Mixed Integer Nonlinear Programming Problem Research Articles

Integrating Energy-Efficient Train Control in railway Vertical Alignment Optimization: A novel Mixed-Integer Linear Programming approach

Incorporating train control into the railway design process enables a practical and comprehensive evaluation of the lifecycle utility of a track profile. This paper proposes a novel integrated approach, termed EETC-VAO, which combines railway track Vertical Alignment Optimization (VAO) and Energy-Efficient Train Control (EETC). Initially formulated as a Mixed-Integer Nonlinear Programming (MINLP) problem, EETC-VAO aims to meet various geometric constraints and simultaneously minimize construction costs, traction energy consumption, and section running times in both directions. The model is subsequently reformulated into an equivalent Mixed-Integer Linear Programming (MILP) model using linearization methods and is further enhanced with valid inequalities, logic cuts, and a warm start algorithm with random velocity generation. The model has been extensively tested across a variety of case studies and train types, from synthetic small-scale scenarios to challenging real-world cases spanning from 3 to 71.2 km. Our findings demonstrate that operational costs can be significantly reduced with only marginal increases in construction costs. The integrated approach achieves reductions in total lifecycle costs of up to 40%, revealing a critical trade-off between construction and operational expenses. Notably, our results also indicate that lower construction costs do not inherently conflict with reduced operational costs, emphasizing the critical importance of integrating the train control scheme into the VAO problem.

Optimal Coordination of Directional Overcurrent Relays in Microgrids Considering European and North American Curves

Protecting AC microgrids (MGs) is a challenging task due to their dual operating modes—grid-connected and islanded—which cause sudden variations in fault currents. Traditional protection methods may no longer ensure network security. This paper presents a novel approach to protection coordination in AC MGs using non-standard features of directional over-current relays (DOCRs). Three key optimization variables are considered: Time Multiplier Setting (TMS), the plug setting multiplier’s (PSM) maximum limit, and the standard characteristic curve (SCC). The proposed model is formulated as a mixed-integer nonlinear programming problem and solved using four metaheuristic techniques: the genetic algorithm (GA), Imperialist Competitive Algorithm (ICA), Harmonic Search (HS), and Firefly Algorithm (FA). Tests on a benchmark IEC MG with distributed generation and various operating modes demonstrate that this approach reduces coordination times compared to existing methods. This paper’s main contributions are threefold: (1) introducing a methodology for assessing the optimal performance of different standard curves in MG protection; (2) utilizing non-standard characteristics for optimal coordination of DOCRs; and (3) enabling the selection of curves from both North American and European standards. This approach improves trip time performance across multiple operating modes and topologies, enhancing the reliability and efficiency of MG protection systems.

Age and energy aware data collection scheme for urban flood monitoring in UAV-assisted Wireless Sensor Networks

Wireless Sensor Networks (WSNs) have become pivotal in numerous applications, including environmental monitoring, precision agriculture, and disaster response. In the context of urban flood monitoring, utilizing unmanned aerial vehicles (UAVs) presents unique challenges due to the dynamic and unpredictable nature of the environment. The primary challenges involve designing strategies that maximize data collection while minimizing the Age of Information (AoI) to ensure timely and accurate decision-making. Efficient data collection is crucial to capturing all relevant information and providing a comprehensive understanding of flood dynamics. Simultaneously, reducing AoI is essential, as outdated data can lead to delayed or incorrect responses, potentially worsening the situation. Addressing these challenges is critical for the effective use of WSNs in urban flood monitoring. Initially, we formulate the problem as a mixed integer non-linear programming (MINLP) problem. Further, it is solved using a Lagrangian-based branch and bound technique by converting it into an unconstrained problem. Then, for large-scale WSN, we propose a hybrid optimization technique which combines a genetic algorithm with a particle swarm optimization technique to simultaneously maximize the data collection and reduce the AoI of the collected data with the constraint of energy consumption of the UAVs. Simulation results demonstrate that our proposed algorithm outperforms existing approaches in terms of both data collection and AoI.

Enhancing the Reliability of Weak-Grid-Tied Residential Communities Using Risk-Based Home Energy Management Systems under Market Price Uncertainty

This paper evaluates the reliability of smart home energy management systems (SHEMSs) in a residential community with an unreliable power grid and power shortages. Unlike the previous works, which mainly focused on cost analysis, this research assesses the reliability of SHEMSs for different backup power sources, including photovoltaic systems (PVs), battery storage systems (BSSs), electric vehicles (EVs), and diesel generators (DGs). The impact of these changes on the daily cost and the balance of energy source contribution in providing electrical energy to household loads, particularly during power outage hours, is also evaluated. To address the uncertainty of electricity market prices, a risk management approach based on conditional value at risk is applied. Additionally, the study highlights the impact of community size on energy costs and reliability. The proposed model is formulated as a mixed-integer nonlinear programming problem and is solved using GAMS. The effectiveness of the proposed risk-based optimization approach is demonstrated through comprehensive cost and reliability analysis. The results reveal that when electric vehicles are used as backup power sources, the energy index of reliability (EIR) is not affected by market price variations and shows significant improvement, reaching approximately 99.9% across all scenarios.

Adaptive Privacy-Preserving Coded Computing with Hierarchical Task Partitioning.

Coded computing is recognized as a promising solution to address the privacy leakage problem and the straggling effect in distributed computing. This technique leverages coding theory to recover computation tasks using results from a subset of workers. In this paper, we propose the adaptive privacy-preserving coded computing (APCC) strategy, designed to be applicable to various types of computation tasks, including polynomial and non-polynomial functions, and to adaptively provide accurate or approximated results. We prove the optimality of APCC in terms of encoding rate, defined as the ratio between the computation loads of tasks before and after encoding, based on the optimal recovery threshold of Lagrange Coded Computing. We demonstrate that APCC guarantees information-theoretical data privacy preservation. Mitigation of the straggling effect in APCC is achieved through hierarchical task partitioning and task cancellation, which further reduces computation delays by enabling straggling workers to return partial results of assigned tasks, compared to conventional coded computing strategies. The hierarchical task partitioning problems are formulated as mixed-integer nonlinear programming (MINLP) problems with the objective of minimizing task completion delay. We propose a low-complexity maximum value descent (MVD) algorithm to optimally solve these problems. The simulation results show that APCC can reduce the task completion delay by a range of 20.3% to 47.5% when compared to other state-of-the-art benchmarks.

ReLU surrogates in mixed-integer MPC for irrigation scheduling

Efficient water management in agriculture is important for mitigating the growing freshwater scarcity crisis. Mixed-integer Model Predictive Control (MPC) has emerged as an effective approach for addressing the complex scheduling problem in agricultural irrigation. However, the computational complexity of mixed-integer MPC still poses a significant challenge, particularly in large-scale applications. This study proposes an approach to enhance the computational efficiency of mixed-integer MPC-based irrigation schedulers by employing Rectified Linear Unit (ReLU) surrogate models to describe the soil moisture dynamics of the agricultural field. By leveraging the mixed-integer linear representation of the ReLU operator, the proposed approach transforms the mixed-integer MPC-based scheduler with a quadratic cost function into a mixed-integer quadratic program, which is the simplest class of mixed-integer nonlinear programming problems that can be efficiently solved using global optimization solvers. The effectiveness of this approach is demonstrated through comparative studies conducted on a large-scale agricultural field across two growing seasons, involving other machine learning surrogate models, specifically Long Short-Term Memory (LSTM) networks, and the triggered irrigation scheduling method. The ReLU-based approach significantly reduces solution times — by up to 99.5% — while achieving comparable performance to the LSTM approach in terms of water savings and Irrigation Water Use Efficiency (IWUE). Moreover, the ReLU-based approach achieves enhanced performance in terms of irrigation water savings and IWUE compared to the triggered approach.

Energy Efficiency Maximization for Multi-UAV-IRS-Assisted Marine Vehicle Systems

Mobile edge computing is envisioned as a prospective technology for supporting time-sensitive and computation-intensive applications in marine vehicle systems. However, the offloading performance is highly impacted by the poor wireless channel. Recently, an Unmanned Aerial Vehicle (UAV) equipped with an Intelligent Reflecting Surface (IRS), i.e., UIRS, has drawn attention due to its capability to control wireless signals so as to improve the data rate. In this paper, we consider a multi-UIRS-assisted marine vehicle system where UIRSs are deployed to assist in the computation offloading of Unmanned Surface Vehicles (USVs). To improve energy efficiency, the optimization problem of the association relationships, computation resources of USVs, multi-UIRS phase shifts, and multi-UIRS trajectories is formulated. To solve the mixed-integer nonlinear programming problem, we decompose it into two layers and propose an integrated convex optimization and deep reinforcement learning algorithm to attain the near-optimal solution. Specifically, the inner layer solves the discrete variables by using the convex optimization based on Dinkelbach and relaxation methods, and the outer layer optimizes the continuous variables based on the Multi-Agent Twin Delayed Deep Deterministic Policy Gradient (MATD3). The numerical results demonstrate that the proposed algorithm can effectively improve the energy efficiency of the multi-UIRS-assisted marine vehicle system in comparison with the benchmarks.

A seismic-risk-based bi-objective stochastic optimization framework for the pre-disaster allocation of earthquake search and rescue units

Accurately predicting earthquakes' time, location and size is nearly impossible with today’s technology. Severe earthquakes require prompt and effective mobilization of available resources, as the speed of intervention has a direct impact on the number of people rescued alive. This, in turn, calls for a strategic pre-disaster allocation of search and rescue (SAR) units, both teams and equipment, to make the deployment of resources as quick and equitable as possible. In this paper, a seismic risk-based framework is introduced that takes into account distance-based contingencies between cities. This framework is then integrated into a mixed-integer non-linear programming (MINLP) problem for the allocation of SAR units under uncertainty. The two minimization objectives considered are the expected maximum deployment time of different SAR units and the expected mean absolute deviation of the fulfillment rates. We recover the best vulnerability-adjusted routes for each size-location scenario as input to the optimization model using the dynamic programming (DP) approach as part of the broader area of reinforcement learning (RL). The results of the hypothetical example indicate that the comprehensive model is feasible in various risk scenarios and can be used to make allocation-deployment decisions under uncertainty. The results of the sensitivity analysis verify that the model behaves reasonably against changes in selected parameters, namely the number of allowed facilities and weights of individual objectives. Under the assumption that the two objectives are equally important, the model achieves a total deviation of %3.5 from the objectives with an expected maximum dispatch time of 1.1327 hours and an expected mean absolute deviation of 0.01.

Integrating machine learning model and computer-aided molecular design toward rational ionic liquid selection for separating fluorinated refrigerants

To reduce emissions and facilitate the reclamation of hydrofluorocarbons (HFCs), which are potent greenhouse gases, ionic liquid (ILs)-based absorption separation of HFCs from fluorinated refrigerant (FR) blends is considered as an effective approach. In this work, computer-aided ionic liquid design (CAILD) is carried out to rationally design ILs for separating FR blends, with the near-azeotropic R-32/R-125 system as an illustrative case study. Firstly, the machine learning algorithm, namely artificial neural network (ANN), is employed to develop a group contribution (GC) model for accurate prediction of FR-in-IL solubility. Then, a mixed-integer nonlinear programming (MINLP) problem is formulated and solved by integrating the ANN-GC model with two available GC models for predicting the melting point and viscosity of ILs. Consequently, the optimal three ILs including [C1C10Im][dca], [C1C11Im][dca], and [C1C12Im][dca] are identified. Finally, quantum chemistry calculations are conducted to disclose the separation mechanism from molecular perspective, further validating the reliability of the CAILD results.

Optimal charging scheduling for Indoor Autonomous Vehicles in manufacturing operations

Indoor Autonomous Vehicles (IAVs) have become instrumental in modern logistics, particularly in dynamic operational environments. Their consistent availability is crucial for both timely task execution and energy efficiency in smart factories. Therefore, an inefficient charging schedule can waste energy, compromising production and warehousing efficiency. In addition, formulating an effective charging schedule is challenging due to the nonlinearity of its design and the uncertainties inherent in smart factory settings. In contrast to previous studies that either simplify the problem using linearization-based methods or approach it with computationally demanding algorithms, this paper efficiently addresses the nonlinear challenge, ensuring a closer alignment with real-world conditions. Hence, a simulation environment is first designed to replicate a smart factory, including validated models of IAVs, Charging Stations (CSs), and a variety of unpredictable static and dynamic obstacles. A comprehensive dataset is then provided from this simulation setup. Utilizing this dataset, a model tailored to ascertain the travel time of IAVs, accounting for inherent uncertainties, is trained using Deep Neural Networks (DNNs). Subsequently, a Mixed-Integer Nonlinear Programming (MINLP) problem is formulated to design the optimization task. Finally, integration of the DNNs’ model with the Branch-and-Bound (BnB) approach, forming the BnBD method, streamlines the determination of the optimal charging schedule. Experimental results highlight significant improvements in charging scheduling, establishing this approach as a viable and promising solution for manufacturing operations.

Optimization-based design of structure and operation parameters simultaneously for real-world distillation column based on rate-based model

Simultaneously optimizing the structure and operating parameters of distillation columns based on rigorous rate-based models is critical for various process intensification technologies to achieve their intended purpose in real-world distillation columns. However, efficiently optimizing a distillation column based on rate-based models while considering the column hydraulics is far from practical. The biggest obstacle in using rigorous models for distillation columns in an equation-oriented environment is ensuring a robust solution for large-scale, strongly coupled nonlinear models, along with the simultaneous optimization of integer and continuous variables. In this study, a new optimization framework was introduced. It incorporated a pseudo-transient simulation method to ensure convergence of the simulation in equation-oriented environment. Additionally, an efficient relaxation method transformed the mixed integer nonlinear programming problem into a nonlinear programming problem, facilitating simultaneous optimization of both the integer variable representing column structure and the continuous variables governing column operation. Three examples, including a complex intensified catalytic distillation column, were optimized to demonstrate the efficiency of the proposed framework. Specifically, its advantages in terms of mathematical optimality and industrial feasibility were illustrated.

End-to-end, decision-based, cardinality-constrained portfolio optimization

Portfolios employing a (factor) risk model are usually constructed using a two step process: first, the risk model parameters are estimated, then the portfolio is constructed. Recent works have shown that this decoupled approach may be improved using an integrated framework that takes the downstream portfolio optimization into account during parameter estimation. In this work we implement an integrated, end-to-end, predict-&-optimize framework to the cardinality-constrained portfolio optimization problem. To the best of our knowledge, we are the first to implement the framework to a nonlinear mixed integer programming problem. Since the feasible region of the problem is discontinuous, we are unable to directly differentiate through it. Thus, we compare three different continuous relaxations of increasing tightness to the problem which are placed as an implicit layers in a neural network. The parameters of the factor model governing the problem’s covariance matrix structure are learned using a loss function that directly corresponds to the decision quality made based on the factor model’s predictions. Using real world financial data, our proposed end-to-end, decision based model is compared to two decoupled alternatives. Results show significant improvements over the traditional decoupled approaches across all cardinality sizes and model variations while highlighting the need of additional research into the interplay between experimental design, problem size and structure, and relaxation tightness in a combinatorial setting.

Pipe merging for transient gas network optimization problems

In practice, transient gas transport problems frequently have to be solved for large-scale gas networks. Gas network optimization problems typically belong to the class of Mixed-Integer Nonlinear Programming Problems (MINLP). However current state-of-the-art MINLP solvers are not yet mature enough to solve large-scale real-world instances. Therefore, an established approach in practice is to solve the problems with respect to a coarser representation of the network and thereby reducing the size of the underlying model. Two well-known aggregation methods that effectively reduce the network size are parallel and serial pipe merges. However, these methods have only been studied in stationary gas transport problems so far. This paper closes this gap and presents parallel and serial pipe merging methods for transient gas transport problems. An empirical evaluation indicates that the developed methods perform very accurately on a huge set of fine-grained real-world data taken from one of the largest transmission system operators in Europe.

Multi-Channel Cooperative Spectrum Sensing and Scheduling for Cognitive IoT Networks

This paper presents a novel multi-channel cooperative spectrum sensing and scheduling (MC$_2$S$_3$) framework for spectrum and energy harvesting cognitive Internet of Things (IoT) networks. We address the challenge of maximizing network throughput by formulating a combinatorial problem that jointly optimizes the sensing scheduling of primary channels (PCs), the assignment of IoT devices for sensing scheduled PCs, and the clustering and allocation of IoT nodes to efficiently use discovered idle PCs; subject to spectrum utilization and collision avoidance constraints. Recognizing the inherent complexity of the underlying NP-hard mixed-integer non-linear programming (MINLP) problem, we propose a decomposition strategy that decouples the master problem into PC exploration and exploitation sub-problems. In the exploration phase, we derive closed-form solutions for optimal sensing durations and detection thresholds that satisfies spectrum utilization and collision avoidance constraints, which are then used to develop a priority metric to rank PCs. The proposed PC ranking informs a sequential PC scheduling and IoT sensing assignment approach that exploits a linear bottleneck assignment (LBA) strategy, proceeding until further scheduling does not enhance network utility. For the exploitation phase, we leverage a non-orthogonal multiple access (NOMA) strategy to multiplex multiple IoT nodes on a single PC, employing an iterative linear sum assignment (LSA) method for efficient allocation to maximize utilization of idle PCs. Numerical results validate the efficacy of our proposed methodologies, reaching an accuracy of approximately 99% in the order of milliseconds, significantly outperforming time complexity of brute-force benchmarks.

Globally optimal sequencing of optimal reactive dispatch control adjustments to minimize operational losses in transmission systems by graph shortest path, parallel computing, and dynamic programming

Minimizing operational losses in transmission systems through the Optimal Reactive Dispatch (ORD), a non-convex mixed-integer nonlinear programming problem, is crucial for operational cost reduction, resource optimization, and greenhouse gas emission mitigation. Besides all intricacies associated with solving ORDs, transmission system operators encounter the challenge of determining sequences in which ORD control adjustments must be implemented before significant changes occur in generators scheduled power output and system loading. Sequencing ORD control adjustments, in spite of not being novel, remains modestly scrutinized in the literature. This paper introduces a two-phase framework that tackles the globally optimal sequencing of n ORD control adjustments over n! potential paths by solving the ORD to minimize operational losses in transmission systems in the first phase, and optimally sequencing ORD control adjustments employing fast power flow calculations, graph shortest path, parallel computing, and dynamic programming in the second phase. We discuss the framework’s second phase asymptotic time complexity, which is exponential over factorial for brute-force approaches, and its capability to guarantee globally optimal paths toward minimal operational losses determined in the framework’s first phase. ORD control adjustments for transmission systems with up to 27 controllable variables are benchmarked against two mixed-integer nonlinear programming solvers: BARON, a global non-convex solver, and Knitro, a local solver (assuming convexity around local optima). Globally optimal sequences of ORD control adjustments over n! potential paths (more than 1028 for sequencing 27 control adjustments) and average algorithm runtimes validate the straightforward application and, more importantly, effectiveness of such a comprehensive framework.

A distributed economic model predictive control-based FPPT scheme for large-scale solar farm

Increasing solar-photovoltaic-power penetration necessitates the implementation of flexible power point tracking (FPPT) in solar farms. However, coordinating multiple solar photovoltaic (PV) power generation systems (SPVPGSs) poses a significant challenge due to their inherent intermittency and varying dynamic characteristics, complicating FPPT implementation. To overcome this challenge, this paper proposes a distributed economic model predictive control (DEMPC) scheme to achieve FPPT, while simultaneously enhancing overall economic performance in solar farms. By integrating solar farm control and local control into one optimal control framework, this scheme eliminates the need for power allocation, PV voltage reference calculation, and pulse width modulation. Leveraging the SPVPGS model and soft power constraint, DEMPC controllers are designed to achieve the economic targets of solar farms, which cooperate through a communication network to realize FPPT and economic optimization. Additionally, the strong nonlinearity of SPVPGS causes a non-convex mixed integer nonlinear programming (MINLP) problem, solved by a MINLP algorithm using finite converter switching states. Simulations under step-changed irradiance and power reference, as well as urgent maintenance conditions, demonstrate that the DEMPC-based FPPT scheme significantly outperforms the traditional hierarchical model predictive control-based FPPT scheme, presenting both superior dynamic response and enhanced economic performance in FPPT implementation.

Access strategies in mmWave cell-free network: A matching and auction theory based approach

The mmWave cell-free network is viewed as an energy-efficient and emerging architecture for beyond 5G systems. However, many practical issues should be solved for this system before the benefits are enjoyed, whereof feasible implementation of access strategy is one of the most vital. To this end, the access strategy including user scheduling and bandwidth allocation is investigated in this paper. Firstly, to maximize the defined utility function, the joint optimization problem of user scheduling and bandwidth allocation is formulated as a mixed integer nonlinear programming problem, which is intractable to search for an optimal solution in polynomial time. Secondly, a two-stage matching and sealed-auction-based access strategy is proposed to obtain a sub-optimal solution, where UEs with high-quality channel conditions are scheduled in the first-stage matching as much as possible, and the rest unmatched UEs are scheduled in the second-stage matching with the sealed-auction based bandwidth allocation. Thirdly, a simplified full-matching based access strategy with lower complexity is proposed for the high-dynamic scenario caused by the high-speed movement of UEs, and the complexities and convergence of the two proposed strategies are quantitatively analyzed. Finally, the performance of proposed strategies is evaluated through abundant simulations in different scenarios, and the superiorities of the proposed strategies are demonstrated through the comparison with reference strategies.

Two-Tier Efficient QoE Optimization for Partitioning and Resource Allocation in UAV-Assisted MEC.

Unmanned aerial vehicles (UAVs) have increasingly become integral to multi-access edge computing (MEC) due to their flexibility and cost-effectiveness, especially in the B5G and 6G eras. This paper aims to enhance the quality of experience (QoE) in large-scale UAV-MEC networks by minimizing the shrinkage ratio through optimal decision-making in computation mode selection for each user device (UD), UAV flight trajectory, bandwidth allocation, and computing resource allocation at edge servers. However, the interdependencies among UAV trajectory, binary task offloading mode, and computing/network resource allocation across numerous IoT nodes pose significant challenges. To address these challenges, we formulate the shrinkage ratio minimization problem as a mixed-integer nonlinear programming (MINLP) problem and propose a two-tier optimization strategy. To reduce the scale of the optimization problem, we first design a low-complexity UAV partition coverage algorithm based on the Welzl method and determine the UAV flight trajectory by solving a traveling salesman problem (TSP). Subsequently, we develop a coordinate descent (CD)-based method and an alternating direction method of multipliers (ADMM)-based method for network bandwidth and computing resource allocation in the MEC system. Extensive simulations demonstrate that the CD-based method is simple to implement and highly efficient in large-scale UAV-MEC networks, reducing the time complexity by three orders of magnitude compared to convex optimization methods. Meanwhile, the ADMM-based joint optimization method achieves approximately an 8% reduction in shrinkage ratio optimization compared to baseline methods.

Task execution latency minimization for energy-sensitive IoTs in wireless powered mobile edge computing: A DRL-based method

Wireless powered mobile edge computing (MEC) has become a vital component of future 6G networks, offering efficient computational capabilities to internet of things (IoT) devices and ensuring a stable energy supply. In this paper, we propose an innovative design for a wireless powered MEC-assisted energy-sensitive IoT systems. The system integrates wireless power technology (WPT), allowing IoT devices to efficiently perform tasks under wireless power supply without consuming their battery energy. Therefore, we formulate a dynamic computational task execution latency minimization (DCTELM) problem to tackle the impacts of unpredictable energy requirements and potential communication latency on resource allocation. We also analyze the differences between time division multiple access and non-orthogonal multiple access communication protocols. Given the complexity and dynamic nature of this problem, we define it as a mixed integer nonlinear programming problem. To address this problem, we introduced an enhanced deep reinforcement learning (DRL) algorithm. First, the DCTELM problem is transformed into a Markov decision process (MDP), which simplifies the structure of the problem. Then, after determining the offloading decision, we reveal the convex relationship between resource allocation and task latency. Furthermore, the action space of MDP is further reduced, and a multi-head proximal policy optimization (PPO) algorithm is proposed to deal with the simplified MDP problem. This process reduces the complexity of problem solving, decreases the consumption of computational resources. Simulation results demonstrate that our method outperforms other baselines in terms of computation latency, particularly with the introduction of an exploration strategy during the training phase, which significantly reduces task latency.

Mathematical optimization modeling for scenario analysis of integrated steelworks transitioning towards hydrogen-based reduction

To reduce carbon dioxide emissions from the steel industry, efforts are made to introduce a steelmaking route based on hydrogen reduction of iron ore instead of the commonly used coke-based reduction in a blast furnace. Changing fundamental pieces of steelworks affects the functions of most every system unit involved, and thus warrants the question of how such a transition could optimally take place over time, and no rigorous attempts have until now been made to tackle this problem mathematically. This article presents a steel plant optimization model, written as a mixed-integer non-linear programming problem, where aging blast furnaces and basic oxygen furnaces could potentially be replaced with shaft furnaces and electric arc furnaces, minimizing costs or emissions over a long-term time horizon to identify possible transition pathways. Example cases show how various parameters affect optimal investment pathways, stressing the necessity of appropriate planning tools for analyzing diverse cases.