Stochastic Mixed-integer Nonlinear Programming Research Articles

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.

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.

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.

A novel chance constrained joint optimization method under uncertainties in distribution networks

The joint optimization in distribution networks considering the uncertainties in wind power or photovoltaic (PV) outputs is a larger scale stochastic mixed integer nonlinear programming (MINLP) problem. However, how to efficient and accurate solve the problem under uncertainties is still a challenge. To handle the uncertainties, a scenario based chance constrained programming model is established. To improve the accuracy, the network reconfiguration and capacitor control are simultaneously performed by optimizing serious voltage and shunt current sources in the model by using equivalent network transformation. To improve calculation efficiency in dealing with chance constraints, a cluster of scenarios satisfying the confidence level is formed for optimization. To avoid introducing plenty of binary variables for the radial constraint and improving calculation efficiency, a heuristic method opening loops one by one is developed. The numerical simulations on distribution networks show the efficiency and accuracy of the proposed algorithm over the existing methods.

Two-Stage emergency material scheduling based on benders decomposition considering traffic congestion after a disaster

This study establishes a stochastic mixed integer nonlinear programming model for a two-stage emergency multiclass material scheduling problem considering traffic congestion with a Bureau of Public Roads (BPR) function. The generalized Benders decomposition (GBD) algorithm is used to solve the model on the basis of the characteristics of the decomposable structure and nonlinear convex function. To prove the effectiveness of the GBD algorithm, we set up a 30-node medium-scale case and a 50-node large-scale case for experimental demonstration and compared the GBD algorithm with the simple branch and bound (SBB) and discrete and continuous optimization solvers. Results show that for the medium-scale experiment, the gap between the solution results of the GBD algorithm and the ones of SBB or other solvers is less than 0.2%. The solution time of the GBD algorithm is only half of the time of SBB and other solvers. The large-scale case of 50 nodes cannot be solved by the SBB or other solvers, whereas the GBD algorithm can solve it normally, and the results are reasonable. In large-scale cases with more than 50 nodes, the GBD algorithm is more effective and efficient compared with SBB and other solvers. The sensitivity of the correlation coefficient α in the BPR function was analyzed. The actual expected total cost increases with the increase in the value of α. Its solving results are accurate and reasonable, proving that the results are more practical and applicable after adding the BPR function.

A sustainable supply chain network considering lot sizing with quantity discounts under disruption risks: centralized and decentralized models.

This study proposes a framework for the main parties of a sustainable supply chain network considering lot-sizing impact with quantity discounts under disruption risk among the first studies. The proposed problem differs from most studies considering supplier selection and order allocation in this area. First, regarding the concept of the triple bottom line, total cost, environmental emissions, and job opportunities are considered to cover the criteria of sustainability. Second, the application of this supply chain network is transformer production. Third, applying an economic order quantity model lets our model have a smart inventory plan to control the uncertainties. Most significantly, we present both centralized and decentralized optimization models to cope with the considered problem. The proposed centralized model focuses on pricing and inventory decisions of a supply chain network with a focus on supplier selection and order allocation parts. This model is formulated by a scenario-based stochastic mixed-integer non-linear programming approach. Our second model focuses on the competition of suppliers based on the price of products with regard to sustainability. In this regard, a Stackelberg game model is developed. Based on this comparison, we can see that the sum of the costs for both levels is lower than the cost without the bi-level approach. However, the computational time for the bi-level approach is more than for the centralized model. This means that the proposed optimization model can better solve our problem to achieve a better solution than the centralized optimization model. However, obtaining this better answer also requires more processing time. To address both optimization models, a hybrid bio-inspired metaheuristic as the hybrid of imperialist competitive algorithm (ICA) and particle swarm optimization (PSO) is utilized. The proposed algorithm is compared with its individuals. All employed optimizers have been tuned by the Taguchi method and validated by an exact solver in small sizes. Numerical results show that striking similarities are observed between the results of the algorithms, but the standard deviations of PSO and ICA–PSO show better behavior. Furthermore, while PSO consumes less time among the metaheuristics, the proposed hybrid metaheuristic named ICA–PSO shows more time computations in all small instances. Finally, the provided results confirm the efficiency and the performance of the proposed framework and the proposed hybrid metaheuristic algorithm.

Dual-spectrum resource allocation for coexisting LTE-advanced and Wi-Fi systems with imperfect channel state information

Long term evolution (LTE) and LTE-Advanced (LTE-A) based transmissions extended from licensed to unlicensed bands, known as LTE-unlicensed and licensed assisted access, can effectively alleviate the congestion of cellular bands. Efficient integration of the dual-spectrum in small-cells can further improve the achievable spectrum efficiency. In this paper, we optimize the power and resource allocation for LTE small-cells by jointly utilizing licensed and unlicensed bands, where practical issues like imperfect channel state information (CSI) and the characteristics of LTE unlicensed traffic are considered. More specifically, we formulate a stochastic mixed-integer non-linear programming (MINLP) problem to maximize the expected sum-rate (ESR) of the system. To solve this NP-hard problem, the so-called reduced-complexity joint licensed and unlicensed resource allocation (RC-JLURA) algorithm is proposed, where the original problem is transformed to a deterministic MINLP problem by employing an analytic approximation. Thanks to the Lagrange dual optimization based on a framework of alternate iteration, the proposed method can provide a near-optimum closed-form solution to the problem concerned, while maintaining a low computational complexity. The achievable performance of the LTE dual-spectrum mechanism is evaluated through simulations, which demonstrate the superiority of the proposed algorithm.

A two-stage stochastic programming model for selective maintenance optimization

The problem of selective maintenance exists in many multicomponent systems carrying out an alternating sequence of missions, with scheduled breaks where only a limited number of components can be maintained due to time limit. In this paper, a stochastic programming approach is proposed for determining an optimum maintenance plan to minimize maintenance costs and expected failure costs, while maximizing the probability of successful accomplishment of the next mission under uncertainties in future operating conditions. Traditionally, future operating conditions that affect failure time distribution when calculating reliability in selective maintenance models were assumed as deterministic. In this study, future operating conditions are assumed to be uncertain. The system is subject to several uncertain condition scenarios of exposure, conditional, usage, stress, etc. Each scenario is modeled with its associated occurrence probability. The presented model is a two-stage stochastic mixed-integer nonlinear programming model with fixed recourse, where the first stage is associated with maintenance decisions made before uncertainties are revealed, and the second stage is modeled as a recourse function which is related to the occurrence probability of system failure. A numerical example of a series-parallel system is used to demonstrate the effectiveness of the suggested model.

Multiple Electricity Markets Competitiveness Undergoing Symmetric and Asymmetric Renewables Development Policies

This paper scrutinizes the impact of different renewable energy sources (RES) development policies on competitiveness within multiple electricity markets (MEMs). Also, the variation in market power indices by increasing the integration of the markets undergoing symmetric and asymmetric RES development policies is investigated. To do so, several stochastic mixed-integer non-linear programming objective functions are used in the agent-based simulation framework to model the power plants’ behavior and markets. The case study shows in the low RES penetrated markets, one can say the more integration level of the markets, the lower potential of exercising market power. The reciprocal judgment is true for a high RES penetrated market. Also, large asymmetry in RES development between markets within MEMs may bring about market power problem for a high RES penetrated market. Unlike the asymmetric RES development policies, adopting homogeneous policies in RES development within MEMs reduces the market power potential in all markets and this potential decreases with the increase in the integration of the markets.

A two-stage decision dependent stochastic approach for airline flight network expansion

Airlines need to expand their flight networks with developing new routes and introducing more flights to increase their market share. In this work, we propose a two-stage stochastic mixed integer nonlinear program (MINLP), which expands an existing flight schedule by operating new flights either with existing fleet resources or a leased aircraft while considering the impact of departure time decisions on the probability distribution of random demand. Moreover, our study helps an airline to link a strategic decision of leasing an aircraft to the tactical aircraft assignment decisions by considering fuel efficiency and seat capacity of the aircraft alternatives in response to new passenger demand. However, the large number of scenarios, nonlinear fuel burn function and nonlinearities due to the decision dependent probabilities become main challenges of solving the problem. In order to deal with the computational requirements of a two-stage stochastic MINLP with decision dependent probabilities, we propose strong conic quadratic and McCormick inequalities, and an exact scenario group wise decomposition algorithm along with a new bounding method. In our computational results, we clearly demonstrate the effectiveness of proposed decomposition algorithm and the strength of the reformulations.

A New Stochastic Model for Bus Rapid Transit Scheduling with Uncertainty

Nowadays, authorities of large cities in the world implement bus rapid transit (BRT) services to alleviate traffic problems caused by the significant development of urban areas. Therefore, a controller is required to control and dispatche buses in such BRT systems.. However, controllers are facing new challenges due to the inherent uncertainties of passenger parameters such as arrival times, demands, alighting fraction as well as running time of vehicles between stops. Such uncertainties may significantly increase the operational cost and the inefficiencies of BRT services. In this paper, we focus on the controller’s perspective and propose a stochastic mixed-integer nonlinear programming (MINLP) model for BRT scheduling to find the optimal departure time of buses under uncertainty. The objective function of the model consists of passenger waiting and traveling time and aims to minimize total time related to passengers at any stop. From the modeling perspective, we propose a new method to generate scenarios for the proposed stochastic MINLP model. Furthermore, from the computational point of view, we implement an outer approximation algorithm to solve the proposed stochastic MINLP model and demonstrate the merits of the proposed solution method in the numerical results. This paper accurately reflect the complexity of BRT scheduling problem and is the first study, to the best of our knowledge, that presents and solves a mixed-integer nonlinear programming model for BRT scheduling.

Lyapunov-Guided Deep Reinforcement Learning for Stable Online Computation Offloading in Mobile-Edge Computing Networks

Opportunistic computation offloading is an effective method to improve the computation performance of mobile-edge computing (MEC) networks under dynamic edge environment. In this paper, we consider a multi-user MEC network with time-varying wireless channels and stochastic user task data arrivals in sequential time frames. In particular, we aim to design an online computation offloading algorithm to maximize the network data processing capability subject to the long-term data queue stability and average power constraints. The online algorithm is practical in the sense that the decisions for each time frame are made without the assumption of knowing the future realizations of random channel conditions and data arrivals. We formulate the problem as a multi-stage stochastic mixed integer non-linear programming (MINLP) problem that jointly determines the binary offloading (each user computes the task either locally or at the edge server) and system resource allocation decisions in sequential time frames. To address the coupling in the decisions of different time frames, we propose a novel framework, named LyDROO, that combines the advantages of Lyapunov optimization and deep reinforcement learning (DRL). Specifically, LyDROO first applies Lyapunov optimization to decouple the multi-stage stochastic MINLP into deterministic per-frame MINLP subproblems. By doing so, it guarantees to satisfy all the long-term constraints by solving the per-frame subproblems that are much smaller in size. Then, LyDROO integrates model-based optimization and model-free DRL to solve the per-frame MINLP problems with very low computational complexity. Simulation results show that under various network setups, the proposed LyDROO achieves optimal computation performance while stabilizing all queues in the system. Besides, it induces very low computation time that is particularly suitable for real-time implementation in fast fading environments.

A three-objective stochastic location-inventory-routing model for agricultural waste-based biofuel supply chain

The utilization of agricultural wastes has visibly emerged as a promising policy towards enhancing the fragile global energy system characterized by a limited fossil fuel reserve. To this aim, this paper proposes a multi-objective model for the design of agricultural waste-based biofuel production with integrated formulation of location, inventory and routing decisions. Our model allows the decision makers to determine the number and location of residue gathering centers and biorefineries, the routes by which a heterogeneous fleet of vehicles collects the different agricultural waste, and the adequate material flow to meet the biofuel demand. Instead of assuming a fixed facility setting along the supply chain planning horizon, we construct a comprehensive mathematical model that includes: (1) the opening of a certain facility at any time period; (2) plant capacity expansion within the planning horizon; and (3) the facility closing not necessarily at last time period. The original model formulation is a three-objective stochastic Mixed-Integer Non-linear Programming. However, we propose a linearization strategy to efficiently convert our model into a MILP formulation. Since the proposed optimization model belongs to the class of NP-hard problems, a two-phase heuristic method is utilized to solve the formulated model. The constructive phase of our heuristic provides an initial solution that is further enhanced by a Simulated Annealing algorithm. We tested the proposed heuristic method for small, medium and large-scale problem instances. The computational results demonstrate that our heuristic is running-time efficient and highly competitive in terms of solution quality, compared to the exact method outcomes.

An expected-cost realization-probability optimization approach for the dynamic energy management of microgrid

Dynamic energy management (DEM) is being recognized as an important tool for the economical and efficient operation of microgrid. This paper proposes an expected-cost realization-probability (ECRP) optimization approach for the DEM of microgrid. In the model, the expected cost is taken as an economic goal to find the schedule with the maximum ECRP (i.e., the maximum probability of obtaining an actual operation cost which is lower than this economic goal). It is a stochastic mixed-integer nonlinear programming (MINLP) problem composed of the AC power flow, battery operation characters and logical variables. To solve this NP-hard problem, an ECRP-based approximate dynamic programming is proposed. The original MINLP problem is decomposed into small nonlinear programming subproblems to facilitate the solution. And the ECRP-based value function is formed to describe the relationship among the decisions of subproblems, which helps to obtain the near-optimal decision in a decomposed manner. Simulation examples presented on a real microgrid indicate that the ECRP of the proposed method is higher than those of the other two comparison algorithms.

An optimal stochastic energy management system for resilient microgrids

This paper presents a stochastic mixed-integer nonlinear programming model for the optimal energy management system of unbalanced three-phase of alternating current microgrids. The proposed model considers the following random variables: nodal demands, nodal renewable generation and voltage reference at the point of common coupling. Furthermore, the proposed model is aimed at providing resilient energy management system solutions via contingency constraints. The proposed mixed-integer nonlinear programming model is transformed into a mixed-integer linear programming model through a set of linearizations that can be solved via off-the-shelf convex programming solvers. The analyzed microgrid comprises photovoltaic generation, energy storage systems, electric vehicle chargers, direct load control, and non-renewable generation, which operates when the microgrid is in islanded mode. The stochastic nature of the problem is considered through a scenario-based approach. The solution to the model determines the day-ahead operation of the microgrid resources that minimizes the average operational cost. An unexpected islanded operation at any given time is considered via contingency constraints. Tests are performed using data of the real microgrid at the Laboratory of Intelligent Electrical Networks (LabREI), at University of Campinas. Results show that the proposed model produces resilient day-ahead energy management system solutions while minimizing the average operational costs and maximizing the use of local renewable energy sources.

Demand Response for Industrial Micro-Grid Considering Photovoltaic Power Uncertainty and Battery Operational Cost

An intelligent demand response (DR) program is developed for multi-energy industrial micro-grid consisting of manufacturing facilities, photovoltaic (PV) panels, and battery energy storage system (BESS). The proposed DR program tackles the practical challenges of components in the micro-grid including industrial process represented by a discrete manufacturing production model, uncertainty of PV generation, and operational cost of the BESS. The proposed DR program optimizes day-ahead production scheduling for manufacturing facilities and operation regime for the BESS in response to time of use electricity price and probabilistic forecasting of PV power. To capture the uncertainty of PV power, a data-driven PV power probabilistic forecasting model is developed and a copula-based approach is deployed for the sampling of temporally correlated scenarios of PV power over the scheduling horizon from probabilistic forecasts. The multi-energy management optimization problem is formulated as a scenario-based stochastic nonconvex mixed integer nonlinear programming (MINLP). A hybrid optimization method integrating the evolutionary algorithm and the branch-and-bound algorithm for mixed integer liner programming is proposed to solve the nonconvex MINLP. Simulation studies illustrate that the proposed DR program efficiently reduces the operational cost for manufacturing production and releases the stress of the main grid by making full use of flexibility of all the components in the micro-grid.

A two-stage stochastic nonlinear integer-programming model for slot allocation of a liner container shipping service

In this study, we propose a container slot allocation problem for a liner shipping service. A liner containership provides a regular shipping service with a fixed itinerary and schedule. In practice, the liner containership may not be fully loaded, which results in a loss of revenue. We therefore segment shippers into two classes: contract shippers and spot shippers. A contract shipper has a contract with the shipping company and negotiates a fixed minimum quantity, so that the shipping company can secure a steady revenue. The remaining containership slots are open to spot shippers, allowing the shipping company to obtain ad hoc revenue. The container slot allocation problem is investigated in this study using a two-stage stochastic mixed-integer nonlinear programming model. We use the sample average approximation based on Lagrangian relaxation and dual decomposition techniques to effectively solve the model. Finally, we conduct a case study to evaluate the applicability and effectiveness of the proposed model and the solution algorithm.