Mixed-integer Optimization Problem Research Articles

Fast robust optimization of ORC based on an artificial neural network for waste heat recovery

Uncertainty in the organic Rankine cycle (ORC) system and the highly complex ORC models pose challenges to optimal operation. A data-driven robust parametric optimization for the ORC system is proposed to ensure high and stable performance under uncertainty. The uncertainty of the cyclic variables is estimated by their distribution parameters, and the average (expected) thermodynamic performance (the net output power of the ORC) is maximized as the optimization objective while minimizing its variance. An artificial neural network with the rectified linear unit is used as a surrogate model for optimization. Then the robust parametric optimization problem can be transformed into a mixed-integer linear optimization problem with a chance-constrained form, and the robust optimal solution can be solved quickly. Case studies show that the solution time of the robust optimization problem using the proposed method is about 1.4 s, which is much less than obtaining the optimal solution based on ORC mechanism models. Meanwhile, the optimal operating condition derived by the proposed robust optimization approach outperforms that obtained by the traditional deterministic strategy. The proposed approach not only improves the robustness of the system but also demonstrates the importance of considering uncertainty in parametric optimization.

Physics-informed deep learning and linear programming for efficient optimization of combined cycle power plants

Efficient operation of combined cycle power plants (CCPP) involves solving mixed-integer nonlinear optimization problems, constrained by thermodynamic differential equations, thermal constraints, and temporal dependencies between variables. The optimization problem aims to maximize profit for day-ahead operation by minimizing short-term and long-term costs, and accounting for volatility in ambient conditions and energy price. In operational terms, the challenge lies in developing an efficient and robust optimization algorithm capable of determining optimal values for thousands of temporal variables. It must also have the capacity to evaluate a multitude of scenarios encompassing various weather and market conditions. To address this challenge, this paper proposes a hybrid optimizer that utilizes deep learning models trained on historical operational data of a real CCPP, along with a surrogate physics-driven piecewise-linear model of CCPP, to determine the optimal values of thousands of control setpoints, with strong temporal co-dependencies, within a two-minute runtime required by industry constraints. A multi-resolution architecture is proposed to efficiently implement this optimizer. The algorithm sequentially solves the problem in coarse and fine resolutions to accelerate convergence. Final results are checked for feasibility using pre-trained deep learning models, ensuring accuracy and validity. The study shows that the proposed model provides comparable results to the exact solution while reducing computational costs, up to 95%. This research has practical implications for power plant system operators, as well as traders and dispatchers, enabling them to make optimal decisions for day-ahead contracting purposes and CCPP operation in a reasonable time, ultimately leading to more efficient and cost-effective power plant management. Furthermore, the methodology introduced in this work could potentially be extended to other industrial applications that require large-scale, complex optimizations, where the precise governing principles of a process can be approximated by piecewise-linear functions.

Supervised Feature Selection via Collaborative Neurodynamic Optimization.

As a crucial part of machine learning and pattern recognition, feature selection aims at selecting a subset of the most informative features from the set of all available features. In this article, supervised feature selection is at first formulated as a mixed-integer optimization problem with an objective function of weighted feature redundancy and relevancy subject to a cardinality constraint on the number of selected features. It is equivalently reformulated as a bound-constrained mixed-integer optimization problem by augmenting the objective function with a penalty function for realizing the cardinality constraint. With additional bilinear and linear equality constraints for realizing the integrality constraints, it is further reformulated as a bound-constrained biconvex optimization problem with two more penalty terms. Two collaborative neurodynamic optimization (CNO) approaches are proposed for solving the formulated and reformulated feature selection problems. One of the proposed CNO approaches uses a population of discrete-time recurrent neural networks (RNNs), and the other use a pair of continuous-time projection networks operating concurrently on two timescales. Experimental results on 13 benchmark datasets are elaborated to substantiate the superiority of the CNO approaches to several mainstream methods in terms of average classification accuracy with three commonly used classifiers.

Frontiers in Operations: Equitable Data-Driven Facility Location and Resource Allocation to Fight the Opioid Epidemic

Problem definition: The opioid epidemic is a crisis that has plagued the United States for decades. One central issue of the epidemic is inequitable access to treatment for opioid use disorder (OUD), which puts certain populations at a higher risk of opioid overdose. Methodology/results: We integrate a predictive dynamical model and a prescriptive optimization problem to compute high-quality opioid treatment facility and treatment budget allocations for each U.S. state. Our predictive model is a differential equation-based epidemiological model that captures the dynamics of the opioid epidemic. We use a process inspired by neural ordinary differential equations to fit this model to opioid epidemic data for each state and obtain estimates for unknown parameters in the model. We then incorporate this epidemiological model into a corresponding mixed-integer optimization problem (MIP) that aims to minimize the number of opioid overdose deaths and the number of people with OUD. We develop strong relaxations based on McCormick envelopes to efficiently compute approximate solutions to our MIPs that have a mean optimality gap of 3.99%. Our method provides socioeconomically equitable solutions, as it incentivizes investments in areas with higher social vulnerability (from the U.S. Centers for Disease Control’s Social Vulnerability Index) and opioid prescribing rates. On average, when allowing for overbudget solutions, our approach decreases the number of people with OUD by [Formula: see text], increases the number of people in treatment by [Formula: see text], and decreases the number of opioid-related deaths by [Formula: see text] after 2 years compared with the baseline epidemiological model’s predictions. Managerial implications: Our solutions show that policymakers should target adding treatment facilities to counties that have significantly fewer facilities than their population share and are more socially vulnerable. Furthermore, we demonstrate that our optimization approach, guided by epidemiological and socioeconomic factors, should help inform these strategic decisions, as it yields population health benefits in comparison with benchmarks based solely on population and social vulnerability. History: This paper has been accepted in the Manufacturing & Service Operations Management Frontiers in Operations Initiative. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2023.0042 .

On Constrained Mixed-Integer DR-Submodular Minimization

Diminishing returns (DR)–submodular functions encompass a broad class of functions that are generally nonconvex and nonconcave. We study the problem of minimizing any DR-submodular function with continuous and general integer variables under box constraints and, possibly, additional monotonicity constraints. We propose valid linear inequalities for the epigraph of any DR-submodular function under the constraints. We further provide the complete convex hull of such an epigraph, which, surprisingly, turns out to be polyhedral. We propose a polynomial-time exact separation algorithm for our proposed valid inequalities with which we first establish the polynomial-time solvability of this class of mixed-integer nonlinear optimization problems. Funding: This work was supported by the Office of Naval Research Global [Grant N00014-22-1-2602].

Computationally efficient solution of mixed integer model predictive control problems via machine learning aided Benders Decomposition

Mixed integer Model Predictive Control (MPC) problems arise in the operation of systems where discrete and continuous decisions must be taken simultaneously to compensate for disturbances. The efficient solution of mixed integer MPC problems requires the computationally efficient online solution of mixed integer optimization problems, which are generally difficult to solve. In this paper, we propose a machine learning-based branch and check Generalized Benders Decomposition algorithm for the efficient solution of such problems. We use machine learning to approximate the effect of the complicating variables on the subproblem by approximating the Benders cuts without solving the subproblem, therefore, alleviating the need to solve the subproblem multiple times. The proposed approach is applied to a mixed integer economic MPC case study on the operation of chemical processes. We show that the proposed algorithm always finds feasible solutions to the optimization problem, given that the mixed integer MPC problem is feasible, and leads to a significant reduction in solution time (up to 97% or 50×) while incurring small error (in the order of 1%) compared to the application of standard and accelerated Generalized Benders Decomposition.

An optimal approach of pure/mixed CO2 sequestration with producing hydrated natural gas: Physical insights and validation

In this contribution, an optimal approach is developed to sequestrate pure/mixed CO2 by dislodging the hydrated natural gas, the cleanest fossil fuel, thereby catching two birds by one stone. For depicting its intricacies, a transient thermo-kinetic swapping model is developed with considering first time the effect of permeability and water saturation, along with porosity involved in sediments of hydrate reservoir. The formulation takes into account various fundamental issues having practical relevance, like porous particles with uneven size and shape, surface dynamics involved in phase transformation, non-unity reaction order (i.e., nth order), among others. A mixed-integer nonlinear multi-objective sequential optimization (MOSO) problem is articulated within the framework of a robust global optimizer (i.e., nondominated sorting genetic algorithm II) to deal with two conflicting objectives, concerning minimization of absolute average relative deviation and maximization of swapping efficiency. This optimal approach is validated with laboratory and field data, showing its outperformance over the latest models at diverse geological conditions. The proposed model framework is employed for lab-scale cases to identify the optimal guest gas (CO2/N2) ratio that secures the maximum swapping efficiency. Simulation results suggest that the presented theoretical model is rigorous in interpreting these swapping dynamics in hydrates.

Generally Applicable Q-Table Compression Method and Its Application for Constrained Stochastic Graph Traversal Optimization Problems

We analyzed a special class of graph traversal problems, where the distances are stochastic, and the agent is restricted to take a limited range in one go. We showed that both constrained shortest Hamiltonian pathfinding problems and disassembly line balancing problems belong to the class of constrained shortest pathfinding problems, which can be represented as mixed-integer optimization problems. Reinforcement learning (RL) methods have proven their efficiency in multiple complex problems. However, researchers concluded that the learning time increases radically by growing the state- and action spaces. In continuous cases, approximation techniques are used, but these methods have several limitations in mixed-integer searching spaces. We present the Q-table compression method as a multistep method with dimension reduction, state fusion, and space compression techniques that project a mixed-integer optimization problem into a discrete one. The RL agent is then trained using an extended Q-value-based method to deliver a human-interpretable model for optimal action selection. Our approach was tested in selected constrained stochastic graph traversal use cases, and comparative results are shown to the simple grid-based discretization method.

Optimization of Clustering and Trajectory for Minimizing Age of Information in Unmanned Aerial Vehicle-Assisted Mobile Edge Computing Network.

With the development of the Internet of Things (IoT) technology, massive amounts of sensor data in applications such as fire monitoring need to be transmitted to edge servers for timely processing. However, there is an energy-hole phenomenon in transmitting data only through terrestrial multi-hop networks. In this study, we focus on the data collection task in an unmanned aerial vehicle (UAV)-assisted mobile edge computing (MEC) network, where a UAV is deployed as the mobile data collector for the ground sensor nodes (SNs) to ensure high information freshness. Meanwhile, the UAV is equipped with an edge server for data caching. We first establish a rigorous mathematical model in which the age of information (AoI) is used as a measure of information freshness, related to both the data collection time and the UAV's flight time. Then a mixed-integer non-convex optimization problem is formulated to minimize the peak AoI of the collected data. To solve the problem efficiently, we propose an iterative two-step algorithm named the AoI-minimized association and trajectory planning (AoI-MATP) algorithm. In each iteration, the optimal SN-collection point (CP) associations and CP locations for the parameter ε are first obtained by the affinity propagation clustering algorithm. The optimal UAV trajectory is found using an improved elite genetic algorithm. Simulation results show that based on the optimized ε, the AoI-MATP algorithm can achieve a balance between data collection time and flight time, reducing the peak AoI of the collected data.

Marginal Probability-Based Integer Handling for CMA-ES Tackling Single-and Multi-Objective Mixed-Integer Black-Box Optimization

This study targets the mixed-integer black-box optimization (MI-BBO) problem where continuous and integer variables should be optimized simultaneously. The CMA-ES, our focus in this study, is a population-based stochastic search method that samples solution candidates from a multivariate Gaussian distribution (MGD), which shows excellent performance in continuous BBO. The parameters of MGD, mean and (co)variance, are updated based on the evaluation value of candidate solutions in the CMA-ES. If the CMA-ES is applied to the MI-BBO with straightforward discretization, however, the variance corresponding to the integer variables becomes much smaller than the granularity of the discretization before reaching the optimal solution, which leads to the stagnation of the optimization. In particular, when binary variables are included in the problem, this stagnation more likely occurs because the granularity of the discretization becomes wider, and the existing integer handling for the CMA-ES does not address this stagnation. To overcome these limitations, we propose a simple integer handling for the CMA-ES based on lower-bounding the marginal probabilities associated with the generation of integer variables in the MGD. The numerical experiments on the MI-BBO benchmark problems demonstrate the efficiency and robustness of the proposed method. Furthermore, in order to demonstrate the generality of the idea of the proposed method, in addition to the single-objective optimization case, we incorporate it into multi-objective CMA-ES and verify its performance on bi-objective mixed-integer benchmark problems.

Unit Commitment Considering Electric Vehicles and Renewable Energy Integration—A CMAES Approach

Global fossil fuel consumption and associated emissions are continuing to increase amid the 2022 energy crisis and environmental pollution and climate change issues are becoming even severer. Aiming at energy saving and emission reduction, in this paper, a new unit commitment model considering electric vehicles and renewable energy integration is established, taking into account the prediction errors of emissions from thermal units and renewable power generations. Furthermore, a new binary CMAES, dubbed BCMAES, which uses a signal function to map sampled individuals is proposed and compared with eight other mapping functions. The proposed model and the BCMAES algorithm are then applied in simulation studies on IEEE 10- and IEEE 118-bus systems, and compared with other popular algorithms such as BPSO, NSGAII, and HS. The results confirm that the proposed BCMAES algorithm outperforms other algorithms for large-scale mixed integer optimization problems with over 1000 dimensions, achieving a more than 1% cost reduction. It is further shown that the use of V2G energy transfer and the integration of renewable energy can significantly reduce both the operation costs and emissions by 5.57% and 13.71%, respectively.

Multi-Compression Scale DNN Inference Acceleration based on Cloud-Edge-End Collaboration

Edge intelligence has emerged as a promising paradigm to accelerate DNN inference by model partitioning, which is particularly useful for intelligent scenarios that demand high accuracy and low latency. However, the dynamic nature of the edge environment and the diversity of end devices pose a significant challenge for DNN model partitioning strategies. Meanwhile, limited resources of the edge server make it difficult to manage resource allocation efficiently among multiple devices. In addition, most of the existing studies disregard the different service requirements of the DNN inference tasks, such as its high accuracy-sensitive or high latency-sensitive. To address these challenges, we propose a Multi-Compression Scale DNN Inference Acceleration (MCIA) based on cloud-edge-end collaboration. We model this problem as a mixed-integer multi-dimensional optimization problem, jointly optimizing the DNN model version choice, the partitioning choice, and the allocation of computational and bandwidth resources to maximize the tradeoff between inference accuracy and latency depending on the property of the tasks. Initially, we train multiple versions of DNN inference models with different compression scales in the cloud, and deploy them to end devices and edge server. Next, a deep reinforcement learning-based algorithm is developed for joint decision making of adaptive collaborative inference and resource allocation based on the current multi-compression scale models and the task property. Experimental results show that MCIA can adapt to heterogeneous devices and dynamic networks, and has superior performance compared with other methods.

Forecast-Based Energy Management for Optimal Energy Dispatch in a Microgrid

This article describes the development of an optimal and predictive energy management system (EMS) for a microgrid with a high photovoltaic (PV) power contribution. The EMS utilizes a predictive long-short-term memory (LSTM) neural network trained on real PV power and consumption data. Optimal EMS decisions focus on managing the state of charge (SoC) of the battery energy storage system (BESS) within defined limits and determining the optimal power contributions from the microgrid components. The simulation utilizes MATLAB R2023a to solve a mixed-integer optimization problem and HOMER Pro 3.14 to simulate the microgrid. The EMS solves this optimization problem for the current sampling time (t) and the immediate sampling time (t+1), which implies a prediction of one hour in advance. An upper-layer decision algorithm determines the operating state of the BESS, that is, to charge or discharge the batteries. An economic and technical impact analysis of our approach compared to two EMSs based on a pure economic optimization approach and a peak-shaving algorithm reveals superior BESS integration, achieving 59% in demand satisfaction without compromising the life of the equipment, avoiding inexpedient power delivery, and preventing significant increases in operating costs.

Fair resource allocation and SBS selection for MC-NOMA-based HetNets supporting D2D communications

This paper studies the application of downlink multi-carrier non-orthogonal multiple access (MC-NOMA) and device-to-device (D2D) in heterogeneous networks (HetNets). In the considered model, the small base stations (SBSs) can reuse the sub-channels (SCs) of the macro base stations (MBS). The paper aims to maximize the system sum-rate while managing co-tier and co-channel interference by optimizing joint SBS selection, fair SC allocation, and power allocation, under the constraints of guaranteeing a certain amount of resources to all cellular users (CUs)and D2D pairs and satisfying CUs’ minimum signal-to-interference-plus-noise ratio (SINR) requirements. However, obtaining an efficient exact solution is challenging due to the non-convexity of the mixed integer optimization problem. To deal with this issue, the problem is decoupled into two sub-problems: 1- SBS selection and fair SC allocation problem and 2- power allocation problem. Then, a four-sided many-to-one matching method is proposed to solve the SBS selection and fair SC allocation problem jointly. The performance of the SC fair allocation algorithm is investigated in terms of the system quality of services (QoS) using a proposed mathematical expression. Furthermore, a novel dynamic search operator is introduced to enhance the performance of the matching-based algorithm. Next, the non-convex power allocation problem is transformed into a deep reinforcement learning (DRL) based framework, and the deep deterministic policy gradient (DDPG) approach is employed to solve the problem. Moreover, a joint SBS selection, SC allocation, and power allocation iterative algorithm is proposed to improve the system sum-rate. Simulation results validate our analysis and indicate the superiority of the proposed algorithms in terms of the system sum-rate compared to the benchmark algorithms.

Model predictive control of bidirectional heat transfer in prosumer-based solar district heating networks

District heating networks with decentralized heat production are ideally suited to include a high share of renewable energy sources for heat production in urban areas with limited space. A new concept is a prosumer-based district heating network in which some or even all buildings are equipped with decentralized building-level heat storages and heat generation plants. To exploit the full potential of the interconnected heating network, buildings with surplus heat are intended to transfer heat to buildings with heating demands to prevent the activation of the central heating plant. This work presents an initial feasibility study on using a model predictive control strategy to manage bidirectional heat transfer between buildings in a district heating network. We formulate this problem as an optimal control problem, incorporating binary decision variables for each potential heat transfer connection. This results in a mixed-integer nonlinear optimization problem that is hard to solve. The problem is solved with a fast gradient-based optimization algorithm in combination with a combinatorial integral approximation strategy. A case study concerning a residential heating network with building-level solar thermal collectors and storage tanks is carried out. The optimized operation is compared to the real operation obtained from one month of measurements. The results show that the optimized strategy with bidirectional heat transfer can exploit the total heat generated in the whole network. The central heating provider is only required when the heat produced or stored by all the buildings is insufficient to meet the total heat demand within the network. In the case study, this leads to a reduction of almost 75% of the required thermal energy from the central heat provider.

3D Trajectory Planning for Real-Time Image Acquisition in UAV-Assisted VR

Nowadays, unmanned aerial vehicles (UAVs), empowered with the capability of high-definition image transmission, are used to capture the rapidly changing physical environment by leveraging its high flexibility to reconstruct an immersive realistic virtual environment for metaverse users. In this paper, we consider a novel UAV-assisted image acquisition system where a UAV is dispatched to take off from an initial location to capture real-time images of multiple ground targets and then transfer the captured images back to the ground user for virtual environment reconstruction. We aim to minimize the time for the UAV to complete the image acquisition task by optimizing the three-dimensional UAV trajectory under the constraints of image quality, information causality and energy consumption. To this end, we first formulate the investigated scenario into a mixed integer optimization problem, which, however, is difficult to solve due to the infinite time-varying variables closely coupled with each other. Then, a three-stage progressive algorithm is proposed to obtain an efficient solution to the formulated mixed integer optimization problem, where the constraints of image quality, information causality and energy consumption can be sequentially satisfied. Finally, comprehensive performance evaluation is conducted to verify the effectiveness of the proposed three-stage progressive trajectory design algorithm, and the results show that the proposed algorithm significantly outperforms the benchmark schemes.

An optimization method of electrostatic sensor array based on Kriging surrogate model and improved non-dominated sorting genetic algorithm with elite strategy algorithm

Array-type electrostatic monitoring is gradually becoming an effective tool for aero-engine fault diagnosis. In this paper, we innovatively apply the surrogate optimization method to the optimization of the sensor array structure in order to meet the need of improving the particle information recognition capability of the electrostatic sensor array (ESA). A structure optimization method of ESA based on the Kriging surrogate model and improved NSGA-II algorithm is proposed. In this paper, a finite element simulation model of ESA is established, and the array optimization problem is abstracted as the solution of a mixed-integer optimization problem. This paper reduces the large-scale numerical simulations in the full-variable space with the help of the Kriging surrogate model. In addition, an improved NSGA-II algorithm for mixed-integer optimization is proposed. The simulation experiment verified that the average absolute error of the sensor before and after optimization for the identification of particle position and charge quantity was reduced by 69.88% and 49.68%, respectively. The array structure optimization method proposed in this paper facilitates the acceleration of the design process of electrostatic sensors and provides a scientific design method for their specific design for airborne application.

Effective hybrid search technique based constraint mixed-integer programming for smart home residential load scheduling

In this paper, the problem of scheduling smart homes (SHs) residential loads is considered aiming to minimize electricity bills and enhance the user comfort. The problem is addressed as a multi-objective constraint mixed-integer optimization problem (CP-MIP) to model the constrained load operation. As the CP-MIP optimization problem is non-convex, a novel hybrid search technique, that combines the Relaxation and Rounding (RnR) approach and metaheuristic algorithms to enhance the accuracy and relevance of decision variables, is proposed. This search technique is implemented through two stages: the relaxation stage in which a metaheuristic technique is applied to get the optimal rational solution of the problem. Whereas, the second stage is the rounding process which is applied via stochastic rounding approach to provide a good-enough feasible solution. The scheduling process has been done under time-of-use (ToU) dynamic electricity pricing scheme and two powering modes (i.e., powering from the main grid only or powering from a grid-tied photovoltaic (PV) residential power system), in addition, four metaheuristics [i.e., Binary Particle Swarm Optimization (BPSO), Self-Organizing Hierarchical PSO (SOH-PSO), JAYA algorithm, and Comprehensive Learning JAYA algorithm (CL-JAYA)] have been utilized. The results reported in this study verify the effectiveness of the proposed technique. In the 1st powering mode, the electricity bill reduction reaches 19.4% and 20.0% when applying the modified metaheuristics, i.e. SOH-PSO and CL-JAYA, respectively, while reaches 56.1%, and 54.7% respectively in the 2nd powering scenario. In addition, CL-JAYA superiority is also observed with regard to the user comfort.

Operation of interconnected power and freshwater networks

As freshwater demand continues to rise and conventional freshwater resources decline, the interaction between the freshwater and power sectors is becoming increasingly relevant. Coordinating both systems entail minimizing the operation costs of the power and freshwater system while supplying the demands of electricity and freshwater and complying with operational constraints. This is particularly relevant in Middle Eastern regions, which have limited availability of freshwater resources. Careful coordination of electricity generation and freshwater production is beneficial for both the power and the freshwater systems. For example, producing excess freshwater at hours of low electricity production cost and storing it for usage at hours of high electricity production cost is economically advantageous. We use detailed models for the operation of the power and freshwater systems. That is, the power system operation is represented through a network-constrained unit commitment model, while the freshwater system model includes nonlinear representations of the freshwater production and transportation. The resulting problem is computationally intractable for large-scale systems. To achieve tractability, nonlinearities in the freshwater model are linearized. The final model is a tractable mixed-integer linear optimization problem that is solved using a state-of-the-art branch-and-cut solver. Simulation results demonstrate that a coordinated scheduling results in lower cost and lower peak as compared with uncoordinated scheduling.

Long-term reliability evaluation of integrated electricity and gas systems considering distributed hydrogen injections

Power-to-gas facilities consume surplus renewable electricity generation to produce alternative gases, such as green hydrogen. They can be injected into, and transported by the gas network for further use, which is a promising way toward a low-carbon energy system. However, injecting alternative gases into the gas systems can adversely affect the gas composition and the lifespan of components (e.g., gas pipelines), and may threaten the reliability of the entire integrated electricity and gas systems (IEGS) in the long term. To address this issue, this paper proposes a long-term reliability evaluation method for IEGS with distributed hydrogen injections. First, new reliability indices are proposed to evaluate both gas adequacy and gas interchangeability under uncertainties. Then, a multi-state reliability model of the pipeline is developed to characterize the corrosion evolution and hydrogen embrittlement in the long term. A contingency management scheme (CMS) is devised to minimize load curtailments and gas interchangeability deviations under component failures. Moreover, several reformulation techniques are tailored to convexify the original two-stage mixed-integer nonlinear CMS optimization problem. An analytical reliability evaluation method embedded with a system state reduction technique is designed to evaluate the long-term reliability of the IEGS more efficiently. Finally, the IEEE 24 bus Reliability Test System and the practical Belgium gas system are used to validate the proposed method. The numerical results show that the injection of alternative gas could jeopardize the reliability of the studied IEGS by 39.73% in the long term. However, we have observed a critical time window (the 8th–9th year), in which if we conduct the inline inspection and maintenance more frequently, the reliability could be improved by up to 53.31%. These results suggest that the injection of alternative gas is beneficial, but should be carefully regulated to maintain the reliability of IEGS.