Two-stage Stochastic Linear Programming Research Articles

Biorefinery superstructure optimization under carbon pricing policies using stochastic programming

The growing utilization of biomass feedstocks for climate change mitigation has led to increased research on biorefineries. Concurrently, environmental policies are evolving as crucial tools to address these global concerns. Cap-and-trade, carbon tax, and carbon cap policies have been established to reduce carbon dioxide emissions. In this work, a mathematical optimization model is developed to integrate biorefinery process design with carbon pricing policies and crediting mechanisms. A two-stage stochastic mixed integer linear programming model is proposed to account for emissions, feedstock supply, chemical demand, and pricing uncertainties. A bi-objective optimization framework is employed to consider economic and environmental metrics. The framework is a valuable tool for governments and businesses to determine pareto-optimal investment strategies under environmental policies. It is applicable for evaluating prospective chemical technologies with carbon pricing policies beyond biorefineries. The results indicate that carbon crediting mechanisms can minimize the financial penalty by up to 50% under a carbon tax policy. Implementing chemical demand constraints within a cap-and-trade policy reduces potential profits, especially when the carbon prices are high. The stochastic programming approach revealed that underestimating the expected carbon cap leads to lower expected profits. Despite the financial implications of these policies, profitable process designs are achievable.

Risk-aware microgrid operation and participation in the day-ahead electricity market

This work examines the daily bidding problem of a grid-connected microgrid with locally deployed resources for electricity generation, storage and its own electricity demand. Trading electricity in energy markets may offer economic incentives but exposes the microgrid to financial risk caused by market commitments. Hence, a multi-objective, two-stage stochastic mixed integer linear programming (MILP) model is formulated, extending prior work of a risk-neutral microgrid bidding approach. The multi-objective model minimises both expected total cost of day-ahead microgrid operations and financial risk from bidding measured by conditional value-at-risk (CVaR). Bidding curves derived as first stage decisions are always feasible under present market rules – including a limitation on the number of break points per submitted curve – while being near optimal for the microgrid’s day-ahead recourse schedule. The proposed optimisation model is embedded in a variant of the ɛ-constrained method to generate bidding curve candidates with different trade-offs between the two conflicting objectives. Moreover, scenario reduction is used to compromise accuracy of the uncertainty set for better computational performance. Particularly, the marginal relative probability distance between initial and reduced scenario set is suggested to make a decision on the extent of scenario reduction. The proposed solution procedure is tested in a computational study to demonstrate its applicability to generate optimal microgrid bidding curve candidates with different emphasis between total cost and CVaR in reasonable computational time.

Multi-resource allocation and care sequence assignment in patient management: a stochastic programming approach

To mitigate outpatient care delivery inefficiencies induced by resource shortages and demand heterogeneity, this paper focuses on the problem of allocating and sequencing multiple medical resources so that patients scheduled for clinical care can experience efficient and coordinated care with minimum total waiting time. We leverage highly granular location data on people and medical resources collected via Real-Time Location System technologies to identify dominant patient care pathways. A novel two-stage Stochastic Mixed Integer Linear Programming model is proposed to determine the optimal patient sequence based on the available resources according to the care pathways that minimize patients’ expected total waiting time. The model incorporates the uncertainty in care activity duration via sample average approximation.We employ a Monte Carlo Optimization procedure to determine the appropriate sample size to obtain solutions that provide a good trade-off between approximation accuracy and computational time. Compared to the conventional deterministic model, our proposed model would significantly reduce waiting time for patients in the clinic by 60%, on average, with acceptable computational resource requirements and time complexity. In summary, this paper proposes a computationally efficient formulation for the multi-resource allocation and care sequence assignment optimization problem under uncertainty. It uses continuous assignment decision variables without timestamp and position indices, enabling the data-driven solution of problems with real-time allocation adjustment in a dynamic outpatient environment with complex clinical coordination constraints.

Resilience-oriented planning for active distribution systems: A probabilistic approach considering regional weather profiles

Recent experiences with increasing climatic variability highlighted the importance of enhanced power distribution system resilience. Conventional methods of power distribution system planning usually focus on persistentinvestments associated with predicted faulty events, aiming to enhance the overall system reliability. Improving distribution system resilience in response to extreme weather events involves reduction of impact of the events having varying probability of occurrence. The probability of an extreme event varies significantly by location due to a combination of geographical, climatological, and meteorological factors. Thus, instead of requiring persistent investments, the solutions for resilience-oriented system upgrades should be guided by the potential effects and the probability of occurrence of extreme weather events in a specific area. To achieve this objective, the present paper proposes a two-stage stochastic mixed-integer linear programming (MILP) model based on regional weather profiles for the long-term resilience planning of distribution systems against extreme weather events. In the planning stage, investment decisions are made regarding overhead distribution line hardening and the distributed generation (DG) installation. During restoration process in the operational stage, dynamic microgrid formation is carried out for advanced distribution grid operations to minimize the cost corresponding to loss of load. The inclusion of regional weather profiles into planning objectives provide added flexibility to the network operators for analysing the trade-off between investment cost and load loss cost in resilient distribution system planning strategies.

Application of Stochastic Programming in Agricultural and Newsvendor Problems and It's Application in Real Life

The study of making the best decision under risk management in a variety of areas of our lives is known as Stochastic Programming. We will go through two-stage Stochastic Linear Programming approaches for a variety of real-world choice issues, as well as how to solve them. We will achieve this by constructing stochastic linear programming models based on real-world situations like the well-known Farmer's situation and News Vendors problems. The influence of pricing, Stochastic Integer Linear Programming problem, second stage Stochastic Integer Linear Programming problem, first stage Stochastic Binary Linear Programming problem, risk aversion problem, and continuous function for random variables based on two-stage SLP with the aid of Farmer's problem will all be examined. We will address the Newsvendor’s problem with Deterministic Equivalent Stochastic Linear Programming, an extension of Deterministic Stochastic Linear Programming for risk aversion with a high number of decision variables and restrictions, utilizing the two-stage Stochastic Linear Programming approach once more. Hand calculation is a challenging way to acquire the solution to the problems. As a result, we will use the programming language AMPL to design computer solutions for tackling both farmer and newsvendor difficulties. We will also utilize MATLAB to create graphs for the farmer's problem's continuous function. Dhaka Univ. J. Sci. 72(1): 30-45, 2024 (January)

Optimal design of aggregated energy systems with (N-1) reliability: MILP models and decomposition algorithms

This work investigates the design optimization of aggregated energy systems (multi-energy systems, microgrids, energy districts, etc.) with (N-1)-reliability requirements. The problem is formulated as a two-stage stochastic Mixed Integer Linear Program which optimizes design (first stage variables) and operation variables (second stage variables) simultaneously considering a set of typical and extreme days. The analysis proposes and compares different approaches to include the (N-1) reliability requirement in the optimization problem. Moreover, the paper proposes two effective decomposition algorithms to solve the large-scale Mixed Integer Linear Program suitable for design problems with and without (N-1) reliability requirements. Depending on the instance, such decomposition algorithms allow reducing the computational time by one or more orders of magnitude (from days to a few hours, in the worst cases tested in this work). The proposed methodology is tested to design the aggregated energy system for a real case study considering both a grid-connected and off-grid installation. Results indicate that the actual reliability of the design solutions depends by the profiles of energy demand and renewable production considered in the failure scenarios included in the design problem. Including N-1 reliability requirements causes an increase in the total annual cost in the range 15–20%, due to the increase in capital costs.

Emergency exit layout planning using optimization and agent-based simulation

Evacuation preparedness includes ensuring proper infrastructure, resources and planning for moving people from a dangerous area to safety. This is especially important and challenging during mass gatherings, such as large concerts. In this paper, we present the Emergency Exit Layout Problem (EELP) which is the problem of locating a given number of emergency exits and deciding their width such that the time it takes to evacuate the crowd from an arena is minimized. The EELP takes into account the geography of the arena and its surroundings, as well as the number of pedestrians in the crowd and the distribution of these within the arena. The EELP is formulated as a two-stage stochastic mixed integer linear program to handle the uncertainty related to the location of the possible incidents and the distribution of the pedestrians. Two cases are studied, a large concert planned at the Leangen trotting track in Trondheim and a smaller indoor arena. For each case, the EELP is solved for different scenarios, and the suggested layouts are evaluated using an agent-based simulation model. In particular, the potential of incorporating detailed assessment regarding the location and probability of specific incidents and the distribution of pedestrians are investigated. The computational study shows that making a more detailed risk assessment has little effect on the large concert, but a significant impact on the location of the emergency exits for the smaller indoor case. The results also indicate that it is more important to consider the location and probability of specific incidents rather than the pedestrian distribution.

Incorporating catastrophe insurance in power distribution systems investment and planning for resilience enhancement

Severe damage and high economic losses caused by frequent extreme disasters are becoming unbearable for distribution system operators (DSOs) under climate change. In order to cope with extreme disasters, this paper proposes an investment portfolio and planning framework of multi-resource for resilience enhancement. In the investment portfolio and planning framework, insurance is incorporated to mitigate high economic losses as a risk sharing and resilience enhancement supplementary strategy, and co-optimized with security investment from the perspective of DSOs. Firstly, the insurance investment is modelled based on Gordon-Loeb model according to insurance industry principles and formulated as constraints by piecewise linearization techniques. Secondly, the investment portfolio and planning framework proposed in this paper is modeled as a scenario-based two-stage stochastic mixed integer linear optimization program (MILP). In the first stage, DSO makes investment and planning decisions including energy storage (ESS) allocation, line hardening strategy and insurance investment. Both operation strategies of normal operation scenarios and extreme disaster scenarios under climate change are optimized in the second stage. Finally, progressive hedging (PH) algorithm is applied to solve the two-stage stochastic MILP model. The proposed model and algorithm are tested on the modified IEEE 33-bus test system. The results verify the validity of investment portfolio and planning framework, and the results show that portfolio investment considering insurance can realize resilience enhancement in both load shedding and economic losses.

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.

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.

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.

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).

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.

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.

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.

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.

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.

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.