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     

Operational optimization of industrial steam systems under uncertainty using data-Driven adaptive robust optimization

This article addresses the operational optimization of industrial steam systems under device efficiency uncertainty using a data-driven adaptive robust optimization approach. A semiempirical model of steam turbine is first developed based on process mechanism and operational data. Uncertain parameters of the proposed steam turbine model are further derived from the historical process data. A robust kernel density estimation method is then used to construct the uncertainty sets for modeling these uncertain parameters. The data-driven uncertainty sets are incorporated into a two-stage adaptive robust mixed-integer linear programming (MILP) framework for operational optimization of steam systems to minimize the total operating cost. Integer variables are introduced to model the on/off decisions of the steam turbines and electrical motors, which are the major energy consumers of the steam system. By applying the affine decision rule, the proposed multilevel optimization model is transformed into its robust counterpart, which is a single-level MILP problem. The proposed framework is applied to the steam system of a real-world ethylene plant to demonstrate its applicability. © 2018 American Institute of Chemical Engineers AIChE J, 65: e16500 2019     

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     

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     

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 unified framework for adjustable robust optimization with endogenous uncertainty

This work proposes a framework for multistage adjustable robust optimization that unifies the treatment of three different types of endogenous uncertainty, where decisions, respectively, (a) alter the uncertainty set, (b) affect the materialization of uncertain parameters, and (c) determine the time when the true values of uncertain parameters are observed. We provide a systematic analysis of the different types of endogenous uncertainty and highlight the connection between optimization under endogenous uncertainty and active learning. We consider decision-dependent polyhedral uncertainty sets and propose a decision rule approach that incorporates both continuous and binary recourse, including recourse decisions that affect the uncertainty set. The proposed method enables the modeling of decision-dependent nonanticipativity and results in a tractable reformulation of the problem. We demonstrate the effectiveness of the approach in computational experiments that cover a wide range of applications. The results show significant benefits from proper modeling of endogenous uncertainty and active learning.     

考虑需求不确定性的化工生产计划与调度集成

生产计划与调度是化工供应链优化中两个重要的决策问题。为了提高生产决策的效率,不仅要对计划与调度进行集成,而且要考虑不确定性的影响。对于多周期生产计划与调度问题,首先在每个生产周期内,分别建立计划与调度的确定性模型,通过产量关联对二者进行集成。然后考虑需求不确定性,使用有限数量的场景表达决策变量,建立二阶段随机规划模型。最后运用滚动时域求解策略,使计划与调度结果在迭代过程中达到一致。实例结果表明,在考虑需求不确定性时,与传统方法相比,随机规划方法可以降低总费用,结合计划与调度的分层集成策略,实现了生产操作性和经济性的综合优化。     

Real‐time dynamic hoist scheduling for multistage material handling process under uncertainties

Multistage material handling processes are broadly used for manufacturing various products/jobs, where hoists are commonly used to transport inline products according to their processing recipes. When multiple types of jobs with different recipes are simultaneously and continuously handled in a production line, the hoist movement scheduling should be thoroughly investigated to ensure the operational feasibility of every job inline and in the meantime to maximize the productivity if possible. The hoist scheduling will be more complicated, if uncertainties of new coming jobs are considered, that is, the arrival time, type, recipe, and number of new jobs are totally unknown and unpredictable before they join the production line. To process the multiple jobs already inline and the newly added jobs, the hoist movements must be swiftly rescheduled and precisely implemented whenever new job(s) come. Because a reschedule has to be obtained online without violating processing time constraints for each job, the solution identification time for rescheduling must be taken into account by the new schedule itself. All these stringent requisites motivate the development of real‐time dynamic hoist scheduling (RDHS) targeting online generation of reschedules for productivity maximization under uncertainties. Hitherto, no systematic and rigorous methodologies have been reported for this study. In this article, a novel RDHS methodology has been developed, which takes into account uncertainties of new coming jobs and targets real‐time scheduling optimality and applicability. It generally includes a reinitialization algorithm to accomplish the seamless connection between the previous scheduling and rescheduling operations, and a mixed‐integer linear programming model to obtain the optimal hoist reschedule. The RDHS methodology addresses all the major scheduling issues of multistage material handling processes, such as multiple recipes, multiple jobs, multicapacity processing units, diverse processing time requirements, and even optimal processing queue for new coming jobs. The efficacy of the developed methodology is demonstrated through various case studies. © 2012 American Institute of Chemical Engineers AIChE J, 59: 465–482, 2013     

Steady‐state process optimization with guaranteed robust stability under parametric uncertainty

Interest in chemical processes that perform well in dynamic environments has led to the development of design methodologies that account for operational aspects of processes, including flexibility, operability, and controllability. In this article, we address the problem of identifying process designs that optimize an economic objective function and are guaranteed to be stable under parametric uncertainties. The underlying mathematical problem is difficult to solve as it involves infinitely many constraints, nonconvexities and multiple local optima. We develop a methodology that embeds robust stability constraints to steady‐state process optimization formulations without any a priori bifurcation analysis. We propose a successive row and column generation algorithm to solve the resulting generalized semi‐infinite programming problem to global optimality. The proposed methodology allows modeling different levels of robustness, handles uncertainty regions without overestimating them, and works for both unique and multiple steady states. We apply the proposed approach to a number of steady‐state optimization problems and obtain the least conservative solutions that guarantee robust stability. © 2011 American Institute of Chemical Engineers AIChE J, 2011     

A data‐driven rolling‐horizon online scheduling model for diesel production of a real‐world refinery

A rolling‐horizon optimal control strategy is developed to solve the online scheduling problem for a real‐world refinery diesel production based on a data‐driven model. A mixed‐integer nonlinear programming (MINLP) scheduling model considering the implementation of nonlinear blending quality relations and quantity conservation principles is developed. The data variations which drive the MINLP model come from different sources of certain and uncertain events. The scheduling time horizon is divided into equivalent discrete time intervals, which describe regular production and continuous time intervals which represent the beginning and ending time of expected and unexpected events that are not restricted to the boundaries of discrete time intervals. This rolling‐horizon optimal control strategy ensures the dimension of the diesel online scheduling model can be accepted in industry use. LINGO is selected to be the solution software. Finally, the daily diesel scheduling scheme of one entire month for a real‐world refinery is effectively solved. © 2012 American Institute of Chemical Engineers AIChE J, 59: 1160–1174, 2013     

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     

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.     

Adversarially robust Bayesian optimization for efficient auto-tuning of generic control structures under uncertainty

The performance of optimization- and learning-based controllers critically depends on the selection of several tuning parameters that can affect the closed-loop control performance and constraint satisfaction in highly nonlinear and nonconvex ways. Due to the black-box nature of the relationship between tuning parameters and general closed-loop performance measures, there has been a significant interest in automatic calibration (i.e., auto-tuning) of complex control structures using derivative-free optimization methods, including Bayesian optimization (BO) that can handle expensive unknown cost functions. Nevertheless, an open challenge when applying BO to auto-tuning is how to effectively deal with uncertainties in the closed-loop system that cannot be attributed to a lumped, small-scale noise term. This article addresses this challenge by developing an adversarially robust BO (ARBO) method that is particularly suited to auto-tuning problems with significant time-invariant uncertainties in an expensive system model used for closed-loop simulations. ARBO relies on a Gaussian process model that jointly describes the effect of the tuning parameters and uncertainties on the closed-loop performance. From this joint Gaussian process model, ARBO uses an alternating confidence-bound procedure to simultaneously select the next candidate tuning and uncertainty realizations, implying only one expensive closed-loop simulation is needed at each iteration. The advantages of ARBO are demonstrated on two case studies, including an illustrative problem and auto-tuning of a nonlinear model predictive controller using a benchmark bioreactor problem.     

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     

Optimal processing network design under uncertainty for producing fuels and value‐added bioproducts from microalgae: Two‐stage adaptive robust mixed integer fractional programming model and computationally efficient solution algorithm

Fractional metrics, such as return on investment (ROI), are widely used for performance evaluation, but uncertainty in the real market may unfortunately diminish the results that are based on nominal parameters. This article addresses the optimal design of a large‐scale processing network for producing a variety of algae‐based fuels and value‐added bioproducts under uncertainty. We develop by far the most comprehensive processing network with 46,704 alternative processing pathways. Based on the superstructure, a two‐stage adaptive robust mixed integer fractional programming model is proposed to tackle the uncertainty and select the robust optimal processing pathway with the highest ROI. Since the proposed problem cannot be solved directly by any off‐the‐shelf solver, we develop an efficient tailored solution method that integrates a parametric algorithm with a column‐and‐constraint generation algorithm. The resulting robust optimal processing pathway selects biodiesel and poly‐3‐hydroxybutyrate as the final fuel and bioproduct, respectively. © 2016 American Institute of Chemical Engineers AIChE J, 63: 582–600, 2017     

Multistage distributionally robust optimization for integrated production and maintenance scheduling

In chemical manufacturing processes, equipment degradation can have a significant impact on process performance or cause unit failures that result in considerable downtime. Hence, maintenance planning is an important consideration, and there have been increased efforts in scheduling production and maintenance operations jointly. In this context, one major challenge is the inherent uncertainty in predictive equipment health models. In particular, the probability distribution associated with the stochasticity in such models is often difficult to estimate and hence not known exactly. In this work, we apply a distributionally robust optimization (DRO) approach to address this problem. Specifically, the proposed formulation optimizes the worst-case expected outcome with respect to a Wasserstein ambiguity set, and we apply a decision rule approach that allows multistage mixed-integer recourse. Computational experiments, including a real-world industrial case study, are conducted, where the results demonstrate the significant benefits from binary recourse and DRO in terms of solution quality.     

Multistage adaptive optimization using hybrid scenario and decision rule formulation

Scenario-based stochastic programming and linear decision rule (LDR)-based robust optimization are prevalent methods for solving multistage adaptive optimization (MSAP) problems. In practical applications such as capacity expansion planning of chemical processes, often multiple sources of uncertainty affect the problem which introduces challenges to traditional stochastic optimization methods. While a large number of uncertain parameters exist in the problem, using scenario-based method results in very large problem size and the solution becomes computationally expensive. In addition, when the constraints include multiplication of uncertain parameters and adaptive variables, the constraints are not linear with respect to uncertain parameters when the LDR method is used. In order to address these challenges, we propose two different hybrid methods where scenario and decision rule methods are combined to solve the MSAP problem. The article demonstrates the computational performance of the proposed hybrid methods using two chemical process planning examples.     

Data‐driven mathematical modeling and global optimization framework for entire petrochemical planning operations

In this work we develop a novel modeling and global optimization‐based planning formulation, which predicts product yields and properties for all of the production units within a highly integrated refinery‐petrochemical complex. Distillation is modeled using swing‐cut theory, while data‐based nonlinear models are developed for other processing units. The parameters of the postulated models are globally optimized based on a large data set of daily production. Property indices in blending units are linearly additive and they are calculated on a weight or volume basis. Binary variables are introduced to denote unit and operation modes selection. The planning model is a large‐scale non‐convex mixed integer nonlinear optimization model, which is solved to ε‐global optimality. Computational results for multiple case studies indicate that we achieve a significant profit increase (37–65%) using the proposed data‐driven global optimization framework. Finally, a user‐friendly interface is presented which enables automated updating of demand, specification, and cost parameters. © 2016 American Institute of Chemical Engineers AIChE J, 62: 3020–3040, 2016     

A branch and bound algorithm to solve large‐scale multistage stochastic programs with endogenous uncertainty

The growth in computation complexity of multistage stochastic programs (MSSPs) with problem size often prevents its application to real‐world size problems. We present two variants of branch‐and‐bound algorithm, which reduce the resource requirements for the generation and solution of large‐scale MSSPs with endogenous uncertainty. Both variants use Knapsack‐problem based Decomposition Algorithm (Christian and Cremaschi, Comput Chem Eng. 2015;74:34–47) to generate feasible solutions and primal bounds. First variant (PH‐KDA) uses a progressive hedging dual‐bounding approach; the second (OSS‐KDA) solves the MSSP removing all nonanticipativity constraints. Both variants were used to solve several instances of the pharmaceutical clinical trial planning problem. The first iteration of both algorithms provides a feasible solution, and a primal bound and a dual bound for the problem. Although the dual‐bounds of OSS‐KDA were generally weaker than PH‐KDA, they are generated considerably faster. For the seven‐product case the OSS‐KDA generated a solution with a gap of 9.92% in 115 CPU seconds. © 2017 American Institute of Chemical Engineers AIChE J, 64: 1262–1271, 2018     

A novel adaptive surrogate modeling‐based algorithm for simultaneous optimization of sequential batch process scheduling and dynamic operations

A novel adaptive surrogate modeling‐based algorithm is proposed to solve the integrated scheduling and dynamic optimization problem for sequential batch processes. The integrated optimization problem is formulated as a large scale mixed‐integer nonlinear programming (MINLP) problem. To overcome the computational challenge of solving the integrated MINLP problem, an efficient solution algorithm based on the bilevel structure of the integrated problem is proposed. Because processing times and costs of each batch are the only linking variables between the scheduling and dynamic optimization problems, surrogate models based on piece‐wise linear functions are built for the dynamic optimization problems of each batch. These surrogate models are then updated adaptively, either by adding a new sampling point based on the solution of the previous iteration, or by doubling the upper bound of total processing time for the current surrogate model. The performance of the proposed method is demonstrated through the optimization of a multiproduct sequential batch process with seven units and up to five tasks. The results show that the proposed algorithm leads to a 31% higher profit than the sequential method. The proposed method also outperforms the full space simultaneous method by reducing the computational time by more than four orders of magnitude and returning a 9.59% higher profit. © 2015 American Institute of Chemical Engineers AIChE J, 61: 4191–4209, 2015