27 research outputs found
Convexification of Queueing Formulas by Mixed-Integer Second-Order Cone Programming: An Application to a Discrete Location Problem with Congestion
Mixed-Integer Second-Order Cone Programs (MISOCPs) form a nice class of
mixed-inter convex programs, which can be solved very efficiently due to the
recent advances in optimization solvers. Our paper bridges the gap between
modeling a class of optimization problems and using MISOCP solvers. It is shown
how various performance metrics of M/G/1 queues can be molded by different
MISOCPs. To motivate our method practically, it is first applied to a
challenging stochastic location problem with congestion, which is broadly used
to design socially optimal service networks. Four different MISOCPs are
developed and compared on sets of benchmark test problems. The new formulations
efficiently solve large-size test problems, which cannot be solved by the best
existing method. Then, the general applicability of our method is shown for
similar optimization problems that use queue-theoretic performance measures to
address customer satisfaction and service quality
Prescriptive PCA: Dimensionality Reduction for Two-stage Stochastic Optimization
In this paper, we consider the alignment between an upstream dimensionality
reduction task of learning a low-dimensional representation of a set of
high-dimensional data and a downstream optimization task of solving a
stochastic program parameterized by said representation. In this case, standard
dimensionality reduction methods (e.g., principal component analysis) may not
perform well, as they aim to maximize the amount of information retained in the
representation and do not generally reflect the importance of such information
in the downstream optimization problem. To address this problem, we develop a
prescriptive dimensionality reduction framework that aims to minimize the
degree of suboptimality in the optimization phase. For the case where the
downstream stochastic optimization problem has an expected value objective, we
show that prescriptive dimensionality reduction can be performed via solving a
distributionally-robust optimization problem, which admits a semidefinite
programming relaxation. Computational experiments based on a warehouse
transshipment problem and a vehicle repositioning problem show that our
approach significantly outperforms principal component analysis with real and
synthetic data sets
Quality vs. Quantity of Data in Contextual Decision-Making: Exact Analysis under Newsvendor Loss
When building datasets, one needs to invest time, money and energy to either
aggregate more data or to improve their quality. The most common practice
favors quantity over quality without necessarily quantifying the trade-off that
emerges. In this work, we study data-driven contextual decision-making and the
performance implications of quality and quantity of data. We focus on
contextual decision-making with a Newsvendor loss. This loss is that of a
central capacity planning problem in Operations Research, but also that
associated with quantile regression. We consider a model in which outcomes
observed in similar contexts have similar distributions and analyze the
performance of a classical class of kernel policies which weigh data according
to their similarity in a contextual space. We develop a series of results that
lead to an exact characterization of the worst-case expected regret of these
policies. This exact characterization applies to any sample size and any
observed contexts. The model we develop is flexible, and captures the case of
partially observed contexts. This exact analysis enables to unveil new
structural insights on the learning behavior of uniform kernel methods: i) the
specialized analysis leads to very large improvements in quantification of
performance compared to state of the art general purpose bounds. ii) we show an
important non-monotonicity of the performance as a function of data size not
captured by previous bounds; and iii) we show that in some regimes, a little
increase in the quality of the data can dramatically reduce the amount of
samples required to reach a performance target. All in all, our work
demonstrates that it is possible to quantify in a precise fashion the interplay
of data quality and quantity, and performance in a central problem class. It
also highlights the need for problem specific bounds in order to understand the
trade-offs at play
A Machine Learning Approach to Two-Stage Adaptive Robust Optimization
We propose an approach based on machine learning to solve two-stage linear
adaptive robust optimization (ARO) problems with binary here-and-now variables
and polyhedral uncertainty sets. We encode the optimal here-and-now decisions,
the worst-case scenarios associated with the optimal here-and-now decisions,
and the optimal wait-and-see decisions into what we denote as the strategy. We
solve multiple similar ARO instances in advance using the column and constraint
generation algorithm and extract the optimal strategies to generate a training
set. We train a machine learning model that predicts high-quality strategies
for the here-and-now decisions, the worst-case scenarios associated with the
optimal here-and-now decisions, and the wait-and-see decisions. We also
introduce an algorithm to reduce the number of different target classes the
machine learning algorithm needs to be trained on. We apply the proposed
approach to the facility location, the multi-item inventory control and the
unit commitment problems. Our approach solves ARO problems drastically faster
than the state-of-the-art algorithms with high accuracy
A Robust Optimization of Capacity Allocation Policies in the Third-Party Warehouse
We study the capacity allocation policies of a third-party warehouse center, which supplies
several different level services on different prices with fixed capacity, on revenue management
perspective. For the single period situation, we use three different robust methods, absolute
robust, deviation robust, and relative robust method, to maximize the whole revenue. Then we give some
numerical examples to verify the practical applicability. For the multiperiod situation, as the
demand is uncertain, we propose a stochastic model for the multiperiod revenue management
problem of the warehouse. A novel robust optimization technique is applied in this model to
maximize the whole revenue. Then we give some numerical examples to verify the practical
applicability of our method