Mixed Integer Nonlinear Programming Model Research Articles

Optimal management in island microgrids using D-FACTS devices with large-scale two-population algorithm

Amidst the increasing complexity of microgrid optimization, characterized by numerous decision variables and intricate non-linear relationships, there is a pressing need for highly efficient algorithms. This study introduces a tailored Mixed Integer Nonlinear Programming (MINLP) model that optimizes the charging and discharging schedules of electric vehicles (EVs) and energy storage systems (ESS) while incorporating Distributed Flexible AC Transmission System (D-FACTS) devices. To address these challenges, a novel approach based on the Large-Scale Two-Population Algorithm (LSTPA) is proposed. The model's effectiveness was evaluated using a 33-node microgrid, where the proposed method achieved a total purchased energy of 1.2 MWh, a voltage deviation of 0.0357 p.u, and a CPU time of 551 s, outperforming traditional methods like NSGA-II, PSO, and JAYA. Additionally, in a 69-node microgrid, the approach resulted in a total purchased energy of 0.3 MWh and a voltage deviation of 0.0078 p.u. These results demonstrate the superior performance of the proposed method in terms of energy efficiency, voltage stability, and computational time, advancing the efficiency of microgrid management.

Data-driven collaborative healthcare resource allocation in pandemics

Severe shortages of healthcare resources are major challenges in pandemics, especially in their early stages. To improve emergency management efficiency, this paper proposes a novel rolling predict-then-optimize framework that includes three interactive modules, i.e., data-driven demand prediction, healthcare resource allocation, and parameter rolling update. Such a framework uses historical data to dynamically update the control parameters of the proposed Net-SEIHRD model, which predicts the healthcare needs of each region by jointly considering government interventions and cross-regional travel behaviors. Based on the forecasted healthcare resource demand in real-time, an optimization model is then formulated to realize coordinated resource allocation across multiple regions by minimizing the total generalized cost. To facilitate model solving, the proposed mixed integer nonlinear programming model is converted into an equivalent mixed integer linear model by using some linearization techniques. Finally, the proposed method is applied to the SARS-CoV-2 emergency response and collaborative allocation of healthcare resources in Shanghai, China. The results show that the proposed prediction model can effectively predict the peak and scale of the spread of the virus. Compared with the traditional LM and SEIHR models, the prediction accuracy of the Net-SEIHRD model is improved by 10.76% and 24.11%, respectively. Moreover, coordinated relief activities across regions, such as patient transfer and drug-sharing can improve the efficiency of pandemic control and save social costs.

Optimizing Supply Chain Design under Demand Uncertainty with Quantity Discount Policy

In typical business situations, sellers usually offer discount schemes to buyers to increase overall profitability. This study aims to design a supply chain network under uncertainty of demand by integrating an all-unit quantity discount policy. The objective is to maximize the profit of the entire supply chain. The proposed model is formulated as a mixed integer nonlinear programming model, which is subsequently linearized into a mixed integer linear programming model and hence able to obtain a global solution. Numerical examples in the manufacturing supply chain where customer demand follows normal distributions are used to assess the effect of quantity discount policies. Key findings demonstrate that the integration of quantity discount policies significantly reduces total supply chain costs and improves inventory management under demand uncertainty, and decision makers need to decide a balance level between service levels and profits.

Last-train timetable synchronization for service compatibility maximization in urban rail transit networks with arrival uncertainties

The last train services of urban rail transit offer all passengers the final chance to reach their destinations. In the context of multimodal transport, considering the random delayed arrivals of late-night vehicles of other transport modes at urban hub stations and their adverse effects on multimodal passengers transferring to urban rail transit, a probabilistic scenario set is established to determine the arrival uncertainties of multimodal passengers. In each scenario, the timetable synchronization problem is formulated as a mixed-integer nonlinear programming model that requires a performance trade-off between destination reachability and the remaining path distance of both non-multimodal and multimodal passengers, integrally termed last train service compatibility. Through linearization techniques, the model is transformed into an equivalent mixed-integer linear programming form, which can be efficiently solved using the Gurobi solver to obtain the optimal last train timetable for each scenario and form an alternative scheme set. An improved probabilistic scenario set-based regret value theory is then developed, in which a novel regret value calculation method is proposed. The scheme with the minimum total weighted opportunity loss in all scenarios is selected as the optimal robust. Real case experiments based on the Chengdu–Chongqing high-speed railway line and Chengdu metro network are conducted to test the performance of our model. The results show that compared with the original timetable, the optimized timetable reduces the number of unreachable passengers by 34.29% and the sum of the average remaining path distance of non-multimodal and multimodal passengers by 57.43% with the help of path planning for all passengers. The proposed approach is proven not only to balance the demands of both reachable and unreachable passengers, but also to significantly improve the robustness of the last train timetable to arrival uncertainties of multimodal passengers and reduce their service inequity.

Day-ahead multi-criteria energy management of a smart home under different electrical rationing scenarios

With the recent global energy crisis, some countries have implemented electrical rationing (ER), making it necessary for smart homes to play a pivotal role in optimizing energy consumption and contributing to sustainable practices. To effectively manage smart home consumption, a stochastic programming approach for a grid-connected smart home energy management system (SHEMS) is proposed in this paper. The system includes PV, battery, diesel, and gas-based heating/cooling systems (HCS). Additionally, a demand response program (DRP) has been employed under time-of-use tariffs in the Syrian energy market. The main objective is to minimize the day-ahead expected cost and consumer discomfort by optimizing the operation of dispatchable units and loads. To manage the risks associated with the expected cost due to potential uncertainties in PV energy generation and electrical rationing programs, the conditional value-at-risk (CVaR) approach is adopted. Two methods are proposed to model the uncertainty in PV energy generation; interval bands and interval-based scenarios. The problem is modeled as a mixed-integer non-linear programming (MINLP) model, and coded in GAMS to test different cases. Based on the results obtained, substantial reductions reached 56.2% in worst-case cost scenarios when employing concurrent DRP-risk management.

Batch-sizing and machinability data systems for milling operations: An optimal sustainable cost of quality approach

Nowadays, manufacturers make every effort to achieve a higher quality of their products at an attractive cost. With all the introduced legislation and incentives in the developed world to address global warming, machining shops in the West also strive to cut greenhouse emissions. This article offers an optimal approach to the micro Computer-Aided Process Planning (CAPP) problem to optimize the internal quality cost and buffering effect while keeping the environmental impact low. To optimize the machining parameters, the mathematical model is developed for different milling operations, face, side, and peripheral. cutting speed, feed rate, axial depth of cut, radial depth of cut, nose radius, and batch sizing while maximizing profit and meeting customer demand. A Mixed-Integer Nonlinear Programming (MINLP) model is formulated and solved using Classical Constrained Nonlinear Optimization (CCNO) and Genetic Algorithms (GAs). Surface roughness, used as a metric to evaluate the desired quality level of a finished machined part type, is modeled as a Gaussian random variable to model the surface roughness of the machined part utilizing a cumulative normal distribution. The ratio of rework and scrap is calculated in terms of the surface roughness of the machined part shifting away from the target and exceeding upper and lower specification limits. The internal failure cost model, addressing both scrap and rework, is developed based on Taguchi’s quadratic loss function. CCNO is employed to validate the results obtained by GAs, relaxing the lot-sizing integrality constraint and, thus, the convexity of the produced relaxed model. An iterative method employing a developed multi-regression model is used to solve for the expended power consumption (an inherent highly nonlinear environmental criterion of the developed model) within both GAs and CCNO. This study reveals that the machining parameters substantially impact the cost components of the objective function as well as the scrap and rework quantities. A stringent quality cost target can force the model to optimize the feed rate and nose radius to minimize the internal failure quality cost while improving the environmental impact, including direct and indirect power consumption and CO2 emissions considerations.

A quadratically constrained mixed-integer non-linear programming model for multiple sink distributions

Rising traffic congestion and fuel costs pose significant challenges for supply chains with numerous retailers. This paper addresses these challenges by optimizing transportation routes for processed tomatoes within a long-haul and intercity distribution network. We use the heterogeneous capacitated vehicle routing problem framework to create a new quadratically constrained mixed-integer non-linear programming model that aims to meet demand at multiple destinations while minimizing transportation costs. Our model incorporates real-time data and route optimization strategies that consider traffic conditions based on freight time and route diversions for expedited deliveries. It aims to devise an optimal transportation schedule that minimizes fuel, operational, and maintenance costs while ensuring efficient delivery of tomato paste. By applying this model to a real-world case study, we estimate a significant 27.59% reduction in transportation costs, dropping them from GH¢20,270 ($1,638.91) to GH¢14,676 ($1,186.61) on average. This highlights the effectiveness of our strategy in lowering costs while maintaining smooth and efficient deliveries.

Collaborative train timetabling and passenger flow control in oversaturated metro lines considering state-dependence

Congestion downgrades the moving speed of passengers in trains and platforms. This in turn affects congestion. That is, the moving speed of passengers interacts with congestion. It is state-dependent and obeys the fundamental diagram. This interaction makes the system unstable and can result in substantial passenger delays and heightened security risks in oversaturated metro lines. This study aims to rectify this by developing a more realistic collaborative optimization approach. Firstly, we formulate a deterministic fluid queuing network model to accurately capture the dynamic propagation of congestion in oversaturated metro lines. This model incorporates the fundamental diagram, thereby effectively considering the interaction between passenger movement speeds and congestion. A recursive algorithm is devised to solve the fluid queuing network model. The complexity of the algorithm is independent of the capacities of the stations and trains. This provides the possibility of analyzing large-scale metro networks. Based on the queuing network model, a mixed-integer nonlinear programming model aimed at minimizing the total passenger waiting time is built. To solve the model effectively, we employ an alternating direction method of multipliers framework to decompose the model into nonlinear programming and integer nonlinear programming problems, which are respectively solved by improved simultaneous perturbation stochastic approximation and adaptive large neighborhood search algorithms. Finally, a small-scale case and a real-world case with data from Chengdu metro lines are implemented to demonstrate the performance and effectiveness of the proposed approach. The experimental results provide some interesting insights. (1) Without consideration of state-dependence, the queuing model underestimates the passenger waiting time and overestimates the service levels. (2) As demand increases, excluding extreme oversaturation scenarios, the optimized results of the state-independent model progressively deteriorate compared to our state-dependent model, with the largest percentage decline being 50.23%. This is especially noticeable when the train capacity is limited. (3) The contribution ratio of passenger flow control in mitigating congestion gradually surpasses that of train timetabling with increasing demand. In the small-scale case, the former contribution ratio to optimization rises from negligible to 98.19%.

Toward green container liner shipping: joint optimization of heterogeneous fleet deployment, speed optimization, and fuel bunkering

AbstractContainer liner shipping companies, under the international shipping carbon reduction indicators proposed by the International Maritime Organization, must transform two key aspects: technology and operations. This paper defines a green liner shipping problem (GLSP) that integrates the deployment of a heterogeneous fleet, speed determination, and fuel bunkering. The objective is to achieve low‐carbon operations in liner shipping, taking into consideration the diversification of power systems, the use of alternative fuels in ships, and the continuous improvement of alternative fuel bunkering systems. For this purpose, we present a bi‐objective mixed integer nonlinear programming model and develop two methodologies: an epsilon‐constraint approach and a heuristic‐based multi‐objective genetic algorithm. We validate the effectiveness of our model and methods through a case study involving container ships of various sizes deployed on intra‐Asian short sea routes by SITC International Holdings Co., Ltd. The experimental results highlight the crucial role of dual‐fuel (DF) ships in the pursuit of low‐carbon strategies by liner companies, with liquefied natural gas and ammonia DF ships being the most widely used. Additionally, fuel cell (FC) ships, particularly those powered by ammonia and hydrogen, demonstrate significant carbon reduction potential. Furthermore, ships with larger container capacities have a greater cost advantage. For the GLSP, speed determination is an auxiliary decision, and the lowest speed is not necessarily the optimal choice. Decision‐makers must carefully balance competing economic and carbon emission reduction objectives, as deploying more alternative fuel ships may increase fuel bunkering and fuel consumption, resulting in a higher total operating cost.

Operation optimization of large-scale natural gas pipeline networks based on intelligent algorithm

With the expansion of natural gas pipelines into larger networks, there is a growing concern regarding energy consumption in the operations of pipeline systems. However, the intricate network structure and hydraulic properties pose challenges for operation optimization. In this paper, a Mixed Integer Nonlinear Programming model is proposed to optimize the steady-state operation of natural gas pipeline network, aiming to minimize the energy consumption of compressor units. The model incorporates constraints related to flow direction, flow balance, pressure drop in pipeline sections, and pressure increase at compressor stations. To solve the complex nonlinear model efficiently, a stochastic optimization algorithm that integrates particle swarm optimization and high-fidelity simulation is proposed. The case study is conducted on a large-scale natural gas pipeline network consisting of forty-three pipeline sections and four compressor stations. The optimal operation scheme is calculated, and the outlet pressures of each compressor are determined. The results demonstrate that the stochastic optimization algorithm proposed in this paper can reduce energy consumption by 21.23 % at most and 19.77 % on average during pipeline operation, which can provide guidance for the operation management of large-scale natural gas pipeline network.

Integrated train timetable with supply–demand interactions at the Chinese high-speed railway

In traditional Chinese railway operations, train timetable and unit assignment are usually separated from pricing and seat allocation, resulting in a decrease in revenue and an increase in operating costs. In this paper, we study the integrated problem of train timetable, unit assignment, pricing, and seat allocation, where passenger choice behavior is modeled by the multinomial logit (MNL) model. This integration enables us to explicitly capture supply–demand interactions. The problem is formulated as a mixed integer nonlinear programming model with the objective of maximizing total profit. By converting the MNL model into its equivalent convex form, we recast the model as a tractable reformulation, allowing the use of a general-purpose solver to solve practical-sized instances. Based on real data from high-speed railway lines in China, three sets of case studies are conducted to validate the effectiveness and efficiency of the proposed approach.

A novel framework for production planning and class-based storage location assignment: Multi-criteria classification approach

Efficient warehouse management is essential for optimizing inventory, minimizing transportation costs, and enhancing overall performance. This research introduces a novel Mixed-Integer Nonlinear Programming (MINLP) model to address the Storage Location Assignment Problem (SLAP) in warehouse management. Integrating multi-criteria decision-making with strategic production planning, our model advances warehouse operations by allocating storage locations to products strategically, focusing on reducing transportation distances and maximizing storage efficiency. The distinctive innovation of this study is the nuanced application of the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) results to strategic storage location assignments, enhancing the model's capability to consider a comprehensive evaluation of inventory attributes, including physical characteristics and perishability. This approach evolves TOPSIS's application in warehouse management, enabling it to consider both physical characteristics and perishability of products. The outcomes of TOPSIS, including product classifications and preferences, serve as vital inputs to the mathematical model, facilitating a comprehensive evaluation of storage locations that encompasses spatial, demand-related, and physical aspects of inventory. Additionally, our research introduces a versatile decision support system, adaptable to various operational requirements. This system enhances practical decision-making in warehouse management, accommodating scenarios based on single or multiple criteria, including the cube-per-order index (COI). The research results highlight the significant impact of this innovative approach in enhancing warehouse management. By addressing the complexities of storage location assignment and integrating multiple criteria, we achieve more efficient and cost-effective warehouse operations. The approach has been shown to be adaptable and practical, making it a valuable contribution to the field of logistics and warehouse management.

Competitive pricing and seed node selection in a two-echelon supply chain

This paper presents a bi-level game model for pricing in a supply chain where the manufacturer (He) is the leader, and the retailer (She) is the follower. The leader decides on the wholesale price, and the follower decides on the selling price and selects seed nodes. The main idea of the model is that the marketing strategy used for promoting the product is focused on giving free samples to potential customers. Hence, the importance of analyzing a social network becomes evident. To maximize her profit, the retailer decides based on three factors: first, the leader's decision about wholesale price; second, the social network structure, which is critical for selecting the seed nodes; and third, people's valuation of the product. Therefore, a bi-level Mixed-Integer Nonlinear Programming model is developed to consider the social binds between potential customers. To solve this model, we employ a meta-heuristic algorithm. Finally, the effect of the model's parameters on decision variables and the objective functions is discussed. Based on the analysis and discussions, the production cost has a prominent impact on the players’ decisions and profits. Furthermore, instead of spending all the marketing budget on increasing seed nodes, it is suggested that they be spent on market research and improving good publicity. Moreover, deciding whether the players want to maximize the profit or market penetration is required before diving into decision-making.

Modelling reservation strategies for managing peak-hour stranding on an oversaturated metro line

Metro is the main travel model for urban commuters in many metropolises around the world. During peak hours, large numbers of passengers pour into metro stations for rail services, but some are unable to board the trains in time and left stranded on the platform or even queuing outside the stations. The trip reservation (TR) strategy, where passengers preplan their trips and reserve their entry time to the stations. This paper develops an entry reservation strategy (ERS) to optimize the commuter flow during peak hours, and construct a multi-objective passenger flow joint optimization model based on many-to-many passenger demand to minimize the total trip cost of passengers at reservation station and the number of stranded passengers at intermediate stations. The passenger flow optimization problem is formulated as a mixed-integer non-linear programming (MINLP) model. We design an iterative sequential search algorithm combined with the GUROBI solver to obtain the parameters of the optimal ERS and the passenger flow distribution in the metro system after disaggregated reformulation of the complex constraints of the model. We also demonstrate the accuracy and effectiveness of the proposed method with two experiments – an illustrative example and a large-scale case study of Beijing Metro. The results of Beijing Metro experiment show that the joint optimization model with entry reservation strategy (JO-ERS) reduces the number of stranded passengers by 88.46 % compared with the original passenger flow from the AFC.

Supply responsive scheduling for ethylene cracking furnace systems based on deep reinforcement learning

AbstractEthylene is one of the most important chemicals, and scheduling optimization is crucial for the profitability of ethylene cracking furnace systems. With the diversification of feedstocks and the high variability in prices, supply chain fluctuations pose significant challenges to the scheduling decisions. Dynamically responding to these fluctuations has become crucial. Traditional mixed integer nonlinear programming (MINLP) models lack the capability of supply chain response, while receding horizon optimization (RHO) models require parameter prediction and repeated optimization solving. To address this challenge, we propose a deep reinforcement learning‐based framework that includes an ethylene dynamic scheduling environment and a decision agent based on deep Q‐network. Across three test cases, compared to the MINLP and RHO models, this framework significantly minimizes losses caused by supply chain fluctuations, thereby increasing daily average net profits by 9%–27%, demonstrating its significant potential for application in responsive scheduling in the presence of supply chain fluctuations.

Optimization Method of Mine Ventilation Network Regulation Based on Mixed-Integer Nonlinear Programming

Mine ventilation is crucial for ensuring safe production in mines, as it is integral to the entire underground mining process. This study addresses the issues of high energy consumption, regulation difficulties, and unreasonable regulation schemes in mine ventilation systems. To this end, we construct an optimization model for mine ventilation network regulation using mixed-integer nonlinear programming (MINLP), focusing on objectives such as minimizing energy consumption, optimal regulation locations and modes, and minimizing the number of regulators. We analyze the construction methods of the mathematical optimization model for both selected and unselected fans. To handle high-order terms in the MINLP model, we propose a variable discretization strategy that introduces 0-1 binary variables to discretize fan branches’ air quantity and frequency regulation ratios. This transformation converts high-order terms in the constraints of fan frequency regulation into quadratic terms, making the model suitable for solvers based on globally accurate algorithms. Example analysis demonstrate that the proposed method can find the optimal solution in all cases, confirming its effectiveness. Finally, we apply the optimization method of ventilation network regulation based on MINLP to a coal mine ventilation network. The results indicate that the power of the main fan after frequency regulation is 71.84 kW, achieving a significant energy savings rate of 65.60% compared to before optimization power levels. Notably, ventilation network can be regulated without adding new regulators, thereby reducing management and maintenance costs. This optimization method provides a solid foundation for the implementation of intelligent ventilation systems.

Integrating resilience and reliability in semiconductor supply chains during disruptions

The semiconductor industry, a cornerstone of modern technology, has been crucial in driving globalization and supporting various sectors, from consumer electronics to automotive industries. However, in recent years, the industry has faced substantial challenges threatening its ability to meet the surging demand for semiconductor chips. Disruptions at any point in the supply chain, from raw material sourcing to end-product delivery, can substantially influence the semiconductor ecosystem. The intricate nature of such SCs makes them highly vulnerable to various disruptions, emphasizing the critical need for building resilient and reliable supply chain strategies. This article presents comprehensive research aimed at addressing critical gaps in the understanding and management of resilience and reliability within the semiconductor supply chain (SSC). This study proposes a multi-objective mixed-integer non-linear programming (MO-MINLP) model to configure an SSC while considering reliability and resilience measures. It emphasizes and draws attention to the importance of resilience and reliability in managing SSC disruptions during a pandemic and potential future epidemic outbreak. Exploring the precise breakdown of batch transportation between two sites shows how disruption can affect product flow along the SC. The applicability of the proposed method is demonstrated through a numerical example of an SSC, solved using the LINGO solver. Finally, a sensitivity analysis is conducted on the model's parameters to assess the capability and effectiveness of the results from managerial viewpoints.

A modelling and solution approach for wind-affected drone-truck routing problem under uncertainty

In the field of logistics and transportation, drones and trucks effectively enhance each other’s capabilities by offering complementary benefits in terms of speed, cargo capacity, and charging frequency. Thus, efficient management of their collaboration is an important task. Although there is a vast literature addressing different aspects of drone-truck combined operations (DTCO), only a few studies incorporate the energy consumption of drones into the optimization model, and the existing ones have made simplified assumptions. This paper proposes an optimization model for DTCO by incorporating a comprehensive energy function affected by the drone speed, cargo weight, wind speed, and wind direction, paying attention to environmental viewpoints. Due to this energy function, the problem is formulated as a mixed-integer nonlinear programming (MINLP) model. To enhance tractability and efficiency, we provide a linear approximation for the MINLP model. Given the stochastic nature of wind conditions throughout the day, we extend the deterministic model as a scenario-based stochastic one. Incorporating uncertainty makes the model more complex and hence, we adopt a modified progressive hedging algorithm (PHA) to efficiently solve the model. Computational results over a variety of instances confirm the effectiveness of the proposed approach.

Energy storage comparison of chemical production decarbonization: Application of photovoltaic and solid oxide electrolysis cells

The fossil fuel driven chemical production leads to significant greenhouse emission, and the low-carbon emission technologies are necessary for carbon neutrality. The integration of solid oxide electrolysis cells (SOEC) and H2-O2 combustion can replace the fossil fuel and supply high-temperature heat for reactions. However, the energy demand keeps consistent but the renewable energy fluctuates with time. Therefore, energy storage is important for such a change. Clean fuel replacement and electrification are applied in a case study of ethylene plant, which requires 147 MW of clean fuel and 91.36 MW of grid power. Photovoltaic (PV) solar energy drives SOEC and liquefied H2, compressed H2, compressed air energy storage (CAES) are compared. A mixed integer nonlinear programming model is proposed to evaluate decarbonization effect and cost, which are balanced by multi-objective optimization. The results show that the liquefied H2 is superior to other choices because of the high efficiency. The hydrogen of 126.27 MW is the optimal point, which requires 415 MW SOEC and PV panels. Also, this study proposes that the power grid should communicate with energy consumers such as chemical plants to ensure the energy storage method, or supply renewable energy directly, which avoids energy loss and unreasonable energy transition.

Game theoretic approach for the synthesis and optimization of economically optimal coalitions in integrated renewable energy and polygeneration networks

This study presents the generation of an integrated resource network that encompasses a bioenergy supply chain and a polygeneration hub. The resource network is designed using a superstructure that comprises 3 layers. The first layer is a supply chain network of seasonally available renewable and non-renewable energy sources and is connected to the second layer by a transport system comprised of rail, road, and pipeline. The second layer, which is the polygeneration hub, comprises a boiler to produce high-pressure steam, steam turbines for generating power, a multi-effect evaporation system for desalinating salt-water and an absorption refrigeration system to produce chilled water. The option of connecting the boiler to a solar-thermal and heat storage subnetwork for feedwater pre-heating is included in the second layer. Piping and electrical cables connect the polygeneration layer to the third layer, which comprises industrial heat demand, represented by heat exchanger network, and electrical power demand. The model is applied to a case study in which the subnetworks in the overall integrated superstructure layers are considered as individual players in an industrial symbioses network with the possibility to be in a cooperative game. The objective function of the model, which is represented as a mixed integer nonlinear programming model, involves the minimization of the summation of the players' marginal contribution, with equal weighting applied for each player, to the coalition's total annual cost. The result obtained involves preference for a coalition among participating subnetworks that include combined heat and power, multi-effect evaporation, and heat exchanger network.