Stochastic Linear Programming Research Articles

Resilience-oriented placement of multi-carrier microgrids in power systems with switchable transmission lines

Extreme events such as floods, hurricanes, earthquakes, and wildfires pose significant threats to the uninterrupted supply of electricity to consumers, as they can cause the failure of numerous power system components. The resilience of a power system is defined by its ability to withstand extreme events and continue meeting demand. While the placement of multi-carrier microgrids (MGs) in power systems can impact resilience, the literature lacks research on resilience-oriented placement. This paper aims to address this gap by proposing a resilience-oriented placement strategy for multi-carrier MGs, utilizing switchable transmission lines to enhance the resilience of power systems. The study focuses on a heat-power-hydrogen MG that obtains gas from a gas network, exchanges power with the main power system, and incorporates combined heat and power (CHP), electrolyzer, and thermal storage. The developed model employs a stochastic mixed-integer linear programming (MILP) approach, ensuring the attainment of the global optimum. Resilience is evaluated using the metric of expected load not supplied (ELNS). The results demonstrate that when the bus connected to the MG is isolated, the MG generates electricity through its CHP unit to meet local demand, reducing the total demand shed in the power system and improving system resilience. Specifically, the MG reduces ELNS from 4955.48 MWh to 2356.64 MWh, indicating a remarkable 52% improvement in ELNS. Furthermore, the study shows that transmission line switching further decreases ELNS from 2356.64 MWh to 848.68 MWh. Several experiments are conducted to analyze the sensitivity of power system resilience to the number of MGs, the MG-power system exchange limit, and the limit on gas import from the gas network.

Service-oriented container slot allocation policy under stochastic demand

The liner shipping industry plays a pivotal role in global cargo transportation, catering to both contract and spot shippers. Proper capacity allocation between these shippers is vital for maintaining service quality and improving revenue. This research investigates the service-oriented container slot allocation problem under stochastic demand, aiming to maximize total freight revenue while providing adequate service levels to contract shippers for sustaining their market loyalty. We use the fill rate (i.e., the proportion of satisfied demand) as a metric for service level and formulate the research problem as a stochastic linear programming model. To solve this model, we convert it into a multi-objective attainability problem by setting a target for total revenue, and apply Blackwell’s Approachability Theorem to theoretically determine the feasibility of any given revenue and service level requirements. Leveraging these insights, we devise near-optimal policies to guide the slot allocation decision under each demand scenario. Numerical experiments demonstrate that our approach outperforms the benchmark policies in the literature. Furthermore, it can also achieve near-optimal performance closely resembling the sampling average approximation (SAA) solution while significantly reducing the computational time.

A Two-Stage Stochastic Linear Programming Model for Tactical Planning in the Soybean Supply Chain

Background: The soybean market is representative of the world. Brazil is the largest producer and exporter of this crop and has low production costs but high logistical costs, which are influenced mainly by transport costs. Added to these characteristics, the disputed grain supply, the possibility of crop failure, and the randomness of some parameters that influence the soybean supply chain make decisions even more challenging. Methods: To mathematically model this problem, we carried out an analysis of the scientific production related to grain supply chain and the models used to address the problem, as well as a document analysis and a case study. Results: This paper proposes a new two-stage stochastic linear programming model with fixed recourse for tactical planning in the soybean supply chain from the perspective of the shipper under take or pay contracts over a one-year time horizon. The first-stage variables are the grain purchasing decisions and the volumes of rail and road transportation hired in advance. The model addresses 243 scenarios derived from four uncertainty sources: the purchase and sale prices of raw agricultural products on the spot market, the probability of crop failure, and the external demand. Conclusions: The model is successfully applied to a soybean trade firm in Brazil with expected gain of US$4,299,720 when using the stochastic model instead of the deterministic model. The stochastic model protected the firm from take or pay fines and crop failures, contracting a smaller volume of rail transport than what the company does.

Platelet inventory management with demand and supply uncertainty and variable pricing considerations

Managing the inventory of blood is a crucial operation in hospitals owing to its significance in medical treatment. At the same time, blood is characterized by unique attributes, such as perishability and the unpredictable nature of its supply and demand. While models have been developed to optimize the said process, gaps in literature exist in terms of considering the possibility of variable pricing and extensively accounting for uncertainties in the supply chain. In this light, the present study proposes a stochastic multi-period mixed integer linear programming cost minimization model that determines the optimal inventory plan for a hospital purchasing platelets, assuming that prices fluctuate along with the blood center’s supply. To implement uncertain supply and demand, the model considers a discrete set of scenarios for each parameter. A study was performed, and the results indicate a promising direction as inventory costs decreased relative to models without the new considerations.

Linear conic and two-stage stochastic optimization revisited via semi-infinite optimization

In this paper, we update the theory of deterministic linear semi-infinite programming, mainly with the dual characterizations of the constraint qualifications, which play a crucial role in optimality and duality. From this theory, we obtain new results on conic and two-stage stochastic linear optimization. Specifically, for conic linear optimization problems, we characterize the existence of feasible solutions and some geometric properties of the feasible set, and we also provide theorems on optimality and duality. Analogously, regarding stochastic optimization problems, we study the semi-infinite reformulation of a problem-based scenario reduction problem in two-stage stochastic linear programming, providing a sufficient condition for the existence of feasible solutions as well as optimality and duality theorems to its non-combinatorial part.

Two-stage distributionally robust optimization for joint system design and maintenance scheduling in high-consequence systems

The failures of high-consequence systems can cause serious harm to humans, including loss of human health, life security, finance, and even social chaos. To protect high-consequence systems, both optimal system design and maintenance activities contribute to improving system reliability and social safety. The existing works generally optimize these two problems sequentially and assume that the degradation process of components is precisely known. However, sequential optimization often results in significant losses due to redundancies, and such a presumption usually cannot be guaranteed in practice, due to limited historical data or a lack of expert knowledge, referred to as epistemic uncertainty. To fill this gap, in this article, we consider an integrated optimization of system design and maintenance scheduling for multi-state high-consequence systems in which the component’s degradation is known with limited distributional information. To address this issue, we utilize the framework of distributionally robust optimization to provide a risk-averse decision to decision-makers even under the worst realizations of random parameters, and develop a two-stage integer distributionally robust model with moment-based ambiguity set to determine the system design and maintenance scheduling simultaneously. The proposed model can be converted to a tractable approximation as an integer linear stochastic programming problem. In order to solve large-scale problems, we develop a sample-based adaptive large neighborhood search algorithm to find the optimal system designs. In the numerical experiments, we present a case study on feedwater heating systems in nuclear power plants and demonstrate that an integrated optimization consideration creates significant benefits in profitability. We also present the out-of-sample performance of the distributionally robust design to avoid extreme risk.

An optimization model for express delivery with high-speed railway

With the expansion of the high-speed railway (HSR) network in China, high-speed rail express delivery (HSReD) is being used to satisfy the increasing demand for express cargo. The decisions on transportation resources arrangement and freight flow allocation are two of the key issues for practical implementation of HSReD. In this study, we examine the above key issues by developing a two-stage stochastic integer linear programming model to maximize the expected net operation profit of HSReD. A meta-heuristic solution approach introduced some tailored tactics is proposed to speed up the process of solving the above model in the large-scale instances. Numerical experiments based on different sizes and practical investigation on China Railway Nanchang Group are conducted to validate the effectiveness of the proposed model and solution approach. Some managerial implications are also obtained based on the sensitivity analysis, which may be potentially useful for optimizing the daily operation management of HSReD.

Towards a responsive-sustainable-resilient tea supply chain network design under uncertainty using big data

Nowadays, issues such as limited natural resources, environmental problems, social matters, and significance of resilience in agricultural supply chain (ASC) have dragged considerable attention worldwide. In this research, a five-level multi-objective stochastic mixed-integer linear programming model is designed for tea supply chain (TSC) in Iran. The objective functions of the suggested network are minimizing total costs of the supply chain (SC), the total water consumption, and non-resilience measures, and maximizing job opportunities of facilities. Literally, considering uncertainty for SC networks is extremely beneficial due to the existence of some variations in different parameters like demand. As a consequence, a robust possibilistic optimization (RPO) is implemented to manage the uncertainty. Due to the nature of the multi-objective optimization problem, the weighted-normalized-extended goal programming (WNEGP) approach is employed to solve the model. In order to credit the model, real data is collected from the tea organization of Iran. It is worth mentioning that parameters are gathered according to three aspects of big data: volume, velocity, and variety. The results validated the functionality of the model regarding planning strategy. In addition, it showed applying more costs on SC triggers an effective sustainable-resilient-responsive network. In terms of managerial insights, this study offers a far-reaching perspective to managers especially in ASC to develop their industries. Finally, some sensitivity analyses are discussed on key parameters such as demand, robustness coefficients, and also the value of the objective functions in various states. It is worth mentioning that sensitivity analyses on different states of the problem show how sustainability and resiliency affect the supply chain efficiency.

Bi-level stochastic energy trading model for technical virtual power plants considering various renewable energy sources, energy storage systems and electric vehicles

The ongoing transition of the energy system towards being low-carbon, digitized and distributed is accelerating. Distributed Energy Resources (DERs) are playing a major role in this transition. These DERs can be aggregated and controlled by Virtual Power Plants (VPPs) to participate in energy markets and make full use of the potential of DERs. Many existing VPP models solely focus on the financial impact of aggregating DERs and do not consider the technical limitations of the distribution system. This may result in technically unfeasible solutions to DERs operations. This paper presents an expanded VPP model, termed the Technical Virtual Power Plant (TVPP), which explicitly considers the technical constraints of the network to provide operating schedules that are both economically beneficial to the DERs and technically feasible. The TVPP model is formulated as a bi-level stochastic mixed-integer linear programming (MILP) optimization model. Two objective functions are used, the upper level focuses on minimizing the amount of power imported into the TVPP from the external grid, while the lower level is concerned with optimally scheduling a mixture of DERs to increase the profit of the TVPP operator. The model considers three TVPPs and allows for energy trading among the TVPPs. The model is applied to several case studies based on the IEEE 119-node test system. Results show improved DERs operating schedules, improved system reliability and an increase in demand response engagement. Finally, energy trading among the TVPP is shown to further reduce the costs of the TVPP and power imported from the upstream electrical network.

Integrating pre and post-disaster activities for designing an equitable humanitarian relief supply chain

The design of an efficient humanitarian relief supply chain (HRSC) for disaster management depends on the integration of pre- and post-disaster decisions, which reduces the suffering of the victims and saves lives. The use of strategies such as quantity flexibility contracts in the pre-disaster phase and evacuation management in the post-disaster phase significantly increase the efficiency of HRSCs for disaster management. Therefore, in this paper, for the first time, a comprehensive two-stage stochastic mixed-integer linear programming model is presented for integrating the pre- and post-disaster activities for possible disasters such as floods and earthquakes by considering quantity flexibility contract and equitable relief goods distribution. In addition, the proposed model determines the location of warehouses and inventory levels in these warehouses in the pre-disaster phase. It also deals with public donations and budget planning, planning vehicles and helicopters, locating hospitals and shelters, and optimizing the inventory flow between echelons in the post-disaster phase. In the proposed model, the quantity flexibility contract is applied to decrease the inventory level in the pre-disaster phase, and reduce the supply risk in the post-disaster phase. The efficiency and performance of the proposed model are evaluated using the data related to a possible earthquake in Karaj (one of the metropolises of Iran).

Optimal Modeling of Sustainable Land Use Planning under Uncertain at a Watershed Level: Interval Stochastic Fuzzy Linear Programming with Chance Constraints

In this paper, an uncertain interval stochastic fuzzy chance constraint land use optimal allocation method is proposed and applied to solve the problem of land use planning in river basins. The UISFCL-LUP method is an aggregation of interval parametric programming, fuzzy linear programming and chance constraint programming which can cope with uncertain problems such as interval value, fuzzy set and probability. In this paper, the uncertain mathematical method is explored and studied in the optimal allocation of land use in the next two planning periods of Nansihu Lake Basin in China. Moreover, it was proved that ISFCL-LUP can deal with the uncertainty of interval, membership function and probability representation and can also be used to solve the land use planning and land use strategy analysis under uncertain conditions. On the basis of model calculations, we obtained the optimal allocation results for six types of land use in four regions over two planning periods based on different environmental constraints. The results show that the optimized λ value (that is, the degree of satisfaction with all the model conditions) is in the range of [0.54, 0.79] and the corresponding system benefits are between [18.4, 20.4] × 1012 RMB and [96.7, 109.3] × 1012 RMB. The results indicate that land managers can make judgments based on the different socio-economic development needs of different regions and determine strategic land use allocation plans under uncertain conditions. At the same time, the model obtained interval solutions under different system satisfaction and constraint violation probabilities, which helps land managers to analyze the importance of land system optimization and sustainable development more deeply.

A stochastic optimization approach for optimal Tail Assignment with knowledge-based predictive maintenance

AbstractRecent developments in the field of data analysis offer enormous potential to derive predictions of future aircraft maintenance necessities from observable aircraft attributes. This procedure is called Predictive Maintenance. The Predictive Maintenance generates considerable business benefits compared to alternatives like reactive or preventive maintenance for various reasons. Nevertheless, recognizing a maintenance necessity is only the first step, since the execution of the maintenance action has to fit into the airlines operational schedule. An essential step to realize financial benefits is to integrate Predictive Maintenance into the airlines optimization processes. To contribute to this task, we expand a well-known Tail Assignment model for the assignment of a potentially heterogeneous fleet of aircraft to a schedule of flights by a number of part failure scenarios. This results in a stochastic mixed integer linear optimization program, for which an optimization algorithm with solution guarantees is developed. This optimization algorithm is based on Benders Decomposition, which is concretized for the Tail Assignment problem and optimized for this task. Using this algorithm, a large proportion of the recovery costs for almost all instances tested is saved. The algorithm solves all instances in a reasonable amount of time. The algorithm uses a state-of-the-art mixed-integer linear solver, implementing a decomposition-based solution procedure for stochastic programs, called the L-shaped method. To demonstrate the potential of the approach, we benchmark our results using four strategies, motivated by commonly used preventive, reactive and passive approaches. Our approach leads to considerable cost savings when compared to each of the four benchmark approaches. The algorithms are tested on a set of instances with up to 80 flights based on a pre-pandemic schedule of a larger German airline. To the best of our knowledge, this article presents the first attempt to implement an exact optimization approach integrating malfunction predictions of the form presented into a Tail Assignment model.

Stochastic service network design: The value of fixed routes

This study introduces a challenging service network design problem with stochastic demands and fixed routes. In this problem, the routes of some service vehicles are fixed during the whole planning horizon, while the remaining vehicles are flexible in that their routes can adapt to different realizations of customer demands each day. The problem is to determine which vehicles should be set as fixed and how their fixed routes should be designed. To solve this problem, we formulate a two-stage stochastic mixed-integer linear program. The fixed routes are designed in the first stage before customer demands are realized, and the routes for flexible vehicles are designed in the second stage after customer demands are observed. A learning-based multiple scenario approach is developed as the solution method. Numerical experiments on real-world operational data from a logistics company show that fixing the routes of some service vehicles may reduce the operational cost by an average of approximately 6.37% compared with that in the case of no fixed routes.

Two-Stage Stochastic Optimization Model for Multi-Microgrid Planning

This paper presents a Two Stage stochastic Programming (TSSP) model for the planning of Multi-Microgrids (MMGs) in Active Distribution Networks (ADNs). The model aims to minimize the total costs while benefiting from interconnections of Microgrids (MGs), considering uncertainties associated with electricity demand and Renewable Energy Sources (RESs). The associated uncertainties are analyzed using Geometric Brownian Motion (GBM) and probability distribution functions (pdfs). The model includes long-term purchase decisions and short-term operational constraints, using Geographical information Systems (GIS) to realistically estimate rooftop solar limits. The planning model is used to study the feasibility of implementing an MMG system consisting of 4 individual Microgrids (MGs) at an ADN in a municipality in the state of São Paulo, Brazil. The results show that the TSSP model tends to be less conservative than the deterministic planning model, which is based on simple and pessimistic reserve constraints, while performing faster than a simple Stochastic Linear Programming (SLP) algorithm, with higher accuracy.

Traffic signal control under stochastic traffic demand and vehicle turning via decentralized decomposition approaches

Traffic congestion is a global pressing issue but can be mitigated via effective traffic signal control schemes. In this paper, based on a cell transmission model we coordinate the control of traffic signals at multiple intersections to maximize vehicle throughput on corridors or road networks, under stochastic traffic demand and vehicle turning. We formulate a two-stage stochastic mixed-integer linear program using finite samples of the uncertain parameter, and combine Benders decomposition with the alternating direction method of multipliers to develop spatially-temporally distributed algorithms for optimizing the problem. We test instances of traffic signal control on corridors and grid networks, generated based on synthetic and real-world traffic data. Our results show that (i) considering traffic uncertainty can significantly improve the signal control quality and (ii) decentralized decomposition approaches can quickly find high-quality signal plans for multiple intersections in complex road networks, and fully utilize the computation and communication technologies in smart-transportation infrastructures.

A Multiple Objective Genetic Algorithm Approach for Stochastic Open Pit Production Scheduling Optimisation

ABSTRACT The conventional approach to mine planning is to use a single estimated orebody model as the basis for production scheduling. This approach, however, does not consider grade uncertainties associated with grade estimation. These uncertainties have a significant impact on the net present value (NPV) and can only be accounted for when modelled as part of the production scheduling optimisation problem. In this research, a set of equally probable simulated orebodies generated through Sequential Gaussian Simulation is used as input to a stochastic optimisation model solved with genetic algorithm (GA). Grade variability is considered as part of the stochastic model. The problem definition and resource constraints are formulated and optimised using a specially designed mining-specific GA. This GA is employed to handle partial block processing through a specialised chromosome encoding technique resulting in near-optimal solutions. Two case studies are presented which compare results from the stochastic model solved with GA (SGA) and a Stochastic Mixed Integer Linear Programming (SMILP) model solved with CPLEX. For the second case study, while the SMILP model was at an optimality gap of 101% after 28 days, the SGA model generated an NPV of $10,045 M at 10.16% optimality gap after 1.5 h.

Resilience enhancement of distribution network under typhoon disaster based on two-stage stochastic programming

The reliability of power supply in distribution network is vulnerable to extreme weather events such as typhoon. Pre-event preparation can effectively mitigate the deterioration of system resilience. Therefore, we propose a distribution network resilience enhancement decision-making framework which is formulated as a two-stage stochastic mixed-integer linear programming (SMILP) model. The first stage invests coordinately in four strategies, including hardening lines, installing distributed generators (DG), allocating mobile emergency generators (MEG), and deploying switches, etc. And the goal at the first stage is to minimize the investment cost of resilience enhancement strategies. The objective at the second stage is to ensure the minimum expected recourse operation cost of the comprehensive strategies for all typical scenarios. The proposed model can combine distribution network long-term resilience planning with short-term post-event recovery. Furthermore, in order to address the uncertain problems of wind field, line damage and load fluctuations under typhoon disaster, this paper proposes the following solutions. For wind field uncertainty, a detailed wind field model considering time transition, sea-land transition and extreme value distribution is established to deal with wind speed prediction. For line damage uncertainty, an improved stress-strength interference model is set up. And for the load fluctuation uncertainty, scenario generation using load random multipliers is applied to address load uncertainty. The proposed framework is tested in IEEE 33-bus distribution system using historical data from 2018 super typhoon “Mangkhut” in China and demonstrates the SMILP model can significantly reduce the post-event expected recourse operation cost and meanwhile improve the distribution network resilience.

A multi-stage stochastic programming model for the unit commitment of conventional and virtual power plants bidding in the day-ahead and ancillary services markets

As more uncontrollable renewable energy sources are present in the power generation portfolio, the need of more detailed and reliable tools for the optimal operation of energy systems has increased in the last years. This work presents a multi-stage stochastic Mixed Integer Linear Program with binary recourse for optimizing the day-ahead unit commitment of power plants and virtual power plants operating in the day-ahead and ancillary services markets. Scenarios reproduce the uncertainty of the ancillary services market requests, and production of photovoltaic panels. A novel decomposition algorithm is proposed to tackle the challenging multistage stochastic program. The methodology is tested on three types of large power plants: a natural gas-fired combined cycle, a combined heat and power combined cycle with thermal storage, and a virtual power plant integrating a combined cycle with battery and photovoltaic fields. Compared to the typical deterministic unit commitment approach, the proposed stochastic optimization approach allows to increase the revenues of the conventional power plant up to 13.58% and, for the combined heat and power and virtual power plant case, it allows finding a feasible and efficient operational scheduling.

Profit-Oriented BESS Siting and Sizing in Deregulated Distribution Systems

Within the deregulation process of distribution systems, the distribution locational marginal price (DLMP) provides effective market signals for future unit investment. In that context, this paper proposes a two-stage stochastic bilevel programming (TS-SBP) model for investors to best allocate battery energy storage systems (BESSs). The first stage obtains the optimal siting and sizing of BESSs on a limited budget. The second stage, a bilevel BESS arbitrage model, maximizes the arbitrage revenue in the upper level and clears the distribution market in the lower level. Karush-Kuhn-Tucker (KKT) optimality conditions, strong duality theory, and the big-M method are utilized to transform the TS-SBP model into a tractable two-stage stochastic mixed-integer linear programming (TS-SMILP) model. A novel statistics-based scenario extraction algorithm is proposed to generate a series of typical operating scenarios. Then, scale reduction strategies for BESS candidate buses and inactive voltage constraints are proposed to reduce the scale of the TS-SMILP model. Finally, case studies on the IEEE 33-bus and 123-bus systems validate the effectiveness of the DLMP in incentivizing BESS planning and the efficiency of the two proposed scale reduction strategies.

Multi-objective two-stage stochastic optimization model for post-disaster waste management

Abstract Post-disaster waste management is one of the most crucial tasks in the recovery phase of the disaster cycle, and it was created to assist affected communities in returning to a stable state following a disaster. To develop an efficient post-disaster waste management strategy, this study presents a multi-objective two-stage stochastic mixed integer linear programming model for post-disaster waste management. The proposed mathematical model was developed based on a mixed strategy of on-site and off-site waste separation in the supply chain. This study aims to minimize not only the total cost and the environmental impact to provide waste flow decisions and choose collection and separation sites, recycling sites, landfill sites, and incineration sites throughout the supply chain under the uncertain situation. To solve a multi-objective problem, a normalized weighted sum method is used to find the solution. A numerical case based on realistic data is presented to validate and verify the proposed model. Based on the numerical example, the results demonstrated that the implementation of the mixed strategy for waste separation with the consideration of uncertain situations can reduce the total cost, balance the environmental impact, and determine the unexpected situation in the post-disaster waste supply chain efficiently.