Mixed-integer Optimization Model Research Articles

Mixed-integer quadratic optimization and iterative clustering techniques for semi-supervised support vector machines

AbstractAmong the most famous algorithms for solving classification problems are support vector machines (SVMs), which find a separating hyperplane for a set of labeled data points. In some applications, however, labels are only available for a subset of points. Furthermore, this subset can be non-representative, e.g., due to self-selection in a survey. Semi-supervised SVMs tackle the setting of labeled and unlabeled data and can often improve the reliability of the results. Moreover, additional information about the size of the classes can be available from undisclosed sources. We propose a mixed-integer quadratic optimization (MIQP) model that covers the setting of labeled and unlabeled data points as well as the overall number of points in each class. Since the MIQP’s solution time rapidly grows as the number of variables increases, we introduce an iterative clustering approach to reduce the model’s size. Moreover, we present an update rule for the required big-M values, prove the correctness of the iterative clustering method as well as derive tailored dimension-reduction and warm-starting techniques. Our numerical results show that our approach leads to a similar accuracy and precision than the MIQP formulation but at much lower computational cost. Thus, we can solve larger problems. With respect to the original SVM formulation, we observe that our approach has even better accuracy and precision for biased samples.

A machine learning approach to analysis of Canadian provincial power system decarbonization

Canada committed to reducing national greenhouse gas emissions by 40–45 % by 2030 and achieving net-zero by 2050, relying on low-carbon electrification across economic sectors. This study investigates the machine learning’s potential to analyze least-cost optimization of Canadian provincial power systems to 2050, to identify robust pathways for net-zero electricity system greenhouse gas emissions, and to provide policy insights respecting wind and solar generation, reducing emissions from natural gas generation, and expanding interprovincial transmission. Surrogate modeling is applied as sensitivities to generation technology capital costs, electricity demand, and carbon pricing to results from an existing capacity expansion mixed-integer linear optimization model. Under almost all future scenarios, six provinces develop new wind generation, with substantial deployment in Alberta (mean 27.3 GW, max. 62.7 GW), Ontario (mean 6.8 GW, max. 19.3 GW), Saskatchewan (mean 3.6 GW, max 8.9 GW) and Manitoba (mean 1.5 GW, max 3.5 GW). With the potential exception of BC, where higher-cost wind generation results in solar generation becoming the primary low-carbon electrical energy source (mean 25.2 GW), future solar generation across Canada is more limited (mean 37.1 GW) and less certain than wind generation in all other Canadian provinces. Existing natural gas generation is maintained in almost all scenarios in all provinces, providing support for maintaining these resources for reliability and reserve requirements. Very limited natural gas is developed under any scenarios, with the exception of British Columbia (mean 4.3 GW) where more than 80 % of scenarios include new natural gas capacity. Additional transmission transfer capacity is developed in most scenarios between BC-Alberta (∼60 % of scenarios), Saskatchewan-Manitoba (∼85 %), and Ontario-Quebec (∼75 %). Carbon prices, along with wind and solar capital costs, act as primary determinants of least-cost decarbonization pathways. These findings offer insights for Canadian policymakers into net-zero pathways for provincial electricity systems that remain robust despite variability in inputs.

Impact of tariff structures on energy community and grid operational parameters

This paper presents a mixed-integer linear programming optimization model of a renewable energy community comprised of members with local generators, battery energy storage systems, electric vehicles, and heat pumps and thermal energy storage, thus representing a local multi-energy system. The goal of the paper is to analyze the impact of different tariff structures for incentivizing energy sharing within energy communities on the distribution grid operational parameters. To achieve this, a comprehensive analysis of a hypothetical renewable energy community is conducted using the presented optimization model. Various tariff structures are evaluated, including flat and dynamic tariffs, reflecting the day-ahead market prices for energy and supply, as well as various network tariff designs, such as regulated charge reduction, time-of-use network tariffs and capacity payments. The results indicate that sharing energy within communities reduces energy losses in the grid, peak loads and reverse power flows and results in a smoother voltage profile. It is also found that tariffs that include dynamic electricity and supply costs can further increase these benefits, while including capacity payments as an incentive mechanism yields most favourable results for all analyzed criteria.

Managing secondary risks with optimal risk response strategy and risk-related resource scheduling

Secondary risks may arise as a result of implementing a risk response plan and may substantially impact the achievement of project objectives. Despite their importance, there is a scarcity of research on scheduling resources to effectively manage these risks. This study presents a pioneering mixed-integer optimization model and a tailored meta-heuristic solution algorithm, specifically designed to determine optimal primary and secondary risk response strategies. To validate our model, we introduce a synthetic data generation framework capable of generating appropriate scenarios based on specific metrics. We apply the model and algorithm to a full factorial numerical experiment constructed using the synthetic data generation algorithm. This allows us to derive managerial insights on secondary risks and the impact that secondary risk responses may have on project objectives. The outcomes of this study provide valuable insights for project managers to better manage both primary and secondary risks.

Modeling insights from the Austrian national gas grid under declining natural gas demand and increasing domestic renewable gas generation by 2040

This paper studies the development of the Austrian gas grid by 2040 under different decarbonization scenarios, ranging from electrifying most energy services to importing large amounts of renewable methane. The Austrian gas grid serves as a case study representative for those gas grids confronted with a drop in natural gas demand and an increase in domestic renewable gas generation in the future. A mixed-integer optimization model with a focus on the detailed representation of the existing grid (gas grid levels, pipeline capacity, route, and age) is used to determine the cost-optimal trade-off decision between expected low-utilized gas pipelines and an off-grid supply alternative (e.g., trucking and on-site gas storage) in a decarbonized Austrian and European energy system. Optimality determines whether to operate, decommission, or make replacement investments in the grid’s pipelines. Based on the required gas grid 2040 and its annual grid costs, estimates for tariffs of end customers are given. The paper’s main conclusion is that the domestic distributed renewable gas generation uptake will be associated with the need for an area-wide gas grid, but one that will be significantly less utilized. Future work could address, among other things, the exploration of the spatial interplay of local production and demand by forming regional clusters of renewable gas.

Fertilizer Logistics in Brazil: Application of a Mixed-Integer Programming Mathematical Model for Optimal Mixer Locations

Background: Brazil is one of the largest consumers of fertilizers and is highly dependent on the international market to meet its demand for agricultural production inputs. The complexity of the fertilizer supply chain motivated us to carry out this study on redesigning the fertilizer logistics chain and evaluate strategies for reducing logistics costs by redesigning the fertilizer mixing network in Brazil, a country that is heavily dependent on imported fertilizers for agriculture. Methods: We introduce a multi-product mixed-integer linear programming optimization model encompassing the logistics network, from import ports to mixing factories and agricultural fertilizer supply centers. This model includes logistics infrastructure and taxes, accounting for greenhouse gas emissions (specifically carbon dioxide) in fertilizer logistics. Results: The results indicate that expanding the port capacity for fertilizer importation can significantly reduce logistics costs and greenhouse gas emissions by up to 22.5%, decreasing by 23.9% compared to the baseline. We also observed that removing taxes on fertilizer importation can reduce logistics costs by approximately 11%, but it increases greenhouse gas emissions by 2.25% due to increased reliance on road transport. We identified 15 highly resilient regions for establishing mixing factories, evaluated various scenarios and determined the importance of these locations in optimizing the fertilizer supply network in the country. Moreover, the results suggest a significant potential to enhance the role of Brazil’s Northern Arc region in fertilizer import flows. Conclusions: Public policies and private initiatives could be directed toward encouraging the establishment of mixing factories in the identified regions and increasing transport capacity in the Northern Arc region. Improving the logistical conditions of the fertilizer network would contribute to food security by reducing the costs of essential inputs in food production and promoting sustainability by reducing greenhouse gas emissions.

Optimizing islanded green ammonia and hydrogen production and export from Saudi Arabia

Hydrogen and its derivatives hold significant promise for decarbonizing power and transport. This study proposes a mixed-integer linear optimization model for the flexible production and transportation of islanded green hydrogen and ammonia. Wind and solar databases of three coastal locations across the Kingdom of Saudi Arabia were utilized to provide a holistic geographic perspective. The results show that plant optimization can lead to production levelized costs of ammonia of 383 USD/tonne NH3, and production levelized costs of hydrogen down to 1.57 USD/kg H2 by 2030. The single largest cost driver is the energy cost, representing 60–70 % of the total project cost. In low-wind resource areas, battery storage technology plays a critical role in supplying consistent energy for continuous ammonia synthesis. Transportation to final demand centers in Europe or Asia can lead to additional costs of less than 40 USD/tonne NH3 and less than 0.8 USD/kg H2 via liquid hydrogen route.

Exploring optimal pathways of the high-CO2 content natural gas source to chemicals and fuels using superstructure multi-objective optimization

This study aims to develop a mathematical formulation of superstructure multi-objective optimization to find optimal pathways of high-CO2 natural gas and PV-hydrogen to chemicals and fuels integrated with CO2 sequestration. A mixed-integer nonlinear programming optimization model is used to compete along 56 pathways in the superstructure optimization network. The model was developed with the general algebraic modeling system and computed with the standard branch and bound (SBB) solver to maximize annual net profit and minimize annual net CO2 emissions. The result shows the selected products in 2020 are liquefied natural gas, methanol, dimethyl ether, formic acid, and acetic acid, with annual net profits and net CO2 emissions of 38.17 million $/year and −8.47 million tons of CO2/year, respectively. In the projection analysis, the annual net profit growth results at a compounding annual growth rate of 7.85% per year between 2020 and 2060, while annual net CO2 emissions decrease until −16.37 million tons of CO2/year in 2060. In 2030, blue acetic acid will be selected via all separation processes. For 2050, the synfuel via partial membrane separation is selected. The other products, formic acid, methanol, and acetic acid via membrane separation (full and partial), will always be selected between 2020 and 2060. Therefore, this could be a future strategy for monetizing high–CO2–content natural gas sources with low-carbon process development.

A linear reduced-order model for the activated sludge process for the integration into a mixed-integer linear energy system optimisation model

Conventional wastewater treatment plants consume significant amounts of electricity. The constant aeration of the wastewater in order to foster the growth of microorganisms or the pumping of wastewater are two examples for energy-intensive processes within a plant. Case studies have shown that switching off blowers and inlet pumps for a certain period of time is possible without a loss in water quality. This yields a potential for wastewater treatment plants to provide demand response (DR) to the power system and thereby increase overall system flexibility. So far, the DR potential has only been quantified for individual plants, while the effects of large-scale DR provision by the wastewater treatment sector for the power system have not yet been studied. One reason for this is the lack of optimisation models which include both the wastewater treatment process and the power system operation in sufficient detail. Our model tackles this gap in the literature by providing a reduced-order linear biochemical model for the activated sludge process within a WWTP that can be incorporated into an operational energy system model. The results show that the effluent concentrations are predicted well by the linear reduced-order model in comparison to the results of the Standard Activated-Sludge model No. 1 (ASM1). Potential model applications are the variation of the airflow rate within a certain range and the variation of liquid influent flow rate to the system, which is a result of electricity load shedding of the inlet pumps and the blowers connected to the activated sludge tank.

Electric vehicle-based express service network design with recharging management: A branch-and-price approach

With soaring fuel prices and the imperative to reduce global carbon emissions, electric vehicles (EVs) possess unparalleled advantages in terms of sustainability. Consequently, numerous express service providers have commenced the implementation of EV-based express service networks. However, designing an effective service network for EV fleets necessitates considering recharging operations during the route planning process. In this study, we explore a problem involving the design of an EV-based express service network under the constraint of limited recharging resources, while taking into account power management for each EV throughout the entire operational cycle. To address this challenge, we present a mixed-integer optimization model incorporating a Dantzig–Wolfe reformulation. Additionally, we develop resource extension functions that enable an efficient dynamic labeling algorithm, incorporating dominance rules, to solve complex subproblems. We propose a branch-and-price framework that integrates an aggregated branching strategy and a heuristic upper-bound technique, enabling the attainment of exact solutions for instances involving large fleet sizes. Computational experiments conducted on realistic instances, involving up to 20 terminals and over 2000 EVs, substantiate the efficiency of our approach.

Valuing the social cost of carbon: Do economists really care about climate change?

The aim of the article is to analyse the scientific literature of the last decade on the social cost of carbon to identify and discuss the best values representing the external cost of CO2 emis-sions calculated with a damage valuation approach, to use in cost-benefit analysis and in other external costs applications, such as Life Cycle Assessment. The social cost of carbon repre-sents the present value of the global damages of climate change attributable to the emission of an additional ton of carbon dioxide into the atmosphere. The measurement of the social cost of carbon is fundamental in the transition to a net zero emissions economy because it allows bal-ancing the costs of investments needed for decarbonization with their benefits in terms of avoided climate damages. The paper develops and applies a method for identifying the best social cost of carbon estimates, that starts from study reviews conducted within institutional processes (IPCC, OECD, US Government bodies), to then analyse the study reviews made in academic or research contexts and only in the end it delves into selected academic studies that provide original estimates of the social cost of carbon. The method allowed to identify and dis-cuss a robust even if conservative estimate of the social cost of carbon for emissions in the period 2020-2080, recently elaborated by the US Environmental Protection Agency, that can be used worldwide, for example to update the reference external cost values for CO2 emis-sions recommended by the European Commission Handbook on external cost of transport.

Bidding strategy of storage hydropower plants in reserve markets

The increasing share of intermittent sources of energy will increase the need for frequency-control reserves. However, the supply from gas and coal-fired power plants might decrease in the following years. Being the procurement of reserves mostly market-based in Europe, the market design should send price signals to encourage participation in these markets. This pa-per analyses the incentives provided by the French market design for seasonal storage and pumped storage hydropower plants to participate in reserve markets. To that end, a determinis-tic mixed-integer linear optimization model is presented. The objective is to maximize profits in the energy and reserve markets according to 2019 market prices. By optimising the trade-offs between the day-ahead and the reserve markets, the storage hydropower plant increase its profits. The pumped storage hydropower plant sometimes chooses the Frequency Contain-ment Reserve market or the day-ahead market only. The apparition of some hours of FCR par-ticipation with the pumped storage plant is explained by its higher number of generating hours and by the higher volatility of reserve energy prices. These two factors also explain the greater response of the pumped storage plant to the incentive measures on the FCR market.

A decentralized production–distribution scheduling problem: Solution and analysis

In modern production–distribution supply chains, decentralization has increased significantly, due to increasing production network efficiency. This study investigates a production scheduling and vehicle routing problem in a make-to-order context under a decentralized decision-making structure. Specifically, two different decision makers hierarchically decide the production and distribution schedules to minimize their incurred costs and we formulate the problem as a bi-level mixed-integer optimization model as a static Stackelberg game between manufacturer and distributor. At the upper level, the manufacturer decides its best scheduling under a flexible job-shop manufacturing system, and at the lower level, the distributor decides its distribution scheduling (routing) which influences the upper-level decisions. The model derives the best production–distribution scheduling scheme, with the objective of minimizing the cost of the manufacturer (leader) at the lowest possible cost for the distributor (follower). As the lower level represents a mixed-integer programming problem, it is challenging to solve the resulting bi-level model. Therefore, we extend an efficient decomposition algorithm based on Duplication Method and Column Generation. Finally, to discuss the decentralization value, the results of the presented bi-level model are compared with those of the centralized approach.

Optimization model for low-carbon supply chain considering multi-level backup strategy under hybrid uncertainty

When constructing a supply chain, the profit, environmental impact, and hybrid uncertainties the supply chain faces should be considered. This research investigates the problem of low-carbon supply chain design under hybrid uncertainty, where demand is regarded as a stochastic variable, supply and transportation disruptions are regarded as uncertain events, and the coefficient of emission reduction capacity of suppliers is regarded as an uncertain variable. Based on the probability theory and uncertainty theory, a mixed-integer optimization model is constructed to handle the disruption risk by utilizing a multi-level backup strategy and to reduce carbon emissions by investing in suppliers. This model guarantees that manufacturers’ demand is satisfied to a given confidence level, manufacturers prefer to construct a supply chain within the acceptable supply chain risk, the emission reduction investment scheme and the supply decision to manufacturers are determined to maximize the profit of the supply chain. To facilitate the solution, we perform deterministic equivalent transformation of stochastic and uncertainty constraints, linearize the nonlinear constraints, and analyze the mathematical properties of the model. Finally, the validity of the proposed model is verified by case studies. The results show that although the larger the supply levels, that is, the more priority levels of suppliers, the more beneficial it is to improve the reliability, too large supply levels will reduce profits. The reasonable setting of the supply levels can optimize the emission reduction investment scheme. In addition, the confidence level of carbon emission should be set within a certain range to avoid the disparity between profit growth and emission reduction. Finally, the greater the belief degree of disruption or the lower the emission reduction capacity of suppliers, the more significant the effect of a multi-level backup strategy.

Innovative approaches to addressing the tradeoff between interpretability and accuracy in ship fuel consumption prediction

Ship fuel consumption is a major component of maritime transport costs and most of its emissions are harmful to the environment. Hence, it is essential to build an accurate ship fuel consumption prediction model, thereby providing reference to the navigation operations. However, maritime industry experts are wary of advanced black-box models since they cannot interpret the outcomes of these models. The application of advanced black-box models in the shipping industry remains limited and it is necessary to develop both accurate and interpretable ship fuel consumption prediction models. This study uses domain knowledge to develop two innovative methods for predicting ship fuel consumption—the first is a physics-informed neural network (PI-NN) model that improves the interpretability of the black-box model while maintaining accuracy and the second is a mixed-integer quadratic optimization (MIO) model that considers more forms of feature variable expressions in an additive white-box model. The proposed approaches address the tradeoff between model interpretability and model accuracy in ship fuel consumption prediction. The experiment results demonstrate that the PI-NN model improves the interpretability of the black-box model while preserving accuracy. The MIO model considers alternative variable expressions, leading to the flexibility of the white-box model. Finally, SHapley Additive exPlanations (SHAP) is used to explain how each feature value contributes to the predictions of the black-box model, thereby providing insights into how each value of feature variables affects fuel consumption. This study provides a solution to the tradeoff between model interpretability and model accuracy and can promote the application of data-driven models in ship fuel consumption prediction. Moreover, this study gives implications for the application of explainable machine learning models in practice.

Hybrid optimization and discrete-event simulation model to reduce waiting times in a primary health center

In the early stages of health attention, such as in the examination rooms, patients are generally treated using a First In First Treat (FIFT) logic. This treatment style is inefficient from an operational view as it leads to long waiting times for critical patients. Based on optimization and discrete-event simulation, this paper introduces a hybrid model to assign patients priorities and exam rooms. Patients arrive to the system following a probability function, and for every discrete interval, a mixed-integer linear optimization model is used to allocate patients, considering health priority, preparation times, and availability of time and resources. Instances of the problem are defined using information from a health system in Mexico. Additional to the linear model, a heuristic algorithm is proposed, composed of two phases and aims to provide good, feasible, and fast solutions without ensuring optimality. Computational experiments showed that the linear model is more appropriate for small instances since the heuristic is faster and the relative deviations with respect to the linear model are below five percent. Finally, the discrete-event simulation is run for two cases: (1) with a FIFT logic and (2) using the optimization model. The hybrid model reduces the global waiting times up to 90% and the number of high health priority patients who are not treated up to 100%.

Competitive integrated airline schedule design and fleet assignment

Airline profits are significantly influenced by competitive timetables that match passenger demand to fleet resources. In this research, we develop an integrated, mixed-integer optimization model to derive a comprehensive flight schedule, fleet assignment, and average airfares jointly. Passenger choice behavior is incorporated through a prospect theory-adjusted, nested, multinomial logit model which estimates market share. The competitive fleet assignment and schedule design (CFSD) formulation is embedded in a differentiated Bertrand game, such that each transport operator optimizes their best response function connected through the market share model. To solve the integrated optimization problem efficiently, a hybrid algorithm is developed that combines stabilized column generation with a large neighborhood search algorithm.The game-theoretic framework is applied to a case study in China involving legacy airlines, a low-cost carrier, and a high-speed rail (HSR) operator. The equilibrium outcomes suggest that the low-cost carrier is likely to stimulate demand and improve the level of service through secondary airports. The further development of high-speed rail service is expected to intensify competition among airlines and reduce the overall fare level. The low-cost carriers and HSR are able to significantly increase overall consumer surplus, providing insights into the potential for low-cost service in the Chinese aviation markets and the entrance of the public service obligation program at minimum cost.

Decarbonizing integrated chlor-alkali and vinyl chloride monomer production: Reducing the cost with industrial flexibility

Industrial demand response will become increasingly important in power grids with high shares of variable renewables, yet the existing knowledge on how the industrial electricity demand and flexibility will change with the decarbonization of chemical processes is limited. Here we develop a mixed-integer linear optimization model, which we use to compare the cost and flexibility of the most relevant decarbonization options for the combined chlor-alkali electrolysis (CAE) and vinyl chloride monomer (VCM) production process. We combine product and energy storage to enable the full flexibility potential of the decarbonized process. Our results show that flexible operation of the CAE process is deemed technically possible but limited by internal process dependencies due to decarbonization of the VCM production. Combining energy and product storage for demand response enables up to 4% operational cost reduction by shifting loads during peak price hours. High overcapacity of PEM electrolyzers is required to release the full flexibility potential in the hydrogen based decarbonization option, while the less flexible direct electrification option shows a potential for OPEX reduction. Full decarbonization of the combined CAE and VCM process without increasing operational cost significantly appears difficult. Our study emphasizes demand response through product and energy storages as a viable pathway for minimizing the added cost, and also enables a significant reduction of electric demand in high-price hours.

Learning and forgetting interactions within a collaborative human-centric manufacturing network

Learning and forgetting (LaF) phenomena are characteristic of labor-intensive production and service industries. To mitigate the effects of LaF in a human-centric manufacturing system integrated with outsourcing, managers need to coordinate their decisions with partners for assigning operations and scheduling processes following a hierarchy. A model that addresses this should consider the expected latency of various tasks across assignments and production sequences and similarities among jobs as that affects learning. This paper develops a novel bi-level LaF model to help determine the leader-follower decisions in a decentralized network. It models the learning concept as a factor of task execution order and task variety. The mixed-integer non-linear optimization model determines the best order coordination and scheduling scheme by minimizing the processing, operating, and holding costs and penalties for missing deadlines. This study also develops an efficient column-and-constraint generation algorithm based on the duplication method, which enables solving bi-level models in which the lower-level model includes integer variables. This study also provides an illustrative real-sized example to validate the model and prove the efficiency of our resolution method. The results indicate that adopting compromise solutions enables preoccupied workers to be released earlier than expected, reducing the costs associated with learning and forgetting (due to latency). Despite the effects of LaF and the decentralized structure of the supply chain, which includes rising network costs, the schedules become more precise, and the cost balance among actors effectively increases.

Optimization of electricity consumption in an industrial zone with a battery system: a case study of Ikitelli industrial zone in Turkey

Industrial power use is considerably high in Turkey. In the case of industrial zones, the three-time tariff is more economical for the owners. However, in industrial zones with considerably high energy consumption during peak hours, an alternative way is needed to reduce electricity costs. A battery-powered model was developed earlier in order to take advantage of the three-time tariff and to reduce the electric charge during peak hours. In this study, the aim was to reduce the power utility costs of industrial consumption and to mitigate CO2 emissions for businesses with high electricity costs by using solar energy. A mixed-integer optimization model was constructed as a combination of solar panels and a battery system to minimize the peak-hour grid-connected energy consumption in industrial zones. Using the consumption data from 2018, a model was implemented for a case study in İkitelli Industrial Zone in terms of 24-h scheduling. Three different alternative energy combination scenarios were studied: a monofacial PV (photovoltaic) panel system with batteries, a bifacial PV panel system with batteries, and electricity received from the grid only at the night tariff. This study found that a combined method could minimize the grid usage during peak hours.