Robust counterpart optimization techniques for linear optimization and mixed integer linear optimization problems are studied in this paper. Different uncertainty sets, including those studied in literature (i.e., interval set; combined interval and ellipsoidal set; combined interval and polyhedral set) and new ones (i.e., adjustable box; pure ellipsoidal; pure polyhedral; combined interval, ellipsoidal, and polyhedral set) are studied in this work and their geometric relationship is discussed. For uncertainty in the left hand side, right hand side, and objective function of the optimization problems, robust counterpart optimization formulations induced by those different uncertainty sets are derived. Numerical studies are performed to compare the solutions of the robust counterpart optimization models and applications in refinery production planning and batch process scheduling problem are presented.

Aim To identify possible novel biomarkers in gingival crevicular fluid (GCF) samples from chronic periodontitis (CP) and periodontally healthy individuals by high-throughput proteomic analysis. Materials and Methods GCF samples were collected from twelve CP and twelve periodontally healthy subjects. Samples were trypically digested, eluted using high-performance liquid chromatography, and fragmented using tandem mass spectrometry (MS/MS). MS/MS were analyzed using PILOT_PROTEIN to identify all unmodified proteins within the samples. Results Using the database derived from Homo sapiens taxonomy and all bacterial taxonomies, 432 human (120 new) and 30 bacterial proteins were identified. The human proteins, angiotensinogen, clusterin and thymidine phosphorylase were identified as biomarker candidates based on their high-scoring only in samples from periodontal health. Similarly, neutrophil defensin-1, carbonic anhydrase-1 and elongation factor-1 gamma were associated with CP. Candidate bacterial biomarkers include 33 kDa chaperonin, iron uptake protein A2 and phosphoenolpyruvate carboxylase (health-associated) and ribulose biphosphate carboxylase, a probable succinyl-CoA:3-ketoacid-coenzyme A transferase, or DNA-directed RNA polymerase subunit beta (CP-associated). Most of these human and bacterial proteins have not been previously evaluated as biomarkers of periodontal conditions and require further investigation. Conclusions The proposed methods for large-scale comprehensive proteomic analysis may lead to the identification of novel biomarkers of periodontal disease.

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Probabilistic guarantees on constraint satisfaction for robust counterpart optimization are studied in this paper. The robust counterpart optimization formulations studied are derived from box, ellipsoidal, polyhedral, “interval+ellipsoidal” and “interval+polyhedral” uncertainty sets (Li, Z., Ding, R., and Floudas, C.A., A Comparative Theoretical and Computational Study on Robust Counterpart Optimization: I. Robust Linear and Robust Mixed Integer Linear Optimization, Ind. Eng. Chem. Res, 2011, 50, 10567). For those robust counterpart optimization formulations, their corresponding probability bounds on constraint satisfaction are derived for different types of uncertainty characteristic (i.e., bounded or unbounded uncertainty, with or without detailed probability distribution information). The findings of this work extend the results in the literature and provide greater flexibility for robust optimization practitioners in choosing tighter probability bounds so as to find less conservative robust solutions. Extensive numerical studies are performed to compare the tightness of the different probability bounds and the conservatism of different robust counterpart optimization formulations. Guiding rules for the selection of robust counterpart optimization models and for the determination of the size of the uncertainty set are discussed. Applications in production planning and process scheduling problems are presented.

Enterprise-wide optimization for the petroleum refining industry involves optimization of the supply chain involving manufacturing and distribution with emphasis on integration of the different decision making levels. The key manufacturing operations include crude oil loading and unloading, mixing of crude oil, production unit operations of conversion and separation, operations of blending, and distribution of products. Other components of the petroleum supply chain network include oil explorations, crude oil procurement, and sales and distribution of products. The main issues present in the petroleum industry across various decision levels (strategic, tactical, and operational) and within oil refinery operations are discussed. This paper presents an extensive literature review of methodologies for addressing scheduling, planning, and supply chain management of oil refinery operations. An attempt is also made to identify the future challenges in efficiently solving these problems.

This paper addresses the uncertainty problem in process scheduling using robust optimization. Compared to the traditional-scenario-based stochastic programming method, robust counterpart optimization method has a unique advantage, in that the scale of the corresponding optimization problem does not increase exponentially with the number of the uncertain parameters. Three robust counterpart optimization formulations;including Soyster's worst-case scenario formulation, Ben-Tal and Nemirovski's formulation, and a formulation proposed by Bertsimas and Sim;are studied and applied to uncertain scheduling problems in this paper. The results show that the formulation proposed by Bertsimas and Sim is the most appropriate model for uncertain scheduling problems, because it has the following advantages: (i) the model has the same size as the other formulations, (ii) it preserves its linearity, and (iii) it has the ability to control the degree of conservatism for every constraint and guarantees feasibility for the robust optimization problem.

The transient, cyclic nature and flexibility in process design make the optimization of pressure swing adsorption (PSA) computationally intensive. Two hybrid approaches incorporating machine learning methods into optimization routines are described. The first optimization approach uses artificial neural networks as surrogate models for function evaluations. The surrogates are constructed in the course of the initial optimization and utilized for function evaluations in subsequent optimization. In the second optimization approach, important design variables are identified to reduce the high-dimensional search space to a lower dimension based on partial least squares regression. The accuracy, robustness, and reliability of these approaches are demonstrated by considering a complex eight-step PSA process for precombustion CO2 capture as a case study. The machine learning-based optimization offers ∼10× reduction in computational efforts while achieving the same performance as that of the detailed models.

In this paper, we study the solution quality of robust optimization problems when they are used to approximate probabilistic constraints and propose a novel method to improve the quality. Two solution frameworks are first compared: (1) the traditional robust optimization framework which only uses the a priori probability bounds and (3) the approximation framework which uses the a posteriori probability bound. We illustrate that the traditional robust optimization method is computationally efficient but its solution is in general conservative. On the other hand, the a posteriori probability bound based method provides less conservative solution but it is computationally more difficult because a nonconvex optimization problem is solved. Based on the comparative study of the two methods, we propose a novel iterative solution framework which combines the advantage of the a priori bound and the a posteriori probability bound. The proposed method can improve the solution quality of traditional robust optimization framework without significantly increasing the computational effort. The effectiveness of the proposed method is illustrated through numerical examples and applications in planning and scheduling problems.