Stochastic Linear Programming Research Articles

Multi-Objective Bus Timetable Coordination Considering Travel Time Uncertainty

This paper proposes a timetable coordination method for transfer problems in a bus transit system. With a given bus network, a stochastic mixed-integer linear programming (MILP) model has been formulated to obtain coordinated bus timetables with the objective of minimizing a weighted sum of the average value of total waiting time and its average absolute deviation value, allowing for random bus travel time. The vital decision variable is the terminal departure offset time of each target bus trip within a certain off-peak period. The robust MILP model can also be used to solve the first-bus transfer problem with the introduction of several new linear constraints. A solution method based on the Monte Carlo simulation has been developed to solve the MILP model. Numerical experiments have been conducted for different scenarios. The results indicate that bus timetables coordinated by the developed model are capable of substantially reducing waiting time for transfer and non-transfer passengers. In addition, the feasibility of simplifying a common sub-route into a single transfer stop in a timetable coordination problem has been explored based on numerical experiments.

Coordinating the day-ahead operation scheduling for demand response and water desalination plants in smart grid

Integrating renewable energy resources (RES) is a challenge for power system operators due to their fluctuations and unpredictability. At the same time, the water shortage problem and the needs to desalinate more freshwater increase the prominence of sufficient energy resources for sustainable operation. Therefore, this paper presents a market-clearing mechanism in a co-optimization model that coordinates the operation of grid-connected reverse osmosis water desalination plants (RO-WDPs) and the operation of renewable-rich power systems. It is assumed that electric demands can participate in the provision of demand response (DR) to the market via multiple DR options. The DR options related to the general loads are demand shifting, load curtailment, distributed generation, and hybrid energy storage systems (HESS). These HESS includes battery storage and hydrogen storage systems. Pertaining to their special characteristics, a new DR option is proposed as a customized option for RO-WDPs. The market clearing mechanism is assumed to be based on the security-constrained unit commitment (SCUC) and formulated as a stochastic mixed-integer linear programming problem (MILP) to optimally schedule the day-ahead operation of the power system. The presented model is applied on 6-bus and IEEE 24-bus reliability test system (RTS) test power systems with significant penetration of RES to demonstrate its merits. The simulation results show that the system efficiency is enhanced by adding the energy flexibility of RO-WDP without endangering the water demand-supply when using the proposed coordinated model. Hence, the total operation cost is minimized, the RES integration is facilitated, and the hourly electricity prices are smoothened.

A stochastic linear programming model for maximizing generation and firm output at a reliability in long-term hydropower reservoir operation

The firm output, usually determined in the design stage of a hydroplant to serve as a threshold to measure system reliability, can be regarded as an unknown parameter to be explored to its maximum in long-term hydropower reservoir operation (LHRO). An unknown firm output to be ensured at certain reliability, however, will make the problem much more nonlinear and then complicate the modeling. This work presents a stochastic linear programming (SLP-1) model that can explicitly incorporate reliability in ensuring an unknown firm output to be maximized by using a probability variable to represent a decision at a state and introducing binary variables to decide whether the decision will ensure the unknown firm output. The present SLP-1 is improved on 1) another previous SLP model (SLP-0) that must have a firm output prespecified at certain reliability, and compared with 2) the stochastic dynamic programming (SDP) model that can only manage to estimate the firm output at a reliability with trial and error. Case studies show the superiority of the present SLP to the SDP that can hardly make the reliability any closer to what the SLP can achieve, especially in ensuring a high firm output, with gaps to desired reliability ranging up to 42.33% for Xiaowan and 31.8% for Nuozhadu. Indeed, the SLP-1 will encounter the dimensional difficulty that needs further efforts to overcome when applied to cascaded reservoirs.

Compromise policy for multi-stage stochastic linear programming: Variance and bias reduction

This paper focuses on algorithms for multi-stage stochastic linear programming (MSLP). We propose an ensemble method named the “compromise policy”, which not only reduces the variance of the function approximation but also reduces the bias of the estimated optimal value. It provides a tight lower bound estimate with a confidence interval. By exploiting parallel computing, the compromise policy provides demonstrable advantages in performance and stability with marginally extra computational time. We further propose a meta-algorithm to solve the MSLP problems based on in-sample and out-of-sample optimality tests. Our meta-algorithm is incorporated within an SDDP-type algorithm for MSLP and significantly improves the reliability of the decisions suggested by SDDP. These advantages are demonstrated via extensive computations, which illustrate the effectiveness of our approach.

The Benders by batch algorithm: Design and stabilization of an enhanced algorithm to solve multicut Benders reformulation of two-stage stochastic programs

This paper introduces a new exact algorithm to solve two-stage stochastic linear programs. Based on the multicut Benders reformulation of such problems, with one subproblem for each scenario, this method relies on a partition of the subproblems into batches. The key idea is to solve at most iterations only a small proportion of the subproblems by detecting as soon as possible that a first-stage candidate solution cannot be proven optimal. We also propose a general framework to stabilize our algorithm, and show its finite convergence and exact behavior. We report an extensive computational study on large-scale instances of stochastic optimization literature that shows the efficiency of the proposed algorithm compared to nine alternative algorithms from the literature. We also obtain significant additional computational time savings using the primal stabilization schemes.

A Two-Stage Stochastic Fuzzy Mixed-Integer Linear Programming Approach for Water Resource Allocation under Uncertainty in Ajabshir Qaleh Chay Dam

Due to the dry climate and unsuitable distribution of rainfall in Iran, sustainable agriculture depends on the proper use of water resources. In this study, the optimal allocation of water at Ajabshir Qaleh Chay Dam in agricultural sector is investigated using an interval parameter two-stage stochastic mixed-integer linear programming approach. Indeed, interval parameters two-stage stochastic programming (ITSP) with fuzzy variables is developed based on mixed-integer programming for a water resource allocation model to retrieve the water shortage of agricultural products and to achieve the optimal allocation of Ajabshir Qaleh Chay Dam water through its river canals between different products under uncertainty conditions. In this developed method, called extended ITSP (EITSP), a number of alternatives are used to compensate for the difference between the amount of promised water allocation targets and the actual allocated water in the optimal allocation of water. Then a new solving approach based on Huang Algorithm, fuzzy chance constrained programming and Zimmermann fuzzy programming will be presented to solve the problems. Furthermore, using a case study in this dam, the results are obtained for the developed approaches to clarify the described methods and to compare these results with each other. Finally, comparing the total system profits of the models shows that in the fuzzy model, the profit and system certainty increase simultaneously. Therefore, due to the lack of water resources in the agricultural sector and the uncertainty, the agricultural authorities of Ajabshir can decrease the unsustainability of water resources using the optimal model while increasing the cost-effectiveness of the farmers.

Improved managerial decisions for fuzzy stochastic bi-objective linear programming models: A Trade-off ratio-based autonomized approach

Improved managerial decisions for fuzzy stochastic bi-objective linear programming models: A Trade-off ratio-based autonomized approach

Improving distribution system resilience by undergrounding lines and deploying mobile generators

To improve the resilience of electric distribution systems, this paper proposes a stochastic multi-period mixed-integer linear programming model that determines where to underground new distribution lines and how to coordinate mobile generators in order to serve critical loads during extreme events. The proposed model represents the service restoration process using the linearized DistFlow approximation of the AC power flow equations as well as binary variables for the undergrounding decisions of the lines, the configurations of switches, and the locations of mobile generators during each time period. The model also enforces a radial configuration of the distribution network and considers the transportation times needed to deploy the mobile generators. It is shown that long-term line undergrounding decisions which are cognizant of short-term mobile generator deployments yield superior results relative to undergrounding decisions made without considering mobile generators. Using an extended version of the IEEE 123-bus test system, numerical simulations show that combining the ability to underground distribution lines with the deployment of mobile generators can significantly improve the resilience of the power supply to critical loads. • Resilience can be improved via mobile generators and underground lines. • Stochastic optimization for siting underground lines and operating mobile generators. • Show benefits from jointly considering undergrounding and mobile generator decisions.

A novel separate chance-constrained programming model to design a sustainable medical ventilator supply chain network during the Covid-19 pandemic

<p style='text-indent:20px;'>Providing new models or designing sustainable networks in recent studies represents a growing trend. However, there is still a gap in the simultaneous modeling of the three dimensions of sustainability in the electronic medical device supply chain (SC). In this paper, a novel hybrid chance-constrained programming and cost function model is presented for a green and sustainable closed-loop medical ventilator SC network design. To bring the problem closer to reality, a wide range of parameters including all cost parameters, demands, the upper bound of the released <inline-formula><tex-math id="M1">\begin{document}$ co_2 $\end{document}</tex-math></inline-formula>, and the minimum percentage of the units of product to be disposed and collected from a customer and to be dismantled and shipped from DCs are modeled as uncertain along with the normal probability distribution. The problem was first formulated into the framework of a bi-objective stochastic mixed-integer linear programming (MILP) model; then, it was reformulated into a tri-objective deterministic mixed-integer nonlinear programming (MINLP) one. In order to model the environmental sustainability dimension, in addition to handling the total greenhouse gas emissions, the total waste products were also controlled. The efficiency and applicability of the proposed model were tested in an Iranian medical ventilator production and distribution network. For sensitivity analyses, the effect of some critical parameters on the values of the objective functions was carefully examined. Finally, valuable managerial insights into the challenges of companies during the COVID-19 pandemic were presented. Numerical results showed that with the increase in the number of customers in the COVID-19 crisis, social responsibility could improve cost mean by up to 8%.</p>

Metamodeling the optimal total revenues of the short-term optimization of a hydropower cascade under uncertainty

This paper deals with the optimization of the short-term production planning of a real cascade of run-of-river hydropower plants. Water inflows and electricity prices are subject to data uncertainty and they are modeled by a finite set of joint scenarios. The optimization problem is written with a two-stage stochastic dynamic mixed-integer linear programming formulation. This problem is solved by replacing the value function of the second stage with a surrogate model. We propose to evaluate the feasibility of fitting the surrogate model by supervised learning during a pre-processing step. The learning data set is constructed by Latin hypercube sampling after discretizing the functional inputs. The surrogate model is chosen among linear models and the dimension of the functional inputs is reduced by principal components analysis. Validation results for one simplified case study are encouraging. The methodology could however be improved to reduce the prediction errors and to be compatible with the time limit of the operational process.

A Stochastic Modeling Approach for Interplanetary Supply Chain Planning

The unique and complex challenges of sustainable interplanetary travel necessitate a novel space logistics paradigm, which is hereafter referred to as the interplanetary supply chain. To lay the groundwork for this anticipated paradigm shift, this work aims to introduce principles and methodologies of supply chain (SC) planning to the field of space logistics. To this end, a stochastic multistage mixed integer linear programming model is proposed that explores the vision of a permanent human presence on Mars from a logistics standpoint. The model optimizes material flows, propellant production, and in-space infrastructure allocations to identify the optimal SC network design. Notably, demand uncertainty at the Mars base is considered, which makes it the first model to investigate the effects of stochasticity in space logistics planning. Several case studies are investigated, which incorporate data on Earth–Mars transfer windows and flight trajectories obtained through the National Aeronautics and Space Administration’s trajectory browser. The results reveal numerous interdependencies and, thus, highlight the need for holistic SC modeling approaches that go beyond the isolated mission planning of the extant literature. For instance, the findings show frequent interactions among spacecraft as they regularly engage in division of labor regarding cargo and propellant transportation. Moreover, complex propellant supply networks that involve strategic in-space infrastructure allocation and propellant production on celestial bodies are established. Overall, the developed model provides valuable insights into the planning of a sustainable interplanetary travel, offers a new SC perspective, and lays the foundation for future research in this area.

Solving multistage stochastic linear programming via regularized linear decision rules: An application to hydrothermal dispatch planning

The solution of multistage stochastic linear problems (MSLPs) represents a challenge for many applications. Long-term hydrothermal dispatch planning (LHDP) materializes this challenge in a real-world problem that affects electricity markets, economies, and natural resources worldwide. No closed-form solutions are available for MSLP and the definition of non-anticipative policies with high-quality out-of-sample performance is crucial. Linear decision rules (LDRs) provide an interesting simulation-based framework for finding high-quality policies for MSLP through two-stage stochastic models. In practical applications, however, the number of parameters to be estimated when using an LDR may be close to or higher than the number of scenarios of the sample average approximation problem, thereby generating an in-sample overfit and poor performances in out-of-sample simulations. In this paper, we propose a novel regularized LDR to solve MSLP based on the AdaLASSO (adaptive least absolute shrinkage and selection operator). The goal is to use the parsimony principle, as largely studied in high-dimensional linear regression models, to obtain better out-of-sample performance for a LDR applied to MSLP. Computational experiments show that overfitting is a non-negligible threat when using the classical non-regularized LDR to solve the LHDP, one of the most studied MSLP with relevant applications in industry. Our analysis highlights the following benefits of the proposed framework in comparison to the non-regularized benchmark: 1) significant reductions in the number of non-zero coefficients (model parsimony), 2) substantial cost reductions in out-of-sample evaluations, and 3) improved spot-price profiles.

Optimizing the aluminum supply chain network subject to the uncertainty of carbon emissions trading market

Aluminum production is characterized by high electricity consumption; hence its decarbonization hinges on upstream low-carbon electricity generation. As a result, the supply chain of aluminum product manufacturing can be strongly affected by the uncertainty of carbon emissions price via electricity generation. This paper investigates these impacts on the strategic design and the tactic operations of the aluminum supply chain by constructing a stochastic mixed-integer linear programming (SMILP) model, in which carbon price uncertainty and demand uncertainty are both accounted for. The supply chain case in the study covers a full process from upstream power generation to end-use consumption. A variety of stochastic carbon price scenarios are designed and assessed. Sensitivity analysis with respect to demand fluctuation is also performed. The results show that carbon price uncertainty significantly affects the configuration of the whole system. The main driving forces are the supplier selection and configuration alteration triggered by the carbon price variation, whereas other metrics such as carbon emissions costs and the responses to demand fluctuations display complicated patterns of change. This study provides a useful analytical framework that can be applied to the planning problem for other industries facing the same challenge.

A chance-constraint optimization model for a multi-echelon multi-product closed-loop supply chain considering brand diversity: An accelerated Benders decomposition algorithm

The closed-loop supply chain (CLSC) network design has become one of the most critical issues due to the importance of resource optimization. Moreover, increasing competition in commercial markets leads to the diversity of a brand’s product portfolio to meet the customer’s demand. Hence, this paper develops a multi-period, multi-brand stochastic mixed-integer linear programming (MILP) model with direct and indirect distribution for the proposed CLSC network. Because of the uncertain nature of demand, the uncertainties for new and second-hand product demand are considered. Besides, a chance-constraint optimization (CCO) approach is applied to deal with uncertainty. Moreover, the accelerated Benders decomposition (BD) algorithm is designed to solve the proposed model. Several test problems are created and used to solve the accelerated BD compared with the conventional BD algorithm to investigate this model. Finally, the results are compared and described analytically, and some future research is suggested.

Optimization-based economic analysis of energy storage technologies in a coupled electricity and natural gas market

This work presents a stochastic mixed-integer linear programming (MILP) optimization framework to investigate the optimal participation and economics of various energy storage technologies, such as pumped-hydro, advanced adiabatic and diabatic compressed air systems and li-ion battery, in a perfectly competitive coupled electricity and natural gas market. The clearing scheme pertains to energy-only markets, aiming to optimize dispatch and maximize social welfare for the integrated energy system, through a two-stage stochastic programming. The first stage presents the day-ahead market clearing procedure, while the second stage illustrates the integrated system operation in the real-time trading floor, through a set of plausible wind power generation realizations. The two markets mainly interact the bilateral operation of the diabatic compressed air energy storage system, both as an electricity producer and as natural gas consumer. The proposed algorithm is applied to a modified IEEE 24-bus power grid and a single-node gas network and provides a thorough analysis of the operational characteristics and profitability of each energy storage technology in the integrated energy system. Results illustrate that electricity storage systems can increase their overall profits under power transmission congestion and while wind power generation volatility increases from 5 % up to 52 %. In contrast, a decrease by 3.1 % in the social welfare is realized in the case of power congestion while this figure is approximately 2.54 % under wind power volatility increase.

STOCHOS: Stochastic opportunistic maintenance scheduling for offshore wind farms

Despite the promising outlook, the numerous economic and environmental benefits of offshore wind energy are still compromised by its high Operations and Maintenance (O&M) expenditures. On one hand, offshore-specific challenges such as site remoteness, harsh weather, transportation requirements, and production losses, significantly inflate the O&M costs relative to land-based wind farms. On the other hand, the uncertainties in weather conditions, asset degradation, and electricity prices largely constrain the farm operator’s ability to identify the time windows at which maintenance is possible, let alone optimal. In response, we propose STOCHOS, short for the stochastic holistic opportunistic scheduler—a maintenance scheduling approach tailored to address the unique challenges and uncertainties in offshore wind farms. Given probabilistic forecasts of key environmental and operational parameters, STOCHOS optimally schedules the offshore maintenance tasks by harnessing the opportunities that arise due to favorable weather conditions, on-site maintenance resources, and maximal operating revenues. STOCHOS is formulated as a two-stage stochastic mixed-integer linear program, which we solve using a scenario-based rolling horizon algorithm that aligns with the industrial practice. Tested on real-world data from the U.S. North Atlantic where several offshore wind farms are in-development, STOCHOS demonstrates considerable improvements relative to prevalent maintenance benchmarks, across various O&M metrics, including total cost, downtime, resource utilization, and maintenance interruptions.

A strategy to enhance the distribution systems recoverability via the simultaneous coordination of planning actions and operational resources

This paper presents a planning and operational strategy to improve the recoverability of distribution systems (DSs) to deal with a set of possible line fault scenarios. The strategy simultaneously optimizes the allocation of dispatchable distributed generation (DG) units while coordinating a dynamic restoration process based on a radial topology reconfiguration, an islanding operation, a demand response program, and the pre-positioning and dispatch of mobile emergency storage units. The uncertainty and variability associated with solar irradiation and demand are captured via a multi-period formulation based on a stochastic mixed-integer linear programming model. The objective function of this model minimizes the investment cost of new dispatchable DG units and the amount of energy shedding within the system. Simulations are performed on adapted 33-node and 53-node test systems to validate the proposed strategy under four different test conditions, numerical results reveal the advantages of simultaneously solving the planning and operational stages to improve the recoverability of the system.

Optimisation models for the day-ahead energy and reserve self-scheduling of a hybrid wind–battery virtual power plant

This work presents a suite of two optimisation models for the short-term self-scheduling and redispatch of a virtual power plant (VPP) composed of a wind farm and a Li-ion battery, that participates in the day-ahead energy and secondary regulation reserve markets of the Spanish power system. First, a two-stage stochastic mixed-integer linear programming model is used to obtain the VPP’s generation and reserve schedule and the opportunity cost of the energy stored in the battery. The model has an hourly resolution and a look-ahead period beyond the markets’ scheduling horizon and considers the hourly battery degradation costs as a function of both the depth of discharge and the discharge rate. Different strategies are evaluated to forecast the real-time use of the committed secondary regulation reserves. Second, a deterministic MILP model is used to determine the redispatch of the VPP using as input the generation and reserve schedule and the VPP’s storage opportunity cost provided by the former model and is executed on an hourly rolling basis. The results obtained show that the proposed models are effective for the short-term scheduling and redispatch of the VPP used with a low computational time, making them tractable and practical for their daily use.

Green-Resilient Supplier Selection and Order Allocation Under Disruption by Utilizing Conditional Value at Risk: Mixed Response Strategies

Two important decisions in supply chains and logistics systems design are the supplier selection and order allocation (SS&OA) problem and the vehicle routing problem (VRP). Supply disruption may reduce the capacity of suppliers, and the transportation network disruption may decrease the number of vehicles in the fleet and disrupt some routes. Also, increasing environmental regulations and environmental awareness makes companies pay more attention to green supply chain management (GSCM). In this paper, we integrate green and resilient supplier selection and order allocation decisions with vehicle routing decisions under disruption. We present a multiproduct two-stage risk-averse mixed-integer stochastic linear programming for the green and resilient supplier selection and order allocation integrated with vehicle routing (G&RSS&OA-V) problem. We consider resilient strategies before disruption, including multiple sourcing, supplier fortification, prepositioned inventory at the protected supplier, and contract with third-party logistics providers (3PLs). The objective function is to minimize the total mean-risk cost and the cost of greenhouse emissions. We use conditional value at risk (CVaR) as a risk measure to control the risk of worst-case cost. The most significant decisions of this model are the strategic decisions of determining the optimal suppliers and the operational decisions of vehicles routing under disruption simultaneously. Other decisions include determining which suppliers should be fortified, the amount to be transported to the hybrid manufacturing-distribution (HMD) center through the supplier or prepositioned emergency inventory, and the amount of lost sales. In order to validate the proposed model and its features, several numerical examples along with sensitivity analysis are performed by GAMS software, which shows the efficiency and application of the developed model, and some managerial insights are reported. The results of the sensitivity analysis show that as α increases from 0.1 to 0.9, the mean-CVaR objective function cost increases to 13.2%. As λ increases from 0.1 to 0.9, the mean-CVaR objective function cost increases to 35.6%. The increase of these two risk factors makes the proposed model more risk-averse. As the expected shortage cost increases by 150%, the mean-CVaR objective function cost increases to 36% while the amount of expected shortage decreases by 56%.

Distributionally Robust Optimal Sensor Placement Method for Site-Scale Methane-Emission Monitoring

Recent research in deterministic sensor placement optimization technologies has improved the capability of monitoring large-scale field environments with a limited budget. In traditional stochastic mixed-integer linear programming formulations, minimizing the expectation of detection time can lead to a detector placement with good average behavior but unexpected worst case behavior. The uncertainty factors in the complex environment and sensor system significantly challenge the effects of the placement strategy provided by stochastic programming (SP). These factors include unknown leakage rate and location, sensor delay, and primary uncertainty of wind conditions. This article introduces a distributionally robust optimization (DRO) formulation of sensor placement under the uncertainty of wind conditions and improves a sensor network’s detection robustness. The method is demonstrated using the atmospheric simulation with site-specific methane-emission scenarios that capture partial natural wind conditions and emission characteristics. DRO techniques are employed to determine sensor locations that minimize the detection time expectation of the emission scenarios with a significantly better worst case behavior. Experiment results show that the proposed DRO method outperforms the sensor placement methods based on SP.