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     

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     

基于马尔可夫链的蒸汽动力系统两阶段随机规划

现有的解决蒸汽动力系统蒸汽需求不确定性的优化方法有随机规划和鲁棒优化,但二者不能同时兼顾稳定性和经济性。本文提出一种基于马尔可夫链的两阶段随机规划去解决这个问题。第一阶段基于空间距离表达划分不确定变量,通过聚类算法划分成不同工况。第二阶段基于状态切换概率构建马尔可夫链,通过场景生成和削减的方法预测蒸汽的需求值。以某煤制气企业蒸汽动力系统为实例建立相应的优化模型,将预测的蒸汽值带入优化模型求解,得出的最优操作方案与随机规划和鲁棒优化法进行对比和分析。结果表明,本优化方法综合了随机规划经济性高和鲁棒优化稳定性高的优点,稳定性和经济性都介于随机规划和鲁棒优化的中间,为解决蒸汽动力系统的不确定优化问题提供了新思路。     

Optimization of a multiperiod refinery planning problem under uncertainty

Uncertainty in refinery planning presents a significant challenge in determining the day-to-day operations of an oil refinery. Deterministic modeling techniques often fail to account for this uncertainty, potentially resulting in reduced profit. The stochastic programming framework explicitly incorporates parameter uncertainty in the problem formulation, thus giving preference to robust solutions. In this work, a nonlinear, multiperiod, industrial refinery problem is extended to a two-stage stochastic problem, formulated as a mixed-integer nonlinear program. A crude-oil sequencing case study is developed with binary scheduling decisions in both stages of the stochastic programming problem. Solution via a decomposition strategy based on the generalized Benders decomposition (GBD) algorithm is proposed. The binary decisions are designated as complicating variables that, when fixed, reduce the full-space problem to a series of independent scenario subproblems. Through the application of the GBD algorithm, a feasible mixed-integer solution is obtained that is more robust to uncertainty than its deterministic counterpart.     

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     

Shale gas pad development planning under price uncertainty

In this article, we study shale gas pad development under natural gas price uncertainty. We optimize the sequence of operations, gas curtailment, and storage on a single pad to maximize the net present value. The optimization problem is formulated as an mixed-integer linear programming model, which is similar to the one proposed by Ondeck et al. We investigate how natural gas price uncertainty affects the operation strategy in the pad development. Both two-stage and multistage stochastic programming are used as the mathematical framework to hedge against uncertainty. Our case study shows that there is value of using stochastic programming when the price variance is high. However, when the variance of the price is low, solving the stochastic programming problems does not create additional value compared with solving the deterministic problem.     

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.     

Stochastic optimization based on a novel scenario generation method for midstream and downstream petrochemical supply chain

A two-stage mixed integer linear programming model (MILP) incorporating a novel method of stochastic scenario generation was proposed in order to optimize the economic performance of the synergistic combination of midstream and downstream petrochemical supply chain. The uncertainty nature of the problem intrigued the parameter estimation, which was conducted through discretizing the assumed probability distribution of the stochastic parameters. The modeling framework was adapted into a real-world scale of petrochemical enterprise and fed into optimization computations. Comparisons between the deterministic model and stochastic model were discussed, and the influences of the cost components on the overall profit were analyzed. The computational results demonstrated the rationality of using reasonable numbers of scenarios to approximate the stochastic optimization problem.     

Mathematical Programming Techniques for Optimization under Uncertainty and Their Application in Process Systems Engineering

In this paper we give an overview of some of the advances that have taken place to address challenges in the area of optimization under uncertainty. We first describe the incorporation of recourse in robust optimization to reduce the conservative results obtained with this approach, and illustrate it with interruptible load in demand side management. Second, we describe computational strategies for effectively solving two stage programming problems, which is illustrated with supply chains under the risk of disruption. Third, we consider the use of historical data in stochastic programming to generate the probabilities and outcomes, and illustrate it with an application to process networks. Finally, we briefly describe multistage stochastic programming with both exogenous and endogenous uncertainties, which is applied to the design of oilfield infrastructures.     

Capacity expansion planning through augmented Lagrangian optimization and scenario decomposition

Stochastic programming is a typical method for addressing the uncertainties in capacity expansion planning problem. However, the corresponding deterministic equivalent model is often intractable with considerable number of uncertainty scenarios especially for stochastic integer programming (SIP) based formulations. In this article, a hybrid solution framework consisting of augmented Lagrangian optimization and scenario decomposition algorithm is proposed to solve the SIP problem. The method divides the solution procedure into two phases, where traditional linearization based decomposition strategy and global optimization technique are applied to solve the relaxation problem successively. Using the proposed solution framework, a feasible solution of the original problem can be obtained after the first solution phase whereas the optimal solution is obtained after the second solution phase. The effectiveness of the proposed strategy is verified through a numerical example of two stage stochastic integer program and the capacity expansion planning examples. © 2011 American Institute of Chemical Engineers AIChE J, 2012     

基于带补偿随机规划的蒸汽动力系统优化设计

在蒸汽动力系统优化设计中,考虑不确定因素的优化策略能避免基于确定性设计策略的保守设计,并能针对不确定因素的实现提出相应的调度调节策略.本研究分析了蒸汽动力系统设计包含的不确定因素的特性及其对蒸汽动力系统优化目标和约束条件的影响.不确定因素的表达分成两类:基于时间变化表达和基于发生概率表达.对基于时间变化表达的因素,转化为多周期问题进行处理;对外部工艺过程变化引起的汽电需求不确定波动等基于发生概率表达的因素,应用随机规划策略,补偿不确定参数的实现可能引起的约束背离.基于本研究建立的多周期带补偿的二阶段随机规划MILP模型,求解蒸汽动力系统结构,同时优化调度调节策略,用调节决策和惩罚不足应对汽电需求等不确定因素的实现,实现系统安全稳定运行和经济效益最优.     

Water networks security: A two-stage mixed-integer stochastic program for sensor placement under uncertainty

This work describes a stochastic approach for the optimal placement of sensors in municipal water networks to detect maliciously injected contaminants. The model minimizes the expected fraction of the population at risk and the cost of the sensors. Our work explicitly includes uncertainties in the attack risk and population density, so that the resulting problem involves optimization under uncertainty. In our formulation, we include the location of a number of sensors as first stage decision variables of a two-stage mixed-integer stochastic linear problem; the second stage evaluates the population at risk for the scenario obtained in the first stage and that information is then used to modify the first stage decisions for the next iteration. Since the model is integer in the first stage, a generalized framework based on the stochastic decomposition algorithm allows us to solve the problem in a reasonable computational time. The paper describes the mixed-integer stochastic model and the algorithmic framework, and compares the deterministic and stochastic optimal solutions. The network used as our case study has been derived through the water network simulator EPANET 1.0; four acyclic water flow patterns are considered. Results show a significant effect of uncertainty in sensor placement and total cost.     

Mid-term supply chain planning under demand uncertainty: customer demand satisfaction and inventory management

This paper utilizes the framework of mid-term, multisite supply chain planning under demand uncertainty to safeguard against inventory depletion at the production sites and excessive shortage at the customer. A chance constraint programming approach in conjunction with a two-stage stochastic programming methodology is utilized for capturing the trade-off between customer demand satisfaction (CDS) and production costs. In the proposed model, the production decisions are made before demand realization while the supply chain decisions are delayed. The challenge associated with obtaining the second stage recourse function is resolved by first obtaining a closed-form solution of the inner optimization problem using linear programming duality followed by expectation evaluation by analytical integration. In addition, analytical expressions for the mean and standard deviation of the inventory are derived and used for setting the appropriate CDS levels in the supply chain. A three-site example supply chain is studied within the proposed framework for providing quantitative guidelines for setting customer satisfaction levels and uncovering effective inventory management options. Results indicate that significant improvement in guaranteed service levels can be obtained for a small increase in the total cost.     

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。     

基于条件场景的不确定性石化供应链优化方法

针对产品需求及其价格存在不确定性的石化供应链计划层最优化问题,本文建立了一种基于条件场景的石化供应链最优化方法。用多个离散场景近似随机变量概率的连续分布,根据随机变量的概率分布特征,对场景发生的概率进行参数估计,进而建立了基于场景的两阶段混合整数线性规划(MILP)模型。利用基于场景的优化结果随离散网格数增加而逐渐趋近连续的随机优化结果这一规律,给出了获得最佳离散网格数的方法,实现了计算时间成本与计算精度之间的平衡。在此基础上引入条件概率方法,利用两个随机变量间的相关性,建立了以产品价格及其需求量为不确定性的石化供应链优化方法。结果表明,与传统未考虑随机变量间相关性的一般场景划分方法相比,本文基于条件场景的随机优化方法可以更快地获得最佳场景数目,进而有效降低了计算量。     

Modular supply chain optimization considering demand uncertainty to manage risk

Supply chain under demand uncertainty has been a challenging problem due to increased competition and market volatility in modern markets. Flexibility in planning decisions makes modular manufacturing a promising way to address this problem. In this work, the problem of multiperiod process and supply chain network design is considered under demand uncertainty. A mixed integer two-stage stochastic programming problem is formulated with integer variables indicating the process design and continuous variables to represent the material flow in the supply chain. The problem is solved using a rolling horizon approach. Benders decomposition is used to reduce the computational complexity of the optimization problem. To promote risk-averse decisions, a downside risk measure is incorporated in the model. The results demonstrate the several advantages of modular designs in meeting product demands. A pareto-optimal curve for minimizing the objectives of expected cost and downside risk is obtained.     

Flexibility assessment and risk management in supply chains

Increased uncertainty in recent years has led the supply chains to incorporate measures to be more flexible in order to perform well in the face of the uncertain events. It has been shown that these measures improve the performance of supply chains by mitigating the risks associated with uncertainties. However, it is also important to assess the uncertainty under which a supply chain network can perform well and manage risk. Flexibility is defined in terms of the bounds of uncertain parameters within which supply chain operation is feasible. A hybrid simulation‐based optimization framework that uses two‐stage stochastic programming in a rolling horizon framework is proposed. The framework enables taking optimum planning decisions considering demand uncertainty while managing risk. The framework is used to study the trade‐offs between flexibility, economic performance, and risk associated with supply chain operation. © 2015 American Institute of Chemical Engineers AIChE J, 61: 4166–4178, 2015     

Multisite supply planning for drug products under uncertainty

In the pharmaceutical industry, the goal of a supply planner is to make efficient capacity allocation decisions that ensure an uninterrupted supply of drug products to patients and to maintain product inventory levels close to the target stock. This task can be challenging due to the limited availability of manufacturing assets, uncertainties in product demand, fluctuations in production yields, and unplanned site downtimes. It is not uncommon to observe uneven distribution of product inventories with some products carrying excess inventories, while other products may be close to a stockout. Maintaining high stock levels can have economic repercussions due to the risk of expiration of unused products (whereas products facing a stockout can adversely affect the treatment regimen of patients). The network complexity of pharmaceutical supply-chains coupled with regulatory constraints and siloed planning systems force supply planners to rely on manual (error-prone) decision-making processes. Such an approach results in suboptimal capacity allocation and inventory management decisions. In this work, we propose a stochastic optimization methodology for the production scheduling of multiple drug products in lyophilization units across multiple sites. The framework leverages information obtained from historical and forecast data to generate scenarios of uncertain parameters (e.g., yield, demand, and downtimes) that can realize in the future. The optimization model determines a product filling schedule that maintains product stock levels close to targets under diverse scenarios. We show that this approach helps in avoiding reactive scheduling and in maintaining a more robust production plan than deterministic procedures (which ignore uncertainty). Specifically, planning under a stochastic optimization approach reduces the number of scenarios under which backlogs are observed and also reduces the magnitude of the backlogs.     

Process optimization with consideration of uncertainties——An overview

Optimization under uncertainty is a challenging topic of practical importance in the Process Systems Engineering. Since the solution of an optimization problem generally exhibits high sensitivity to the parameter variations, the deterministic model which neglects the parametric uncertainties is not suitable for practical applications. This paper provides an overview of the key contributions and recent advances in the field of process optimization under uncertainty over the past ten years and discusses their advantages and limitations thoroughly. The discussion is focused on three specific research areas, namely robust optimization, stochastic programming and chance constrained programming, based on which a systematic analysis of their applications, developments and future directions are presented. It shows that the more recent trend has been to integrate different optimization methods to leverage their respective superiority and compensate for their drawbacks. Moreover, data-driven optimization, which combines mathematical programming methods and machine learning algorithms, has become an emerging and competitive tool to handle optimization problems in the presence of uncertainty based on massive historical data.     

Medium-term maintenance turnaround planning under uncertainty for integrated chemical sites

Plant maintenance poses extended disruptions to production. Maintenance effects are amplified when the plant is part of an integrated chemical site, as production levels of adjacent plants in the site are also significantly influenced. A challenge in dealing with turnarounds is the difficulty in predicting their duration, due to discovery work and delays. This uncertainty in duration affects two major planning decisions: production levels and maintenance manpower allocation. The latter must be decided several months before the turnarounds occur. We address the scheduling of a set of plant turnarounds over a medium-term of several months using integer programming formulations. Due to the nature of uncertainty, production decisions are treated through stochastic programming ideas, while the manpower aspect is handled through a robust optimization framework. We propose combined robust optimization and stochastic programming formulations to address the problem and demonstrate, through an industrial case study, the potential for significant savings.