Mixed Integer Nonlinear Programming Model Research Articles

A Modified Simulated Annealing and Enhanced Harmony Search Algorithms for the Continuous Facility Layout Problem (CFLP)

This paper focuses on solving the Continuous Facility Layout Problem (CFLP), which aims to minimize material handling costs by strategically placing facilities in a known or unknown area. The objective is to find optimal or near-optimal facility arrangements while adhering to non-overlapping and spatial constraints, thus enhancing the efficiency of production systems. The study addresses various layout scenarios, including single-row facility layout (with and without clearance), continuous layout, and unequal-area facility layout problems. To achieve this, the authors propose a mixed-integer nonlinear programming (MINLP) model tailored for continuous layouts. Two meta-heuristic algorithms are developed to optimize this model: a hybrid Simulated Annealing-Genetic Algorithm (SA-GA), which leverages genetic crossover operations, and an Enhanced Harmony Search Algorithm (EHSA), featuring dynamic parameters and a novel improvisation technique. The proposed methods provide flexible and efficient solutions for both small and large-scale layout problems, offering decision-makers practical tools for real-world applications.

A systematic methodology for selecting optimal technology for waste heat utilization in food processing industry

Utilizing waste heat that is rejected from thermal processes can greatly enhance the industry's energy balance. Identifying potential waste heat sources in the food and beverage processing industry and making estimations for their possible utilization can be an extremely challenging undertaking. To determine waste heat potentials in the food processing industry and to select an optimal technology for its utilization a novel systematic methodology based on the comprehensive energy audit and mathematical optimization has been developed. The developed mixed integer nonlinear programming model incorporates all novel and the most commonly employed technologies to utilize waste heat.The developed systematic methodology is tested on a case study, of a milk and dairy production facility. Obtained results have shown that up to 53 % of the available waste heat can be used in the scenario of limited investment costs. Otherwise, in the scenario when investment costs are not set as a limitation, 75 % of waste heat can be used, according to plant demands. Harmonising production processes is necessary to use all the waste heat. The developed systematic methodology can be applied to any food processing industry and other facilities producing waste heat because it presents a universal approach.

How the establishment of carbon emission trading system affects ship emission reduction strategies designed for sulfur emission control area

As the backbone of the international seaborne trade, the pollution and emission from shipping are highly concerned by governments and international organizations as well. The International Maritime Organization has designated five emission control areas (ECAs) to limit sulfur oxides emissions of ships. Meanwhile, the European Union has included the shipping industry in the EU Emission Trading System since 2024 to control their carbon emissions. The strategies like switching fuel, installing scrubber or detour, are commonly used to comply with the ECA regulation, but they may increase carbon emissions and possibly incur extra carbon trading cost. In this case, the shipping company should investigate the influence of marine emission trading system (METS) on emission reduction strategies used under ECA regulation. Therefore, this paper focuses on this problem and formulates it as a mixed-integer nonlinear programming model. Several numerical experiments are conducted to show the applicability of the proposed model. The results show that ship’s detour behavior is inevitable under the ECA regulation, which incurs more carbon emissions. When METS regulation is not considered, the optimal emission reduction strategy is installing scrubber. In contrast, when METS regulation is considered, the optimal choice may be installing scrubber or fuel switching depending on different sailing route, which indicates the effect of METS on ship emission reduction strategy. Moreover, the fleet deployment and sailing speed of scrubber installation strategy are affected by METS regulation, while the strategies of fuel switching and using LNG-powered ship are almost unaffected. This indicates that installing scrubber is more easily affected by emission control policies.

Bus scheduling with heterogeneous Fleets: Formulation and hybrid metaheuristic algorithms

This paper focuses on optimizing mixed-fleet bus scheduling (MFBS) with vehicles of different sizes in public transport systems. We develop a novel mixed-integer nonlinear programming (MINLP) model to address the MFBS problem by optimizing vehicle assignment and dispatching programs. The model considers user costs, operator costs, and the crowding inconvenience of standing and sitting passengers. To tackle the complexity of the MFBS problem, we employ Genetic Algorithm (GA) and Grey Wolf Optimizer (GWO). Besides, we develop two hybrid metaheuristics, including GA-SA [a combination of GA and Simulated Annealing (SA)] and GWO-SA (a combination of GWO and SA), to improve optimization capabilities for the MFBS problem. We also employ a Taguchi approach to fine-tune the metaheuristics’ parameters. We widely examine and compare the metaheuristics’ performance across various-sized samples (small, medium, and large), considering solution quality, computational time, and the result stability of each algorithm. We also compare the metaheuristics’ solutions with the optimal solutions acquired by GAMS software in small and medium-scale samples. Our findings show that the GWO-SA outperforms the other metaheuristics. Applying our model to a real bus corridor in Santiago, Chile, we find that precise dispatching plans generated by more sophisticated/advanced algorithms (GA-SA and GWO-SA) lead to larger cost savings and improved performance compared to simpler algorithms (GA and GWO). Interestingly, utilizing more advanced algorithms makes a difference in terms of fleet planning in crowded scenarios, whereas for low and medium-demand cases, simpler dispatching algorithms could be used without a drop in accuracy.

Covariate balancing for high-dimensional samples in controlled experiments

In controlled experiments, achieving covariate balancing across all groups is crucial as it ensures that the estimated treatment effects are not confounded by the effects of covariates. This study proposes a mixed-integer nonlinear programming model to address the covariate balancing problem. Specifically, we introduce a new covariate imbalance measure, which is the maximum discrepancy in both the first and second central moments between any two groups. The second central moment can effectively capture the correlation of covariates in a physical sense, which is crucial for partitioning high-dimensional samples. A mixed-integer nonlinear programming model is constructed to minimize the proposed measure to obtain the optimal partitioning results. The nonlinear model is then linearized to accelerate the optimization process. We conduct computational experiments based on simulated datasets, including one-dimensional, two-dimensional, and three-dimensional Gaussian distributed samples, and a real clinic trial dataset. Compared to the conventional discrepancy-based method, our method achieves a 54.81% and a 40.6% reduction in the maximum discrepancy of partitioning results in the two-dimensional simulated Gaussian samples and the real clinic trial dataset, respectively. These results demonstrate the superiority of the proposed model in partitioning high-dimensional samples with correlated covariates compared with the conventional discrepancy-based method.

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.