Mixed Integer Linear Programming Approach Research Articles

Reallocation of chemotherapy appointments in a large health system using a mixed integer linear programming approach

ABSTRACT Outpatient chemotherapy scheduling has significant implications for both patients and health systems. Consideration of treatment location preference is important for patient satisfaction and outcomes, and it is a complex decision impacted by travel distance. In health systems with one treatment site that stands out from the rest as a destination medical center (the primary site), there are financial and resource utilization incentives to free up as much space as possible for appointments at that site. In this study, we demonstrate that leveraging the underutilized health system sites allows decompression of appointment volume at the primary site, and it takes full advantage of valuable resources such as oncology nurses and chair availability. A Mixed Integer Linear Programming approach was used to develop a model under four scenarios which reallocates appointments from the primary site to other health system sites based on patient travel distance to the sites. This approach was applied to data from the Mayo Clinic Health System Minnesota region, which demonstrated that the health system has the potential to move approximately 50% of eligible appointments out of the primary site, resulting in an overall volume change of approximately 30%. Implications for scheduling policies and infrastructure are discussed.

Optimization of the electric vehicle charging strategy problem for sustainable intercity travels with multiple refueling stops

Electric vehicle (EV) drivers considering long-distance trips still face range anxiety due to the limited range of EVs and the scarcity of charging stations. Thus, it becomes important to ensure the feasibility of the selected route and determine an optimal charging strategy. As a crucial aspect of decision support for EV drivers, this study proposes a mixed integer linear programming (MILP) approach for the EV charging strategy problem (EVCSP), incorporating a piecewise linear approximation technique to address the challenges posed by nonlinear charging times. The proposed optimization model, namely CSPM determines where, when, and how much to charge an EV for a specified route to minimize travel time and cost. The solution time of large-scale test problems and a case study on Türkiye reveal the robustness and reliability of the CSPM. Furthermore, two multi-objective optimization methods (the weighted sum and the lexicographic method) are applied to the case study, and the results are analyzed. The results indicate that the travel cost is more sensitive to the selected charging strategy, with a range of 46.09 % across the applied charging strategies, whereas travel time remains more resilient, with a maximum fluctuation of 19.77 %. A comparative analysis with a full charging strategy reveals that the CSPM reduces the travel time by 60.1 % and improves the cost efficiency by 105.72 %.

Optimisation of Ammonia Production and Supply Chain from Sugarcane Ethanol and Biomethane: A Robust Mixed-Integer Linear Programming Approach

Optimisation of Ammonia Production and Supply Chain from Sugarcane Ethanol and Biomethane: A Robust Mixed-Integer Linear Programming Approach

Applying the Simulated Annealing Algorithm to the Set Orienteering Problem with Mandatory Visits

This study addresses the set orienteering problem with mandatory visits (SOPMV), a variant of the team orienteering problem (SOP). In SOPMV, certain critical sets must be visited. The study began by formulating the mathematical model for SOPMV. To tackle the challenge of obtaining a feasible route within time constraints using the original MILP approach, a two-stage mixed-integer linear programming (MILP) model is proposed. Subsequently, a simulated annealing (SA) algorithm and a dynamic programming method were employed to identify the optimal route. The proposed SA algorithm was used to solve the SOP and was compared to other algorithms, demonstrating its effectiveness. The SA was then applied to solve the SOPMV problem. The results indicate that the solutions obtained using SA are superior and more efficient compared to those derived from the original MILP and the two-stage MILP. Additionally, the results reveal that the solution quality deteriorates as the ratio of the set of mandatory visits increases or the maximum allowable travel time decreases. This study represents the first attempt to integrate mandatory visits into SOP, thereby establishing a new research direction in this area. The potential impact of this research is significant, as it introduces new possibilities for addressing complex combinatorial optimization problems.

Optimal design of concentrated solar power-based hydrogen refueling station: Mixed integer linear programming approach

This paper developed a mixed integer linear programming model to optimally design hydrogen refueling station coupled with an on-grid concentrated solar power. The model aims to minimize the total life cycle cost of the integrated system addressing various constraints. The utility of the proposed model is validated using hydrogen refueling station under the weather conditions of Dhahran city, Saudi Arabia. The station fulfills the daily hydrogen requirements of taxi-fleet in Dhahran city, estimating a daily demand of 4202 kg of hydrogen. The proposed hydrogen refueling station requires a solar field of 71,721 square meters to generate 50,233 MWh of energy annually. The on-grid concentrated solar power system is determined to produce an excess energy of 10,515 MWh which is exported to the grid, highlighting positive economic impact. The levelized cost of hydrogen of the proposed system is 7.17 $/kg. The achieved results validated the feasibility and efficiency of the proposed system, positioning it as a sustainable solution for meeting the rising demand for clean and green transportation in the region.

Optimization of production, distribution, natural resources and capacity planning throughout process sector worldwide supply chains with various goals

This study delves into the intricate dynamics of green transformation within global mining and other process industries, focusing on production, distribution, and capacity planning under the framework of the Asian Mineral Vision. By addressing sustainability, resource efficiency, and cost reduction, the research highlights the multifaceted challenges faced by industry participants in achieving green objectives. In addition to financial efficiency, the study emphasizes the importance of customer service quality and responsiveness. To tackle these challenges, we propose a novel multiobjective mixed-integer linear programming (MILP) approach. The model incorporates key goals such as minimizing total cost, total flow time, and total missed sales, reflecting the interconnected nature of financial efficiency, operational agility, and customer satisfaction. To further enhance the model's flexibility, we integrate discrete strategies for plant capacity expansion, recognizing the crucial role of capacity management in responding to evolving demand dynamics. The multiobjective optimization problem is addressed using the lexicographic minimax technique and the ε-constraint method. A comprehensive numerical example illustrates the practical relevance and effectiveness of our proposed model and solution methods, providing valuable insights into improving the resilience and performance of green supply chain networks in the process industry.

Real-time optimization for integrated R2 charging and refueling stations in a multi-carrier energy system considering hydrogen chain management

This study presents a real-time optimization-based strategy for sustainable energy management using an energy hub that integrates combined heat and power (CHP) units, heat pumps, and fuel-cell technology to meet community electricity, heating, and cooling demands. It incorporates renewable-based R2 stations that leverage photovoltaic and wind power, integrated with hydrogen production and storage, to supply stationary energy consumers and the transportation sector. The system provides hydrogen for fuel cell electric vehicles (FCEVs) and electricity for electric vehicles (EVs). Integrating the R2 station with the primary energy hub optimizes the use of locally generated power, reducing dependence on external sources. Hydrogen fuel is produced on-site and off-site, focusing on solving the capacitated routing problem for efficient delivery. The system's architecture is modeled using a mixed-integer linear programming (MILP) approach. According to the results, no energy is drawn from the grid to meet the end-user's electricity demand.

Closed-loop supply chain network of end-of-life wind turbines mathematical model proposal considering environmental, employment, and cost reduction aspects

The worldwide climate issue makes wind energy an attractive renewable energy source. Wind energy reduces carbon emissions, energy prices, creates jobs, improves power access, and ensures energy security. The mass of employed materials in wind turbines indicates a lot of potential raw materials when the whole mass is evaluated. Additionally, a new wind turbine costs $2–$4 million. Remanufacturing or using recycled raw materials from end-of-life wind turbines is practical due to the decreased cost of new wind turbines. To ensure sustainability, CO2 emissions and social advantages like employment must be considered. Given the typical life expectancy of wind turbines at 20–25 years, this issue must be addressed globally in the mid-term. The objective of the conducted study is to provide a multi-objective mathematical framework that encompasses the environmental, social, and economic aspects of wind turbines at the end of their operational lifespan. Hence, this study presents a methodical framework for achieving sustainability in the wind energy sector through the implementation of a closed-loop supply chain. Furthermore, a lexicographic solution approach is offered for each objective of the mathematical model, with the aim of providing decision makers with a comprehensive overview of solutions. The methodology employed in this work involves the utilization of a mixed-integer linear programming (MILP) approach to address the closed-loop supply chain for end-of-life wind turbines. Additionally, a lexicographic solution approach is provided to effectively solve the problem at hand. The novelty of this article lies in its examination of the closed loop supply chain network for end-of-life wind turbines, encompassing the social, environmental, and economic dimensions. This study represents the first of its kind to explore these aspects comprehensively. The proposed mathematical model has three objectives which cover the remanufacturing and recycling and aims to minimize CO2 emissions, unemployment, and transportation cost. In order to evaluate the proposed model, Türkiye is utilized as case with real-life data and Gurobi 9.1.2 is utilized. The locations of wind turbines could not be collected individually, and Euclidian distance was used instead of road distance due to a lack of data. These are the main limitations of the proposed study. The study revealed that the TEC and TCE lexicographic approach has the best recycling and remanufacturing cost-to-manufacturing cost ratio, 52.54 %. The least desired result is 74.50 % from CTE.

Integral techno-economic design & operational optimization for district heating networks with a Mixed Integer Linear Programming strategy

The Netherlands, with over 90% of homes heated by natural gas is currently in the early phases of the heat transition to find alternative solutions towards 2050. According to ambitions of the Dutch government up to 50% of future implemented heating systems in the built environment will use District Heat Networks. The investment-heavy nature for District Heating System (DHS) makes it challenging to establish viable business-cases as supply-side parties require security of demand. However, resident participation is lacking as there is no integral estimate for the cost of heat over the lifetime in the early planning phase. This paper proposes a network integral techno-economic optimization with minimal a-priori assumptions. An integral network cost optimization enables to achieve a considerably more reliable cost of heat in the early planning phase. Both the investments and operational strategy are optimized with a Mixed Integer Linear Programming approach that captures the physics as well as the financial choices. Linearization of the physics are chosen to have a conservative impact on the costs estimates. The end scenario network is designed where the placement and size of sources, storage components and pipes are optimized together with the operational strategy, e.g. thermal allocation time-series, for all assets. The workflow was applied to a greenfield network for the Dutch municipality of Rijswijk. It was shown that a 18% reduction in Total Cost of Ownership in the primary grid could be achieved by introducing decentralized sources, decentralized storages and seasonal storage.

Reliability enhancement through optimal placement of photovoltaic power plant and battery energy storage in distribution system

Integration of distributed generation (DG) units could reduce the duration of power outage for a certain number of consumers in distribution networks after a network failure. Prerequisites are that i) the DG unit has the technical characteristics that enable its islanded operation, the degree of automation of the distribution network allows it and ii) it is in accordance with the current technical regulations of the country. The extent to which integration of distributed sources can improve reliability depends on the DG size, type, and the point of common coupling. Besides, an energy storage system could be installed along with DG unit, so that energy supply availability during fault periods can be secured. This paper proposes an algorithm based on mixed-integer linear programming (MILP) approach for optimal placement of photovoltaic power plant (PVPP) and battery energy storage system (BESS). The optimal location is determined so that expected non supplied energy is minimized. The uncertainties in the estimation of production of PVPP, load and BESS state of charge (SoC) were taken into account by Monte Carlo simulations (MCS).

A multi-objective mixed integer linear programming approach for simultaneous optimization of cost and resilience of power distribution networks

In recent years, customers have experienced a significant increase in weather-related power outages. The power distribution network (PDN), a subset of the power system, in particular is more susceptible to extreme events. Therefore, ensuring the resilient and cost-effective operation of PDNs following extreme weather conditions poses a significant challenge for distribution network operators. This paper presents an approach for simultaneously optimizing the cost and load restoration for resilience enhancement of power distribution networks while determining the optimal positioning and generation levels of mobile emergency generators. The proposed method, in addition, employs a distribution network reconfiguration to improve the load restoration process, by optimally determining the status of switches. The multi-objective formulation involves the minimization of load shedding to increase the resilience of the system, while the other objective is formulated to minimize the cost of load restoration. A weighted sum method is employed to address the multi-objective mixed-integer linear programming (MILP) model. A set of non-dominated solutions determined using the proposed formulation provides opportunities to the distribution system operator in choosing a resilience improvement strategy according to the availability of the operational budget. The proposed model is implemented on 33-bus distribution system to validate the efficacy of the proposed model.

Multi-objective optimization of a hydrogen supply chain network: Wind and solid biomass as primary energy sources for the public transport in Sicily

Europe’s heavy reliance on diesel power for nearly half of its railway lines, for both goods and passengers, has significant implications for carbon emissions. To address this challenge, the European Union advocates for a shift towards hydrogen-based mobility, necessitating the development of robust and cost-effective hydrogen supply chains at regional and national levels. Leveraging renewable energy sources such as wind farms and solid biomass could foster the transition to sustainable hydrogen-based transportation. In this study, a mixed-integer linear programming approach integrated with an external heavy-duty refueling station analysis model is employed to address the optimal design of a new hydrogen supply chain. Through multi-objective optimization, this study aimed to minimize the overall daily costs and emissions of the supply chain. By applying the model to a case study in Sicily, different scenarios with varying supply chain configurations and wind curtailment factors were explored. The findings revealed that increasing the wind curtailment factor from 1% to 2% led to reductions of 12% and 15% in the total daily emission costs and network costs, respectively. Additionally, centralized biomass-based plants dominated hydrogen production, accounting for 96% and 94% of the total production under 1% and 2% wind curtailment factors, respectively. Furthermore, transporting gaseous hydrogen via tube trailers proved more cost effective than using tanker trucks for liquid hydrogen when compressed gaseous hydrogen is required at the dispenser of forecourt refueling stations. Finally, the breakdown of the levelized cost for the hydrogen refuelling station strongly depends on the form of hydrogen received at the gate, namely, liquid or gaseous. Specifically, for the former, the dispenser accounts for 60% of the total cost, whereas for the latter, the compressor is responsible for 58% of the total cost. This study highlights the importance of preliminary and quantitative analyses of new hydrogen supply chains through model-based optimization.

Mixed-integer linear programming for computing optimal experimental designs

The problem of computing an exact experimental design that is optimal for the least-squares estimation of the parameters of a regression model is considered. We show that this problem can be solved via mixed-integer linear programming (MILP) for a wide class of optimality criteria, including the criteria of A-, I-, G- and MV-optimality. This approach improves upon the current state-of-the-art mathematical programming formulation, which uses mixed-integer second-order cone programming. The key idea underlying the MILP formulation is McCormick relaxation, which critically depends on finite interval bounds for the elements of the covariance matrix of the least-squares estimator corresponding to an optimal exact design. We provide both analytic and algorithmic methods for constructing these bounds. We also demonstrate the unique advantages of the MILP approach, such as the possibility of incorporating multiple design constraints into the optimization problem, including constraints on the variances and covariances of the least-squares estimator.

Inter-array cabling optimization in offshore wind power plants, a new reliability indicator for radial grids

The inter-array grid relates to a significant share of the investments into an offshore wind power plant (OWPP). Optimizing the cable connections regarding costs and reliability is a mathematically complex task due to the high variety of possible wind and component (wind turbine or cable) failure scenarios. This paper presents a novel mixed integer linear programming approach to support investment decisions into OWPPs by trading off cabling purchase and installation costs with power capacity risk (PCR), which is defined as a length-weighed cumulative power flow summation that reflects the consequences of cable failures. Then, quasi-random Monte Carlo simulations assess the optimized collection grids (CGs) to quantify their levelized cost of energy (LCOE). To construct relevant case studies, this work investigates the real OWPPs Ormonde, Horns Rev 1, Thanet, and London Array, which contain 30, 80, 100, and 175 wind turbines. The results reveal Pearson correlation coefficients around 0.99 between the proposed PCR and the expected energy not supplied. Furthermore, this paper’s findings indicate that minimum-cost CGs do not necessarily present the lowest LCOE.

Optimally integrated waste heat recovery through combined emerging thermal technologies: Modelling, optimization and assessment for onboard multi-energy systems

This paper investigates waste heat (WH) valorisation on maritime vessels from a systemic perspective. It aims to demonstrate how a set of innovative waste heat recovery (WHR) technologies can synergistically work together to valorise waste heat in multiple ways. It proposes and develops a dedicated techno-economic analysis framework and model, surpassing previous literature that focused on individual technologies or specific combinations. Employing a Mixed Integer Linear Programming (MILP) approach, this study evaluates the dynamic and flexible operation of the WHR system throughout the round-trip journey of a vessel. Identifying the most profitable WHR system layout, determining the capacity of technologies, optimising the interconnections between technologies, and establishing strategic WH dispatching are all among the key objectives of the study.An average-scale vessel with a 36 MW diesel engine, is selected for this study which involves a 17-day journey with multiple stops in Northern Europe. The vessel adequately represents the complexity of onboard energy systems in terms of types and variability of demands, as well as WH availability. The proposed WHR system, which is tailored for the selected vessel, integrates three active technologies: an isobaric expansion engine (IEE) to contribute to mechanical power demand, a sorption system for providing cooling, and an advanced Organic Rankine Cycle (ORC) for trigeneration of power, heating, and cooling. All technologies are supported by the passive WHR technology of Thermal Energy Storage (TES). The results show that deployment of the optimised WHR system onboard the selected vessel enhances energy efficiency by 5 to 7.5 percentage points and reduces fuel consumption by 13%. The study also explores economic key performance indicators (KPIs), such as the Internal Rate of Return (IRR) which is found at about 15%, clearly evidencing a compelling solution to ship owners. Additionally, the discussion includes a detailed analysis of the contribution of individual technologies in covering onboard demands, as well as their synergistic interactions.This work clarifies the role, value, and benefit of WHR technologies onboard, advancing the understanding from individual WH recovery interventions to a system-level approach. This is especially valuable in practice, considering its adaptability across various vessel types within the global fleet.

A stochastic programming approach to the antibiotics time machine problem

Antibiotics Time Machine is an important problem to understand antibiotic resistance and how it can be reversed. Mathematically, it can be modeled as follows: Consider a set of genotypes, each of which contain a set of mutated and unmutated genes. Suppose that a set of growth rate measurements of each genotype under a set of antibiotics is given. The transition probabilities of a ‘realization’ of a Markov chain associated with each arc under each antibiotic are computable via a predefined function given the growth rate realizations. The aim is to maximize the expected probability of reaching to the genotype with all unmutated genes given the initial genotype in a predetermined number of transitions, considering the following two sources of uncertainties: (i) the randomness in growth rates, (ii) the randomness in transition probabilities, which are functions of growth rates. We develop stochastic mixed-integer linear programming and dynamic programming approaches to solve static and dynamic versions of the Antibiotics Time Machine Problem under the aforementioned uncertainties. We adapt a Sample Average Approximation approach that exploits the special structure of the problem and provide accurate solutions that perform very well in an out-of-sample analysis.

A hybrid metaheuristic to optimize electric first-mile feeder services with charging synchronization constraints and customer rejections

This paper addresses the on-demand meeting-point-based feeder electric bus routing and charging scheduling problem under charging synchronization constraints. The problem considered exhibits the structure of the location routing problem, which is more difficult to solve than many electric vehicle routing problems with capacitated charging stations. We propose to model the problem using a mixed-integer linear programming approach based on a layered graph structure. An efficient hybrid metaheuristic solution algorithm is proposed. A mixture of random and greedy partial charging scheduling strategies is used to find feasible charging schedules under the synchronization constraints. The algorithm is tested on instances with up to 100 customers and 49 bus stops/meeting points. The results show that the proposed algorithm provides near-optimal solutions within less one minute on average compared with the best solutions found by a mixed-integer linear programming solver set with a 4-hour computation time limit. A case study on a larger sized case with 1000 customers and 111 meeting points shows the proposed method is applicable to real-world situations.

Efficient configuration of heterogeneous multistatic sonar networks: A mixed-integer linear programming approach

The work undertaken in this paper pertains to the optimal spatial configuration of a heterogeneous Wireless Sensor Network (WSN) for the Area Coverage (AC) problem. Specifically, this research falls under the heading of Anti-Submarine Warfare (ASW) with an emphasis on active sonar systems and, more pointedly still, on a specific type of sensor: sonobuoys (portmanteau word formed by “sonar” and “buoy”). These buoys are further divided into three main categories: transmitter-only (Tx), receiver-only (Rx) and transmitter–receiver (TxRx). In this paper, we will therefore try to determine the geographical location of the different buoys comprising a Multistatic Sonar Network (MSN), special case of WSN, so as to maximize the overall surface area covered. To do this, we discretize an Area of Interest (AoI) into regular cells using bathymetric and altimetric data, and place a deployment position and a fictitious target at the center of each cell so that we can evaluate the network’s performance. More precisely, we are taking into account a limited number of sensors (buoys) with possible pairwise incompatibilities, variable performances, probabilistic detection models, an adverse masking effect (direct blast) as well as coastlines features. Finally, in order to solve this problem, we have developed several efficient Mixed-Integer Linear Programs (MILPs), all of which have been thoroughly tried-and-tested on a benchmark set of 100 instances derived from real elevation data. This has led us to identify an ideal model, i.e. one that is significantly better than all the others in the statistical sense.

English

English

Operational strategies and integrated design for producing green hydrogen from wind electricity

Realizing the potential of renewable hydrogen production requires flexible operation of electrolysis systems to integrate with intermittent power sources. This work develops an optimization model to assess flexible operational strategies for alkaline and proton exchange membrane (PEM) electrolysers powered by wind energy. The model quantitatively analyses trade-offs between electrolyser shutdown strategies, overloading capacities, and battery integration to identify optimal regimes balancing efficiency, flexibility, and economics. The results reveal a mixed-integer linear programming approach can optimize system configurations and control strategies to minimize the levelized cost of hydrogen production. Optimal near-minimum load operation is achieved by independently optimizing the load of each electrolyser block, while avoiding shutdowns above a critical load level. Strategic electrolyser overloading can provide economic benefits by reducing installed capital costs, if technical feasibility and accelerated degradation are addressed. Battery energy storage integration significantly improves economics by enhancing asset utilization, provided excess renewable energy is available. The model provides novel insights on integrating alkaline and PEM electrolysis with intermittent wind power to advance renewable hydrogen production. Quantifying trade-offs between operational flexibility and economics will help guide flexible design and control strategies for cost-optimal renewable electrolysis systems.