexity bound of an earlier O(n^3) algorithm for concave-cost and bounded inventory.

A Branch-and-Cut Method for Dynamic Decision Making Under Joint Chance Constraints

We consider a class of packing problems with uncertain data, which we refer to as the chance-constrained binary packing problem. In this problem, a subset of items is selected that maximizes the total profit so that a generic packing constraint is satisfied with high probability. Interesting special cases of our problem include chance-constrained knapsack and set packing problems with random coefficients. We propose a problem formulation in its original space based on the so-called probabilistic covers . We focus our solution approaches on the special case in which the uncertainty is represented by a finite number of scenarios. In this case, the problem can be formulated as an integer program by introducing a binary decision variable to represent feasibility of each scenario. We derive a computationally efficient coefficient strengthening procedure for this formulation, and demonstrate how the scenario variables can be efficiently projected out of the linear programming relaxation. We also study how methods for lifting deterministic cover inequalities can be leveraged to perform approximate lifting of probabilistic cover inequalities. We conduct an extensive computational study to illustrate the potential benefits of our proposed techniques on various problem classes.