Kyungsik Lee

Typically, two-stage stochastic programs have been modeled and solved based on the finite support assumption, but the large number of scenarios makes it hard to solve, and there also are potential risks of inaccurate estimation of underlying distribution. In this paper, to mitigate the drawbacks, we present a novel risk-averse two-stage stochastic program with finite support, which we call partition-based risk-averse two-stage stochastic program. In the program, a set of scenarios is partitioned into several groups, and the second-stage cost is defined as the expectation of risk levels for all of the groups. In particular, the conditional value-at-risk is considered as a risk measure for each group, and so the risk level of the model is affected by a quantile parameter or a partition of a given set of scenarios. In order to solve the model exactly for a given partition, a column-and-constraint generation algorithm is proposed. In addition, a scenario partitioning algorithm to enable the risk level of the model to be close to a given target is devised, and partitioning schemes for combining it with the proposed column-and-constraint generation algorithm are proposed. Extensive numerical experiments were performed that demonstrated the effectiveness of the proposed partitioning schemes and the efficiency of the proposed solution approach.

In this paper, we consider the problem of production capacity estimation for a semiconductor wafer fabrication facility. Capacity estimation involves determining the maximum achievable throughput of a wafer fabrication facility during a given planning horizon in consideration of both product mix and target cycle time. The wafer fabrication facility (fab) is one of the most complex production systems, consisting of hundreds of process steps for each product as well as thousands of processing machines and re-entrant process flows wherein products must visit the same workcenter multiple times. In this regard, estimating production capacity by modeling the wafer manufacturing process is a challenging problem. To properly capture the dynamics of the process, we propose a flexible-lead-time-based optimization model that considers both the state of work-in-process (WIP) over time and the relationship between WIP levels and lead times. The results of simulation experiments using a real-sized instance demonstrate the advantages of the proposed model over existing alternatives.

Principal component analysis (PCA) has been a widely used technique for dimension reduction while retaining essential information. However, the ordinary PCA lacks interpretability, especially when dealing with large scale data. To address this limitation, sparse PCA (SPCA) has emerged as an interpretable variant of ordinary PCA. However, the ordinary SPCA relies on solving a challenging non-convex discrete optimization problem, which maximizes explained variance while constraining the number of non-zero elements in each principal component. In this paper, we propose an innovative least angle SPCA technique to address the computational complexity associated with SPCA, particularly in ultrahigh dimensional data, by sequentially identifying sparse principal components with minimal angles to their corresponding components extracted through ordinary PCA. This sequential identification enables solving the optimization problem in polynomial time, significantly reducing computational challenges. Despite its efficiency gains, our proposed method also preserves the main attributes of SPCA. Through comprehensive experimental results, we demonstrate advantages of our approach as a viable alternative for dealing with the computational difficulties inherent in ordinary SPCA. Notably, our method emerges as an efficient and effective solution for conducting ultrahigh dimensional data analysis, enabling researchers to extract meaningful insights and streamline data interpretation.

In this study, we theoretically compare integer programming models for the two-dimensional two-staged knapsack problem. Including the well-known level packing model, we introduce two pattern-based models called the strip packing model and the staged pattern model derived from integer programming models for the two-dimensional two-staged cutting stock problem. We show that the level packing model provides weaker linear programming (LP) relaxation bounds than pattern-based models. Furthermore, we also present upper bounds on the LP-relaxation bound of the level packing model, which can be obtained from the LP-relaxation bounds of the pattern-based models.

We consider the chance-constrained binary knapsack problem (CKP), where the item weights are independent and normally distributed. We introduce a continuous relaxation for the CKP, represented as a non-convex optimization problem, which we call the non-convex relaxation. A comparative study shows that the non-convex relaxation provides an upper bound for the CKP, at least as tight as those obtained from other continuous relaxations for the CKP. Furthermore, the quality of the obtained upper bound is guaranteed to be at most twice the optimal objective value of the CKP. Despite its non-convex nature, we show that the non-convex relaxation can be solved in polynomial time. Subsequently, we proposed a polynomial-time 1/2-approximation algorithm for the CKP based on this relaxation, providing a lower bound for the CKP. Computational test results demonstrate that the non-convex relaxation and the proposed approximation algorithm yields tight lower and upper bounds for the CKP within a short computation time, ensuring the quality of the obtained bounds.