Data‐driven adaptive nested robust optimization: General modeling framework and efficient computational algorithm for decision making under uncertainty

A novel data‐driven adaptive robust optimization framework that leverages big data in process industries is proposed. A Bayesian nonparametric model—the Dirichlet process mixture model—is adopted and combined with a variational inference algorithm to extract the information embedded within uncertainty data. Further a data‐driven approach for defining uncertainty set is proposed. This machine‐learning model is seamlessly integrated with adaptive robust optimization approach through a novel four‐level optimization framework. This framework explicitly accounts for the correlation, asymmetry and multimode of uncertainty data, so it generates less conservative solutions. Additionally, the proposed framework is robust not only to parameter variations, but also to anomalous measurements. Because the resulting multilevel optimization problem cannot be solved directly by any off‐the‐shelf solvers, an efficient column‐and‐constraint generation algorithm is proposed to address the computational challenge. Two industrial applications on batch process scheduling and on process network planning are presented to demonstrate the advantages of the proposed modeling framework and effectiveness of the solution algorithm. © 2017 American Institute of Chemical Engineers AIChE J, 63: 3790–3817, 2017     

Resilient supply chain design and operations with decision-dependent uncertainty using a data-driven robust optimization approach

To addresses the design and operations of resilient supply chains under uncertain disruptions, a general framework is proposed for resilient supply chain optimization, including a quantitative measure of resilience and a holistic biobjective two-stage adaptive robust fractional programming model with decision-dependent uncertainty set for simultaneously optimizing both the economic objective and the resilience objective of supply chains. The decision-dependent uncertainty set ensures that the uncertain parameters (e.g., the remaining production capacities of facilities after disruptions) are dependent on first-stage decisions, including facility location decisions and production capacity decisions. A data-driven method is used to construct the uncertainty set to fully extract information from historical data. Moreover, the proposed model takes the time delay between disruptions and recovery into consideration. To tackle the computational challenge of solving the resulting multilevel optimization problem, two solution strategies are proposed. The applicability of the proposed approach is illustrated through applications on a location-transportation problem and on a spatially-explicit biofuel supply chain optimization problem. © 2018 American Institute of Chemical Engineers AIChE J, 65: 1006–1021, 2019     

改进PSO算法在蒸汽动力系统多周期运行优化中的应用

为了满足石化企业工艺过程对蒸汽和电力不断变化的要求,实现企业降低成本、节能降耗的目的,必须保证蒸汽动力系统在最优的状态下运行。针对这一问题在以往研究的基础上提出了包括设备维护和启停费用的改进的混合整数线性规划模型,利用改进的PSO算法对其求解,并通过实例证明了利用该模型、使用改进的PSO算法能很快得到最优的方案,并节省了大量的运行成本。     

A data‐driven multistage adaptive robust optimization framework for planning and scheduling under uncertainty

A novel data‐driven approach for optimization under uncertainty based on multistage adaptive robust optimization (ARO) and nonparametric kernel density M‐estimation is proposed. Different from conventional robust optimization methods, the proposed framework incorporates distributional information to avoid over‐conservatism. Robust kernel density estimation with Hampel loss function is employed to extract probability distributions from uncertainty data via a kernelized iteratively reweighted least squares algorithm. A data‐driven uncertainty set is proposed, where bounds of uncertain parameters are defined by quantile functions, to organically integrate the multistage ARO framework with uncertainty data. Based on this uncertainty set, we further develop an exact robust counterpart in its general form for solving the resulting data‐driven multistage ARO problem. To illustrate the applicability of the proposed framework, two typical applications in process operations are presented: The first one is on strategic planning of process networks, and the other one on short‐term scheduling of multipurpose batch processes. The proposed approach returns 23.9% higher net present value and 31.5% more profits than the conventional robust optimization method in planning and scheduling applications, respectively. © 2017 American Institute of Chemical Engineers AIChE J, 63: 4343–4369, 2017     

Proactive scheduling of batch processes by a combined robust optimization and multiparametric programming approach

We address short‐term batch process scheduling problems contaminated with uncertainty in the data. The mixed integer linear programming (MILP) scheduling model, based on the formulation of Ierapetritou and Floudas, Ind Eng Chem Res. 1998; 37(11):4341–4359, contains parameter dependencies at multiple locations, yielding a general multiparametric (mp) MILP problem. A proactive scheduling policy is obtained by solving the partially robust counterpart formulation. The counterpart model may remain a multiparametric problem, yet it is immunized against uncertainty in the entries of the constraint matrix and against all parameters whose values are not available at the time of decision making. We extend our previous work on the approximate solution of mp‐MILP problems by embedding different uncertainty sets (box, ellipsoidal and budget parameter regulated uncertainty), and by incorporating information about the availability of uncertain data in the construction of the partially robust scheduling model. For any parameter realization, the corresponding schedule is then obtained through function evaluation. © 2013 American Institute of Chemical Engineers AIChE J, 59: 4184–4211, 2013     

Data-driven distributionally robust optimization of shale gas supply chains under uncertainty

This article aims to leverage the big data in shale gas industry for better decision making in optimal design and operations of shale gas supply chains under uncertainty. We propose a two-stage distributionally robust optimization model, where uncertainties associated with both the upstream shale well estimated ultimate recovery and downstream market demand are simultaneously considered. In this model, decisions are classified into first-stage design decisions, which are related to drilling schedule, pipeline installment, and processing plant construction, as well as second-stage operational decisions associated with shale gas production, processing, transportation, and distribution. A data-driven approach is applied to construct the ambiguity set based on principal component analysis and first-order deviation functions. By taking advantage of affine decision rules, a tractable mixed-integer linear programming formulation can be obtained. The applicability of the proposed modeling framework is demonstrated through a small-scale illustrative example and a case study of Marcellus shale gas supply chain. Comparisons with alternative optimization models, including the deterministic and stochastic programming counterparts, are investigated as well. © 2018 American Institute of Chemical Engineers AIChE J, 65: 947–963, 2019     

多因素不确定条件下的间歇生产调度优化

不确定条件下的间歇生产调度优化是生产调度问题研究中具有挑战性的课题。提出了一种基于混合整数线性规划(MILP)的鲁棒优化模型,来优化不确定条件下的生产调度决策。考虑到生产过程中的操作成本和原料成本,建立了以净利润最大为调度目标的确定性数学模型。然后考虑需求、处理时间、市场价格三种不确定因素,建立可调整保守程度的鲁棒优化模型并转换成鲁棒对应模型。实例结果表明,鲁棒优化的间歇生产调度模型较确定性模型利润减少,但生产任务数量增加,设备空闲时间缩短,从而增强了调度方案的可靠性,实现了不确定条件下生产操作性和经济性的综合优化。     

Multistage adaptive stochastic mixed integer optimization under endogenous and exogenous uncertainty

To solve multistage adaptive stochastic optimization problems under both endogenous and exogenous uncertainty, a novel solution framework based on robust optimization technique is proposed. The endogenous uncertainty is modeled as scenarios based on an uncertainty set partitioning method. For each scenario, the adaptive binary decision is assumed constant and the continuous variable is approximated by a function linearly dependent on endogenous uncertain parameters. The exogenous uncertainty is modeled using lifting methods. The adaptive decisions are approximated using affine functions of the lifted uncertain parameters. In order to demonstrate the applicability of the proposed framework, a number of numerical examples of different complexity are studied and a case study for infrastructure and production planning of shale gas field development are presented. The results show that the proposed framework can effectively solve multistage adaptive stochastic optimization problems under both types of uncertainty.     

Nonlinear robust optimization for process design

A novel robust optimization framework is proposed to address general nonlinear problems in process design. Local linearization is taken with respect to the uncertain parameters around multiple realizations of the uncertainty, and an iterative algorithm is implemented to solve the problem. Furthermore, the proposed methodology can handle different categories of problems according to the complexity of the problems. First, inequality‐only constrained optimization problem as studied in most existing robust optimization methods can be addressed. Second, the proposed framework can deal with problems with equality constraint associated with uncertain parameters. In the final case, we investigate problems with operation variables which can be adjusted according to the realizations of uncertainty. A local affinely adjustable decision rule is adopted for the operation variables (i.e., an affine function of the uncertain parameter). Different applications corresponding to different classes of problems are used to demonstrate the effectiveness of the proposed nonlinear robust optimization framework. © 2017 American Institute of Chemical Engineers AIChE J, 64: 481–494, 2018     

On the relation between flexibility analysis and robust optimization for linear systems

Flexibility analysis and robust optimization are two approaches to solving optimization problems under uncertainty that share some fundamental concepts, such as the use of polyhedral uncertainty sets and the worst‐case approach to guarantee feasibility. The connection between these two approaches has not been sufficiently acknowledged and examined in the literature. In this context, the contributions of this work are fourfold: (1) a comparison between flexibility analysis and robust optimization from a historical perspective is presented; (2) for linear systems, new formulations for the three classical flexibility analysis problems—flexibility test, flexibility index, and design under uncertainty—based on duality theory and the affinely adjustable robust optimization (AARO) approach are proposed; (3) the AARO approach is shown to be generally more restrictive such that it may lead to overly conservative solutions; (4) numerical examples show the improved computational performance from the proposed formulations compared to the traditional flexibility analysis models. © 2016 American Institute of Chemical Engineers AIChE J, 62: 3109–3123, 2016     

Multi‐stage adjustable robust optimization for process scheduling under uncertainty

Variations in parameters such as processing times, yields, and availability of materials and utilities can have a detrimental effect in the optimality and/or feasibility of an otherwise “optimal” production schedule. In this article, we propose a multi‐stage adjustable robust optimization approach to alleviate the risk from such operational uncertainties during scheduling decisions. We derive a novel robust counterpart of a deterministic scheduling model, and we show how to obey the observability and non‐anticipativity restrictions that are necessary for the resulting solution policy to be implementable in practice. We also develop decision‐dependent uncertainty sets to model the endogenous uncertainty that is inherently present in process scheduling applications. A computational study reveals that, given a chosen level of robustness, adjusting decisions to past parameter realizations leads to significant improvements, both in terms of worst‐case objective as well as objective in expectation, compared to the traditional robust scheduling approaches. © 2016 American Institute of Chemical Engineers AIChE J, 62: 1646–1667, 2016     

Optimal supply chain design and operations under multi‐scale uncertainties: Nested stochastic robust optimization modeling framework and solution algorithm

Although strategic and operational uncertainties differ in their significance of impact, a “one‐size‐fits‐all” approach has been typically used to tackle all types of uncertainty in the optimal design and operations of supply chains. In this work, we propose a stochastic robust optimization model that handles multi‐scale uncertainties in a holistic framework, aiming to optimize the expected economic performance while ensuring the robustness of operations. Stochastic programming and robust optimization approaches are integrated in a nested manner to reflect the decision maker's different levels of conservativeness toward strategic and operational uncertainties. The resulting multi‐level mixed‐integer linear programming model is solved by a decomposition‐based column‐and‐constraint generation algorithm. To illustrate the application, a county‐level case study on optimal design and operations of a spatially‐explicit biofuel supply chain in Illinois is presented, which demonstrates the advantages and flexibility of the proposed modeling framework and efficiency of the solution algorithm. © 2016 American Institute of Chemical Engineers AIChE J, 62: 3041–3055, 2016     

Theoretical and computational comparison of continuous‐time process scheduling models for adjustable robust optimization

Coping with uncertainty in system parameters is a prominent hurdle when scheduling multi‐purpose batch plants. In this context, our previously introduced multi‐stage adjustable robust optimization (ARO) framework has been shown to obtain more profitable solutions, while maintaining the same level of immunity against risk, as compared to traditional robust optimization approaches. This paper investigates the amenability of existing deterministic continuous‐time scheduling models to serve as the basis of this ARO framework. A comprehensive computational study is conducted that compares the numerical tractability of various models across a suite of literature benchmark instances and a wide range of uncertainty sets. This study also provides, for the first time in the open literature, robust optimal solutions to process scheduling instances that involve uncertainty in production yields. © 2018 American Institute of Chemical Engineers AIChE J, 64: 3055–3070, 2018     

A computational framework and solution algorithms for two‐stage adaptive robust scheduling of batch manufacturing processes under uncertainty

A novel two‐stage adaptive robust optimization (ARO) approach to production scheduling of batch processes under uncertainty is proposed. We first reformulate the deterministic mixed‐integer linear programming model of batch scheduling into a two‐stage optimization problem. Symmetric uncertainty sets are then introduced to confine the uncertain parameters, and budgets of uncertainty are used to adjust the degree of conservatism. We then apply both the Benders decomposition algorithm and the column‐and‐constraint generation (C&CG) algorithm to efficiently solve the resulting two‐stage ARO problem, which cannot be tackled directly by any existing optimization solvers. Two case studies are considered to demonstrate the applicability of the proposed modeling framework and solution algorithms. The results show that the C&CG algorithm is more computationally efficient than the Benders decomposition algorithm, and the proposed two‐stage ARO approach returns 9% higher profits than the conventional robust optimization approach for batch scheduling. © 2015 American Institute of Chemical Engineers AIChE J, 62: 687–703, 2016     

Simultaneous design and operation optimization of renewable combined cooling heating and power systems

Combined cooling, heating, and power (CCHP) systems are promising solutions for conserving energy and reducing emissions. This article proposes a new mixed-integer linear programming (MILP) model for simultaneous design and operation optimization of a renewable CCHP system, considering component nonlinear operating characteristics and performance degradation with time. A bi-objective MILP problem is solved to achieve a trade-off between total annual cost (TAC) and greenhouse gas emissions (GHGe). A case study of a commercial region is employed to demonstrate our proposed methodology. The results shows, in comparison with conventional cost minimization, our solution features a tardy increase of 12.8% in TAC and a sharp reduction of 75.5% in GHGe. Moreover, we find that ignoring performance degradation leads to an over-estimation of 2.3–13.7% in system economic performance. The proposed methodology provides an effective and flexible framework for optimal design and operational analysis of renewable CCHP systems.     

A generalized cutting-set approach for nonlinear robust optimization in process systems engineering

We propose a novel computational framework for the robust optimization of highly nonlinear, non-convex models that possess uncertainty in their parameter data. The proposed method is a generalization of the robust cutting-set algorithm that can handle models containing irremovable equality constraints, as is often the case with models in the process systems engineering domain. Additionally, we accommodate general forms of decision rules to facilitate recourse in second-stage (control) variables. In particular, we compare and contrast the use of various types of decision rules, including quadratic ones, which we show in certain examples to be able to decrease the overall price of robustness. Our proposed approach is demonstrated on three process flow sheet models, including a relatively complex model for amine-based CO₂ capture. We thus verify that the generalization of the robust cutting-set algorithm allows for the facile identification of robust feasible designs for process systems of practical relevance.     

Distributionally robust chance constrained programming with generative adversarial networks (GANs)

This paper presents a novel deep learning based data-driven optimization method. A novel generative adversarial network (GAN) based data-driven distributionally robust chance constrained programming framework is proposed. GAN is applied to fully extract distributional information from historical data in a nonparametric and unsupervised way without a priori approximation or assumption. Since GAN utilizes deep neural networks, complicated data distributions and modes can be learned, and it can model uncertainty efficiently and accurately. Distributionally robust chance constrained programming takes into consideration ambiguous probability distributions of uncertain parameters. To tackle the computational challenges, sample average approximation method is adopted, and the required data samples are generated by GAN in an end-to-end way through the differentiable networks. The proposed framework is then applied to supply chain optimization under demand uncertainty. The applicability of the proposed approach is illustrated through a county-level case study of a spatially explicit biofuel supply chain in Illinois.     

工艺蒸汽加热网络与蒸汽系统的集成优化

蒸汽动力系统作为过程工业的重要组成部分,由蒸汽系统、分配系统和蒸汽加热网络组成。文中提出对蒸汽系统、分配系统和蒸汽加热网络进行集成优化。首先通过夹点分析对工艺蒸汽加热网络进行优化匹配,得到各等级蒸汽需求量及凝结水回收参数;然后根据建立的蒸汽系统和分配系统的运行优化模型得到产汽及分配方案。通过对某化工厂的能量利用系统进行集成优化,得到的优化方案与传统设计方案相比,锅炉燃料消耗量减少16.38%,同时系统所需要的外界补给水量减少16.27%。提出的集成优化方法对蒸汽动力系统的设计优化具有一定的指导意义。     

Scheduling of crude oil operations under demand uncertainty: A robust optimization framework coupled with global optimization

Scheduling of crude oil operations is an important component of overall refinery operations, because crude oil costs account for about 80% of the refinery turnover. The mathematical modeling of blending different crudes in storage tanks results in many bilinear terms, which transform the problem into a challenging, nonconvex, mixed‐integer nonlinear programming (MINLP) optimization model. In practice, uncertainties are unavoidable and include demand fluctuations, ship arrival delays, equipment malfunction, and tank unavailability. In the presence of these uncertainties, an optimal schedule generated using nominal parameter values may often be suboptimal or even become infeasible. In this article, the robust optimization framework proposed by Lin et al. and Janak et al. is extended to develop a deterministic robust counterpart optimization model for demand uncertainty. The recently proposed branch and bound global optimization algorithm with piecewise‐linear underestimation of bilinear terms by Li et al. is also extended to solve the nonconvex MINLP deterministic robust counterpart optimization model and generate robust schedules. Two examples are used to illustrate the capability of the proposed robust optimization approach, and the extended branch and bound global optimization algorithm for demand uncertainty. The computational results demonstrate that the obtained schedules are robust in the presence of demand uncertainty. © 2011 American Institute of Chemical Engineers AIChE J, 58: 2373–2396, 2012     

A new superstructure optimization paradigm for process synthesis with product distribution optimization: Application to an integrated shale gas processing and chemical manufacturing process

We propose a novel process synthesis framework that combines product distribution optimization of chemical reactions and superstructure optimization of the process flowsheet. A superstructure with a set of technology/process alternatives is first developed. Next, the product distributions of the involved chemical reactions are optimized to maximize the profits of the effluent products. Extensive process simulations are then performed to collect high‐fidelity process data tailored to the optimal product distributions. Based on the simulation results, a superstructure optimization model is formulated as a mixed‐integer nonlinear program (MINLP) to determine the optimal process design. A tailored global optimization algorithm is used to efficiently solve the large‐scale nonconvex MINLP problem. The resulting optimal process design is further validated by a whole‐process simulation. The proposed framework is applied to a comprehensive superstructure of an integrated shale gas processing and chemical manufacturing process, which involves steam cracking of ethane, propane, n‐butane, and i‐butane. © 2017 American Institute of Chemical Engineers AIChE J, 63: 123–143, 2018