Stochastic Nonlinear Programming Research Articles

Energy expansion planning with a human evolutionary model

This study presents a novel method for planning the expansion of transmission lines and energy storage systems while considering the interconnectedness of electricity and gas networks. We developed a two-level stochastic planning model that addresses both the expansion of transmission and battery systems in the electrical grid and the behavior of the gas network. This research explores the challenges and effects of integrating high levels of renewable energy sources while ensuring security within both networks. Our model uses a stochastic mixed-integer non-linear programming approach. To solve this complex model, we applied the Human Evolutionary Model (HEM). We tested our approach on two case studies: a simple 6-node network and the more complex IEEE RTS 24-bus network for the electricity grid, combined with 5-node and 10-node gas networks, respectively. The results demonstrate the effectiveness of our model, particularly in scenarios where connections in the power and gas networks are disrupted, preventing load shedding even when integrated network connections are cut.

Design and operation of modular biorefinery supply chain under uncertainty using generalized Benders decomposition

AbstractBiomass supply chain performance is heavily affected by uncertainties stemming from supply, demand, or unexpected disruptions. Unlike petrochemical plants that use crude oil, biorefineries often have to deal with the uneven spatial‐temporal distribution of feedstock supply. The modular production strategy provides more flexibility in chemical manufacturing by allowing fast capacity expansion and unit movement. However, modeling and optimizing modular biomass supply chain under uncertainties becomes challenging due to high dimensionality and the existence of discrete decisions. This work optimizes the multiperiod biomass supply chain using the rolling horizon planning and two‐stage stochastic programming framework. We then applied generalized Benders decomposition to reduce the computational complexity of the stochastic mixed integer nonlinear programming supply chain optimization. Furthermore, the solution of the stochastic programming could be used to quantitatively describe the life‐cycle assessment uncertainties of the biomass supply chain performance, demonstrating seasonality and random variability.

A Mutual Catastrophe Insurance Framework for Horizontal Collaboration in Prepositioning Strategic Reserves

Improving Disaster Preparedness Through Mutual Catastrophe Insurance In “A Mutual Catastrophe Insurance Framework for Horizontal Collaboration in Prepositioning Strategic Reserves,” H. Zbib, B. Balcik, M.-È. Rancourt, and G. Laporte present an innovative approach to collaborative disaster preparedness. The novel framework considers a risk-averse mutual insurer offering multiyear insurance contracts with coverage deductibles and limits to a portfolio of risk-averse policyholders. It is designed to foster horizontal collaboration among policyholders for joint disaster preparedness by effectively integrating operational and financial functions. The problem is modeled as a large-scale nonlinear multistage stochastic program and solved by using an effective Benders decomposition algorithm. The framework is validated with real data from 18 Caribbean countries focusing on hurricane preparedness. Given the predicted impacts of climate change, the proposed multiyear mutual catastrophe insurance framework promises to reshape global disaster preparedness and make a profound societal impact by providing a transparent disaster financing plan to protect vulnerable regions. The study’s findings stress the importance of long-term cooperation, prenegotiation of indemnification policies, and strategic setting of deductibles and limits by taking into account the correlation between policyholders.

Stochastic ambulance dispatching and routing in mass casualty incident under road vulnerability

This paper investigates the stochastic ambulance dispatching and routing (SADR) problem with multiple casualty collection points in the mass casualty incidents (MCI) under uncertainty of road vulnerability and traffic congestion caused by an earthquake. A two-stage stochastic mixed-integer nonlinear programming model is formulated to derive a reliable and high-quality ambulance scheduling, dispatching and routing solution for maximizing expected number of survivors of all casualties. An integrated simulation optimization approach combining a data-driven travel time scenario generation algorithm, sample average approximation, and a column-generation-based heuristic method is developed to efficiently solve this complicated two-stage SADR model with a nonlinear objective function and continuous travel time distributions. Collaborating with the National Science and Technology Center for Disaster Reduction (NCDR), an empirical study using a potential real-world earthquake scenario occurring in Taiwan is conducted to demonstrate the usefulness and effectiveness of our proposed model and solution approach compared to the deterministic model and two current-practice heuristics.

Model and solution approach to coordinate production-inventory strategies considering nonlinear price-sensitive demand: application to Canadian pulp and paper industry

This study addresses a practical problem within a multi-level supply chain where a wide range of customers can be served through different strategies such as make-to-stock, make-to-order, or vendor-managed inventory. The customer demand is stochastic, and sensitive to pricing associated with different production-inventory strategies. We propose a two-stage stochastic mixed-integer non-linear programming model. In the first stage, decisions are made regarding the selection of production-inventory strategies and pricing to maximise the expected profit. The second stage involves decisions related to production, inventory, and distribution, which are used to evaluate the first-stage decisions under various scenarios with different levels of accuracy. To solve the model, a metaheuristic approach based on the Simulated Annealing algorithm is developed. To showcase the practical applicability of our model and solution approach, we use a real case study in a Canadian pulp and paper supply chain. The results revealed that both the production-inventory strategy assigned to customers and the sales price underwent changes across scenarios. Furthermore, we demonstrated that by implementing the SA algorithm, we could improve the initial profit by up to 1.43% through slight adjustments in the sales price and assigned strategies for customers.

Retrofit Planning and Flexible Operation of Coal-Fired Units Using Stochastic Dual Dynamic Integer Programming

The continuous growth of wind power requires greater power system operational flexibility owing to the variability and uncertainty of wind power generation. Retrofitting existing coal-fired units is a demonstrated cost-effective method for improving system flexibility. This study proposes a decision support tool that provides optimal investment decisions to enhance power system operational flexibility. A novel flexibility retrofit planning (FRP) model is proposed in which the embedded operational problem is a stochastic mixed-integer nonlinear programming (SMINLP) unit commitment problem. In the operational unit commitment model, novel mixed-integer nonlinear formulations are proposed to represent the coal-fired unit attribute (minimum power output, startup and shutdown times, and maximum ramp rate) modifications after retrofitting. A new solution method based on the Benders decomposition (BD) and stochastic dual dynamic integer programming (SDDIP) algorithms is proposed to solve the proposed FRP model. The modified IEEE 39-bus, 57-bus, and 118-bus test systems were used to verify the effectiveness of the proposed model and solution method. The results demonstrate that the proposed FRP model can balance the retrofitting and operational costs to minimize the total cost and accommodate more wind generation. The proposed solution method is computationally efficient for solving FRP models.

Optimization of crude oil operations scheduling by applying a two-stage stochastic programming approach with risk management

This paper focuses on the problem of crude oil operations scheduling carried out in a system composed of a refinery and a marine terminal, considering uncertainty in the arrival date of the ships that supply the crudes. To tackle this problem, we develop a two-stage stochastic mixed-integer nonlinear programming (MINLP) model based on continuous-time representation. Furthermore, we extend the proposed model to include risk management by considering the Conditional Value-at-Risk (CVaR) measure as the objective function, and we analyze the solutions obtained for different risk levels. Finally, to evaluate the solution obtained, we calculate the Expected Value of Perfect Information (EVPI) and the Value of the Stochastic Solution (VSS) to assess whether two-stage stochastic programming model offers any advantage over simpler deterministic approaches.

Carsharing equitable relocation problem: A two-stage stochastic programming approach with learning-embedded endogenous uncertainty in demand

Uncertain demand may exacerbate the imbalance of the supply–demand for a one-way carsharing system and complicate vehicle relocation decisions. To consider the effect of the uncertainty, this study proposes a two-stage stochastic nonlinear programming model, integrating long-term and short-term decisions and maximizing the profit of the carsharing companies. Specifically, in the first stage, tactical decisions of fleet sizing and initial vehicle distribution are determined before the realization of the uncertain demand. Operational decisions of both operator-based and user-based relocation are optimized in the second stage. Moreover, this paper first studies the user-based relocation incentives, which affect the distribution of uncertain demand with an endogenous relationship. A learning-embedded optimization method is introduced to learn such a distribution, enabling the decision-making optimization model to achieve higher performance under the guidance of the demand uncertainty. Second, we envision an equitable relocation issue that considers an uneven distribution of the unsatisfied demand with two different equity criteria measured from the aspects of stations and OD pairs, respectively. Third, the large problem scale, the nonlinear objective function and constraints, and the endogenous demand uncertainty constitute the nontrivial challenges to the feasible solution. For solving the problem efficiently, we linearize the nonlinear terms and develop a dedicated two-phase solution algorithm with a learning-embedded trust-region method in phase I to solve the continuous relaxation problem and a mixed-integer linear programming guided iterative rounding in phase II to obtain the integer solutions of carsharing operations. The solution algorithm adaptively bridges the learning and optimization process via the trust-region method with flexible sample generation. Finally, we conduct numerical experiments based on a real-world one-way carsharing system in Beijing to demonstrate the effectiveness and applicability of the proposed method and reveal some insights for the carsharing service.

Towards sustainable UAV operations: Balancing economic optimization with environmental and social considerations in path planning

Unmanned Aerial Vehicles (UAVs), i.e., drones, are expected to be widely used in various applications, such as parcel delivery and passenger transport, with the benefits of mitigating traffic congestion and reducing carbon emissions. In this paper, we study a UAV path planning problem under uncertain weather conditions, and design a data-driven dynamic decision support system for multiple types of UAVs. To this end, we categorize all relevant costs into three types, namely, economic, environmental, and social costs, and formulate a nonlinear two-stage stochastic programming model to establish optimal paths for UAV missions under weather uncertainty. We then discretize the nonlinear model and propose a tight linear approximation for the discretized problem to allow for a near real-time implementation. To quantify weather uncertainty, we propose a weather scenario generation algorithm to map ensemble-based weather forecast information to airspace blockage maps. With comprehensive computational studies through simulations, we show that our proposed stochastic approach can lower operating costs by an average of around 6%, where the savings increase as weather conditions become more severe and complex. We also find that, for missions operated by small UAVs, it is not sufficient to determine a path solely based on economic cost minimization, but it should rather be through total cost minimization, which involves environmental and social costs. Considering only the economic cost in the optimization may lead to much higher non-economic costs. However, for missions operated by large UAVs, it is sufficient to determine paths through economic cost optimization, as including environmental and social costs in the optimization process does not result in solutions that are much different from those obtained by considering only the economic costs. For both small and large UAVs, a path established solely through environmental or social cost minimization may not be economically sustainable, as doing so would imply very high economic costs.

Stochastic multi-benefit planning of mobile energy storage in reconfigurable active distribution systems

This paper proposes a multi-benefit planning framework for mobile energy storage systems (MESSs) in reconfigurable active distribution systems (DSs). The goal of this framework is to improve the DS operation and reliability through achieving four objectives: (1) minimizing the DS costs, (2) minimizing the DS energy losses, (3) improving the DS voltage profiles, and (4) minimizing the expected energy not supplied (EENS) in the system. The developed framework determines optimal decisions associated with the network reconfiguration as well as the MESS sizes and locations, taking into account the variations in DS loads, energy prices, and renewable energy sources. The planning problem in this study is formulated as a stochastic multi-objective mixed-integer nonlinear programming problem, which is solved using a genetic algorithm (GA)-based hybrid optimization method. The proposed framework is tested using a case study of an active DS that includes various types of energy sources and load demands. Comparing the results of the developed planning framework with the base-case system demonstrates the efficacy of the planning framework and the techno-economic benefits of MESSs, where the proposed framework with MESSs successfully achieved significant improvements in the DS operating cost, energy losses, voltage profiles, and EENS. The advantages of MESSs are also verified through a comparison with stationary energy storage systems.

Optimization under uncertainty of a hybrid waste tire and natural gas feedstock flexible polygeneration system using a decomposition algorithm

Market uncertainties motivate the development of flexible polygeneration systems that are able to adjust operating conditions to favor production of the most profitable product portfolio. However, this operational flexibility comes at the cost of higher capital expenditure. A scenario-based two-stage stochastic nonconvex Mixed-Integer Nonlinear Programming (MINLP) approach lends itself naturally to optimizing these trade-offs. This work studies the optimal design and operation under uncertainty of a hybrid feedstock flexible polygeneration system producing electricity, methanol, dimethyl ether, olefins or liquefied (synthetic) natural gas. A recently developed C++ based software framework (named GOSSIP) is used for modeling the optimization problem as well as its efficient solution using the Nonconvex Generalized Benders Decomposition (NGBD) algorithm. Two different cases are studied: The first uses estimates of the means and variances of the uncertain parameters from historical data, whereas the second assesses the impact of increased uncertain parameter volatility. The value of implementing flexible designs characterized by the value of the stochastic solution (VSS) is in the range of 260–405 M$ for a scale of approximately 893 MW of thermal input. Increased price volatility around the same mean results in higher expected net present value and VSS as operational flexibility allows for asymmetric exploitation of price peaks.

Short-term hydropower optimization in the day-ahead market using a nonlinear stochastic programming model

Short-term hydropower optimization in the day-ahead market using a nonlinear stochastic programming model

Optimization with Neural Network Feasibility Surrogates: Formulations and Application to Security-Constrained Optimal Power Flow

In many areas of constrained optimization, representing all possible constraints that give rise to an accurate feasible region can be difficult and computationally prohibitive for online use. Satisfying feasibility constraints becomes more challenging in high-dimensional, non-convex regimes which are common in engineering applications. A prominent example that is explored in the manuscript is the security-constrained optimal power flow (SCOPF) problem, which minimizes power generation costs, while enforcing system feasibility under contingency failures in the transmission network. In its full form, this problem has been modeled as a nonlinear two-stage stochastic programming problem. In this work, we propose a hybrid structure that incorporates and takes advantage of both a high-fidelity physical model and fast machine learning surrogates. Neural network (NN) models have been shown to classify highly non-linear functions and can be trained offline but require large training sets. In this work, we present how model-guided sampling can efficiently create datasets that are highly informative to a NN classifier for non-convex functions. We show how the resultant NN surrogates can be integrated into a non-linear program as smooth, continuous functions to simultaneously optimize the objective function and enforce feasibility using existing non-linear solvers. Overall, this allows us to optimize instances of the SCOPF problem with an order of magnitude CPU improvement over existing methods.

Decision Rule Approaches for Pessimistic Bilevel Linear Programs Under Moment Ambiguity with Facility Location Applications

We study a pessimistic stochastic bilevel program in the context of sequential two-player games, where the leader makes a binary here-and-now decision, and the follower responds with a continuous wait-and-see decision after observing the leader’s action and revelation of uncertainty. We assume that only the information regarding the mean, covariance, and support is known. We formulate the problem as a distributionally robust (DR) two-stage problem. The pessimistic DR bilevel program is shown to be equivalent to a generic two-stage distributionally robust stochastic (nonlinear) program with both a random objective and random constraints under proper conditions of ambiguity sets. Under continuous distributions, using linear decision rule approaches, we construct upper bounds on the pessimistic DR bilevel program based on (1) a 0-1 semidefinite programming (SDP) approximation and (2) an exact 0-1 copositive programming reformulation. When the ambiguity set is restricted to discrete distributions, an exact 0-1 SDP reformulation is developed, and explicit construction of the worst-case distribution is derived. To further improve the computation of the proposed 0-1 SDPs, a cutting-plane framework is developed. Moreover, based on a mixed-integer linear programming approximation, another cutting-plane algorithm is proposed. Extensive numerical studies are conducted to demonstrate the effectiveness of the proposed approaches on a facility location problem. History: Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0168 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0168 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Pulse width modulation control for stochastic Lagrange systems

The pulse width modulation (PWM) control for Lagrange systems driven by servo motors in stochastic environment is researched in this paper. Principles of PWM for stochastic nonlinear systems are proved, a continuous control bounded in probability of stochastic Lagrange system is designed, and the stochastic closed-loop nonlinear programming is modeled in the same form as that in the deterministic case. Sure boundedness of the control is required as a hard constraint, and the stochastic quadratic programming problem is presented for which an analytic solution is constructed and the expression of the optimal duty ratio is obtained. Finally, the main procedures of design and implementation of the stochastic PWM are summarized, and the effectiveness of the proposed scheme is demonstrated by a simulation of articulated manipulator.

Continuous monitoring scheduling for moving targets by Earth observation satellites

The studies of earth observation satellite (EOS) scheduling for stationary targets have been increasing rapidly in recent years. However, these studies ignore EOS scheduling for moving targets (EOSSMT), which is urgently needed in many situations. EOSSMT is more complicated due to two factors, 1) location prediction of the moving targets and 2) algorithm design of EOSSMT optimization model for highly-nonlinear characteristics. In this article, we present a novel continuous monitoring scheduling methodology for moving targets by EOSs. Firstly, in order to predict the location of moving targets, a prediction-capture method, which can calculate the capture probability (CP) of the targets, is proposed. Then, based on the concept CP, we regard the EOSSMT as a stochastic integer nonlinear programming problem. Aiming to maximize observation times and duration, two objective functions with low complexity are proposed correspondingly. Finally, to solve our highly-nonlinear optimization model more efficiently, a dispersion-based heuristic (DBH) is proposed. In the computational experiments, a number of instances (scenarios), in which moving targets are successfully observed, verify the correctness and efficiency of our novel methodology for EOSSMT. Computational experiments also show that DBH can achieve better results than the Genetic Algorithm and Greedy Algorithm, with significantly less algorithm complexity.

Two-stage data-driven dispatch for integrated power and natural gas systems by using stochastic model predictive control

The optimal dispatch of the integrated power and natural gas systems can increase the utilization rate of renewable energy and energy efficiency while decreasing operation costs. The common prediction errors of wind power and electric load have the potential to negatively impact the normal operation of the integrated power and natural gas systems. A two-stage data-driven dispatch strategy is proposed to reduce this effect, consisting of the day-ahead dispatch stage and the intraday rolling dispatch stage using stochastic model predictive control (MPC). In the day-ahead dispatch stage, the data-driven chance constraints of tie-line power and reserve of gas-fired generators are built, and the day-ahead tie-line power is obtained and regarded as input parameters to the intraday dispatch stage. In the intraday dispatch stage, the data-driven chance constraints of tie-line power and reserve of gas-fired generators with the latest rolling prediction data are built, and the remaining control variables are obtained. The distribution characteristics of the stochastic prediction errors of wind power and electric load are captured and described by the variational Bayesian Gaussian mixture model with massive historical data. Then the original stochastic mixed-integer nonlinear programming problem is converted to a tractable deterministic one by the quantile-based analytical reformulation and convex relaxation technique. Finally, the proposed strategy is verified by the numerical experiments based on a modified IEEE 33-bus system integrated with a 10-node natural gas system and a micro hydrogen system. The numerical results demonstrate that the proposed strategy reduces the actual costs and decreases the violation rate caused by the stochastic prediction errors of wind power and electric load.

An inexact multi-objective mixed-integer nonlinear programming approach for water-soil-fertilizer management under uncertainty considering “footprint family-planetary boundary” assessment

Water, energy, and food (WEF) are interlinked and create a dynamic system that impacts both human well-being and ecology. Given the importance of ecology within the WEF system, water resources serve as the core issue. The allocation of water resources in irrigation districts is a challenging problem for the coordinated development of agricultural production, water resources, and the ecological environment. The integration of stochastic multi-objective programming, fuzzy credibility-constrained programming, and mixed integer programming offers a solution to this issue, with the construction of a fuzzy credibility-constrained stochastic multi-objective mixed-integer nonlinear programming model. The applicability and validity of this model were verified by applying it to the Kaikong Irrigation District (KID) of the Tarim River Basin in northwest China, with notable findings indicating that the optimized system reduces agricultural costs by 5.82%, increases irrigation water use efficiency by 1.80%, and reduces global warming potential by 6.45%. This study investigates the effects of diverse allocation strategies of water and land resources on the social, economic, and ecological subsystems and their interactions by downscaling four subprocesses PBs to the KID scale. The optimization model reveals that only the nitrogen footprint of Kuerle City surpasses the nitrogen boundary of the KID. The proposed solutions based on the model can encourage the green and ecologically-friendly development of agricultural production and can be applied to agricultural systems in arid regions with similar conditions.

Two-stage stochastic nonlinear winner determination for logistics service procurement auctions under quantity discounts

Quantity discount is a frequently adopted scheme that has not been explicitly investigated in logistics service procurement auctions. This paper focuses on a revised winner determination problem under quantity discounts and demand uncertainty for a fourth-party logistics (4PL) provider in a combinatorial reverse auction. To characterize our research problem, a two-stage stochastic nonlinear programming model is constructed. Inspired by the idea of sample average approximation (SAA), the nonlinear model is reformulated as a deterministic mixed integer linear programming model by using a linearization technique with superior expressions. Since the reformulation has a large number of decision variables and constraints, we integrate SAA with a dual decomposition Lagrangian relaxation technique (DDLR) to develop a solution method called SAA-DDLR. Numerical experiments are conducted to illustrate the effectiveness and applicability of our model and method. Sensitivity analysis reveals that both the 4PL and 3PLs can benefit from the quantity discount scheme. Managerial insights are drawn for the 4PL to run a cost-effective logistics system in the presence of quantity discounts.

Matching Supply and Demand with Lead-Time Dependent Price and with Safety Stocks in a Make-to-Order Production System

We studied the ability to reduce the supply–demand mismatch of a periodic Make-to-Order (MTO) production system using safety stocks with marketing managing demand using lead-time guarantee and price as levers. The aim is to understand the interdependencies between lead-time guarantee, price, and safety stocks. We modeled the problem as an unconstrained stochastic non-linear programming problem, maximizing the expected profit per-unit time and obtaining a closed-form solution. The price is a function of the lead-time guarantee. Based on the sensitivity analysis of problem parameters, we found that lead-time competitiveness is adversely affected by a low safety stock level, MTO production rate (i.e., low supply capability), and product price (i.e., high demand volume). A shorter lead-time requires higher safety stock through reduced product and inventory holding costs. A higher price for a shorter lead-time in a lead-time-sensitive market reduces the safety stock. In a price-sensitive market, lead-time is decreased instead of the price. Demand variation results in longer lead-time and higher safety stock (provided the holding cost is low). For a higher price premium, price increases and lead-time decrease (safety stock increases). The integrated operation-marketing model captures the complex trade-offs not seen in a hierarchical model to produce better solutions.