Mixed Integer Nonlinear Programming Model Research Articles

Heuristic algorithms for placing geomagnetically induced current blocking devices

We propose a new heuristic approach for solving the challenge of determining optimal placements for geomagnetically induced current blocking devices on electrical grids. Traditionally, these determinations are approached by formulating the problem as mixed-integer nonlinear programming models and solving them using optimization solvers based on the spatial branch-and-bound algorithm. However, computing an optimal solution using the solvers often demands substantial computational time due to their inability to leverage the inherent problem structure. Therefore, in this work we propose a new heuristic approach based on a three-block alternating direction method of multipliers algorithm, and we compare it with an existing stochastic learning algorithm. Both heuristics exploit the structure of the problem of interest. We test these heuristic approaches through extensive numerical experiments conducted on the EPRI-21 and UIUC-150 test systems. The outcomes showcase the superior performance of our methodologies in terms of both solution quality and computational speed when compared with conventional solvers.

A penalty-free hybrid algorithm framework based on feasible stream matching principle for large-scale heat exchanger networks synthesis

In this work, a penalty-free hybrid stochastic-deterministic algorithm framework is proposed for large-scale heat exchanger networks (HENs) synthesis (HENS), formulated as a computationally-hard mixed-integer nonlinear programming (MINLP) problem. In the outer level, an improved genetic algorithm (GA) is developed to optimize process stream matches represented by integer variables whose values are generated by a unique heat exchanger vector. Unlike previous researches, the improved GA does not rely on any penalty terms, because we propose a feasible stream matching principle to exclude all infeasible process stream matches and only feasible matches are considered in optimization process. In the inner level, a reduced-size MINLP model is solved using deterministic methods to minimize total annualized costs (TACs), which are then used to evaluate the fitness of candidate HENs. Through this way, the proposed framework combines deterministic and stochastic methods to enhance optimization efficiency and global search capability. Illustrative tests on six benchmark cases demonstrate that the framework can efficiently achieve lower-cost solutions compared to deterministic, stochastic, or hybrid methods. The results show a decrease in TAC for all six cases and a reduction in solution time ranging from 11.1% to 97.2%. Importantly, the proposed framework can be extended to solve MINLP problems in other process networks.

Packing spheres with quasi-containment conditions

A novel sphere packing problem is introduced. A maximum number of spheres of different radii should be placed such that the spheres do not overlap and their centers fulfill a quasi-containment condition. The latter allows the spheres to lie partially outside the given cuboidal container. Moreover, specified ratios between the placed spheres of different radii must be satisfied. A corresponding mixed-integer nonlinear programming model is formulated. It enables the exact solution of small instances. For larger instances, a heuristic strategy is proposed, which relies on techniques for the generation of feasible points and the decomposition of open dimension problems. Numerical results are presented to demonstrate the viability of the approach.

Modeling a Multimodal Routing Problem with Flexible Time Window in a Multi-Uncertainty Environment

In this study, we extend the research on the multimodal routing problem by considering flexible time window and multi-uncertainty environment. A multi-uncertainty environment includes uncertainty regarding the demand for goods, the travel speed of the transportation mode, and the transfer time between different transportation modes. This environment further results in uncertainty regarding the delivery time of goods at their destination and the earliness and lateness caused by time window violations. This study adopts triangular fuzzy numbers to model the uncertain parameters and the resulting uncertain variables. Then, a fuzzy mixed integer nonlinear programming model is established to formulate the specific problem, including both fuzzy parameters and fuzzy variables. To make the problem easily solvable, this study employs chance-constrained programming and linearization to process the proposed model to obtain an equivalent credibilistic chance-constrained linear programming reformulation with an attainable global optimum solution. A numerical case study based on a commonly used multimodal network structure is presented to demonstrate the feasibility of the proposed method. Compared to hard and soft time windows, the numerical case analysis reveals the advantages of the flexible time window in reducing the total costs, avoiding low reliability regarding timeliness, and providing confidence level-sensitive route schemes to achieve flexible routing decision-making under uncertainty. Furthermore, the numerical case analysis verifies that it is necessary to model the multi-uncertainty environment to satisfy the improved customer requirements for timeliness and enhance the flexibility of the routing, and multimodal transportation is better than unimodal transportation when routing goods in an uncertain environment. The sensitivity analysis in the numerical case study shows the conflicting relationship between the economic objective and the reliability regarding the timeliness of the routing, and the result provides a reference for the customer to find a balance between them.

Optimizing Route and Speed under the Sulfur Emission Control Areas for a Cruise Liner: A New Strategy Considering Route Competitiveness and Low Carbon

In order to reduce pollution caused by ship emissions, the International Maritime Organization (IMO) implemented sulfur emission control areas (SECAs). In comparison to ordinary vessels, cruise ships with dual attributes of transportation and tourism generate a greater amount of marine pollution, which poses a significant threat to the marine environment in both berthing ports and the sailing area. In light of the fierce competition of the cruise tourism market, cruise lines are looking for strategies, such as designing more attractive cruise routes, to maintain their core competencies under the emission control policy. In order to achieve this goal, this paper presents a mixed-integer non-linear programming (MINP) model with two objectives and is derived from the traditional route optimization problem. The primary objective is to optimize the route and speed of a cruise liner, while simultaneously enhancing route competitiveness and minimizing carbon emissions both within and outside the SECAs. Subsequently, the multi-objective particle swarm optimization (MOPSO) algorithm was used to reach the objective, and simulations were carried out to verify the effectiveness of the model and method. The results show that speed and sailing route optimization can affect carbon emissions. This paper has a certain application value and guiding significance for cruise line decision makers that will be beneficial for the environment.

Multi-objective location-routing model and algorithm study of material reserve bases for large-scale railway engineering construction projects

Material supply is an important support for engineering construction and an important factor affecting the progress and quality of engineering construction. A multi-objective mixed-integer nonlinear programming model is proposed for the location routing problem of the material reserve base of large-scale railway engineering construction projects, considering realistic factors like the construction period, the cycle of the material reserve, the variety of materials, the heterogeneous fleet, and demand splitting. An improved Non-dominated Sorting Genetic Algorithm-II (NSGA- II) is developed for solving this model. This paper optimises obtaining the initial population based on the linear feature of railway engineering, improves the fitness function, and designs a two-stage crossover operator. In addition, the algorithm is compared with the augmented ϵ-constraint method with the help of a practical example. Three metrics are used to assess the performance of the algorithm's outputs, and two metrics are used to assess the similarity of the Pareto fronts obtained by the two algorithms. The results show that the proposed algorithm performs better in terms of ‘quantity’ and ‘spacing’, that the Pareto solutions obtained are accurate, and that the speed of solution is almost 10 times that of the augmented ϵ-constraint (AUGMECON) method.

A MINP model and hybrid heuristic algorithm for railway cold chain service network design problem: a case study of China

Railway Cold Chain Service Network Design (RCC-SND) aims to optimise resource allocation and utilisation to ensure the efficient transportation of perishable foods. In this study, we propose a Mixed Integer Nonlinear Programming (MINP) model to solve the hub selection, service frequency determination, wagon flow organisation and routing problems in RCC-SND. The application of the model is illustrated using the real network comprising 163 cities in China. The PSO-GA algorithm is proven effective in solving the problem. Furthermore, we introduce two future scenarios to assess the carbon reduction potential and economic costs of railway cold chain operations. There are some main findings: as the hub number increases from 30 to 40, the total cost decreases, as the hub continues to rise, this reduction is offset by the increased hub operational costs. Increasing hubs can enhance direct train frequency while causing railway capacity redundancy, this redundancy can be addressed by freight volume increase, as demonstrated in this study, 4.5 times increase in freight volume can boost train utilisation by up to 12.5%. By 2030, railway cold chain carbon emissions are projected to exceed 1.1 million tons, the adoption of hydrogen as alternative energy is an economically viable solution for emission reduction.

Model for Intercity Railway Schedule Optimization Incorporating Metrorail Transfers

The promotion of multimodal coordination is of great significance for the joint development of diverse urban transportation systems. Focusing on a comprehensive urban passenger transport hub that converges both intercity modes and multiple inner-city modes, our study aims to incorporate intercity railway and Metrorail coordination into train scheduling optimization to enhance transfer efficiency and provide a seamless service for passengers in urban railway networks. By considering the variety of passenger demands in different transport modes (e.g., rail transit and traditional mainline railway), train loading capacity, and potential disruption scenarios, we specifically developed a mixed-integer nonlinear programming (MINLP) model to simultaneously generate the coordination strategies, arrival/departure times of trains at each station, and the number of carriages in service in the network. To enhance computational efficiency, we proposed several improvement strategies involving the model linearization to reformulate the MINLP model into a mixed-integer linear programming (MILP) model and the determination of appropriate big- M values. Finally, numerical experiments based on historical passenger data from a realistic urban railway network in China were implemented to verify the effectiveness of our approach. The results demonstrate that our intercity railway and Metrorail coordination strategies could, in practice, significantly improve passenger transfer efficiency by over 35% compared with the train schedule without coordination.

Heterogeneous joint vaccine allocation and quarantine restriction planning under uncertainty: The COVID-19 pandemic

Once effective vaccines are discovered for a pandemic and initial distribution infrastructure, and mechanisms are implemented, the next concern is an optimal vaccination campaign to stop the disease or minimize the casualties. This paper studies the optimization of a Vaccine Distribution Network (VDN) through a single-objective Mixed-Integer Non-Linear Programming (MINLP) model for minimizing the total cost in terms of governmental aid (to maintain the economy), logistics, vaccination, and hospitalization costs. The proposed model develops the optimal joint vaccination-quarantine strategy under equity considerations to (1) determine the optimal allocation of available vaccine stockpiles to different population classes in different regions and (2) seek to impose quarantine restrictions to different regions optimally. The proposed model was applied to a real case study, the vaccine distribution network in France. Through a comprehensive numerical analysis, the optimal vaccine allocation is calculated, and the impact of equity and quarantine on the performance of the distribution network is investigated. Moreover, fractional dosing on the number of infections is examined. The results show that equity considerations, counter-intuitively, lead to an increased number of infections. Furthermore, it is illustrated that fractional dosing helps policymakers to control the pandemic better under a limited supply of vaccines.

Integrated truck drone delivery services with an optimal charging stations

Battery-powered drones, or unmanned aerial vehicles (UAVs), present significant market potential for faster, economical, and eco-friendly urban delivery solutions. Many firms are investing in drone logistics ventures to capitalize on their capabilities. However, the limited range of drone deliveries, dictated by battery capacity, poses a significant challenge. Hybrid delivery systems combining trucks and drones have gained attention to overcome this challenge. Traditional models assume trucks park at customer sites while drones handle additional deliveries, yet this approach may hinder drone efficiency due to constraints in parking availability. Alternative approaches like battery swapping and mobile charging stations on trucks have been explored but exhibit limitations such as blind spots and substantial battery reserves. We propose establishing dedicated drone charging stations and optimizing drone routing for efficient deliveries to address these issues We present a MINLP (Mixed Integer Non-Linear Programming) model aimed at identifying the most cost-effective solution that optimizes both transportation efficiency and charging infrastructure investment. Quantitative experiments demonstrate the efficacy of this approach, alongside sensitivity analysis aiding economic decision-making across various operational scenarios. Notably, our experiments show a 15% reduction in total costs when extending the operational lifespan of charging stations from 2 to 4 years.

Supply chain network design based on Big Data Analytics: heuristic-simulation method in a pharmaceutical case study

Digital technology is a pivotal aspect of the ongoing digital transformation upon which supply chain networks (SCNs) heavily rely. Through the utilization of Big Data Analytics (BDA), organizations can gain insights into sales, production and formulate decision support systems. The pharmaceutical supply chain emerges as a crucial player in producing and distributing medications vital for treating COVID-19 patients. In this study, we have innovatively designed a multi-echelon Pharmaceutical Supply Chain Network (PSCN) using BDA during the COVID-19 pandemic. We have developed a novel Mixed-Integer Non-Linear Programming model addressing allocation, production planning, and inventory control decisions, known as the LAPI problem. A transformation method is employed to convert the non-linear model into a linear one to enhance computational efficiency. The primary objective of our mathematical model is to minimize the total cost and CO2 emissions of the transportation system and established centres while maximizing the shelf life of medicines. Our experiments reveal that medium- and large-scale problems, characterized by the three main attributes of BDA – variety, velocity and volume (3 V‟s), pose challenges for optimal solutions. We present a case study of Iran to validate the mathematical model for large-scale problems. Employing a simulation approach, we estimate the demand for required pharmaceuticals, underscoring our proposed model’s real-world applicability and effectiveness. This research contributes to the growing necessity of eco-friendly and efficient supply chain strategies, especially in the critical domain of pharmaceuticals during pandemic situations.

Optimal emergency hospitals construction in an unexpected epidemic with considering the interactive effect

AbstractConstructing emergency hospitals is one of the most critical measures to defeat an unexpected epidemic. However, existing operations research (OR) studies rarely consider the interactive effect between the construction of emergency hospitals and the dynamics of epidemic transmission. Inspired by this gap, we propose a new modeling framework for decision‐making in emergency hospital construction. In our optimization model, we address the pandemic evolution functions as constraints. We also consider the heterogeneity among infected individuals, distinguishing between those with mild and severe symptoms, each requiring treatment in different types of emergency hospitals. We formulate the problem as a mixed integer nonlinear programming model. Our model can envision the current and future evolution of the epidemic and the impact of different decisions regarding emergency hospital construction on epidemic development. Simultaneously, it provides the optimal strategy to build hospitals and minimize the total number of untreated patients due to the disease. The proposed model is tested using the Covid‐19 outbreak case in Wuhan. The results can provide precise guidelines for emergency hospitals construction, including timing and capacity, and offer decision boundaries for policymakers considering the uncertainty of disease transmission. Furthermore, our decision‐making framework is general and can be adapted to study other epidemics.

Joint optimization of train timetable and passing facility: A capacity allocation strategy

To addreess the challenge of overcrowding in the urban rail transit system, this paper proposes a novel joint optimization strategy that optimizes the timetable and the passing facility’s working status (PFWS) simultaneously according to the passenger demands. To this end, a mixed-integer nonlinear programming (MINLP) model is formulated to jointly minimize overcrowding on trains and platforms, as well as the passenger travel time. The passing facility setting and the passenger flow activities depiction are particularly taken into account by the rigorous constraints. Subsequently, we design a simulation-based hybrid variable neighborhood search (SHVNS) algorithm to efficiently generate near-optimal solution. This algorithm incorporates a variable neighborhood search (VNS) algorithm, a randomly reset strategy, and a simulation-based algorithm. The VNS algorithm includes five neighborhood operators to enhance search efficiency. Finally, various experiments with the real-world data from Chongqing rail transit line 6 in China are tested to prove the applicability and the performance of our proposed model and algorithm. The computational results show that the joint optimization strategy outperforms a current operation strategy in terms of the overcrowded situation on trains and platforms, often by more than 60%, and some management implications and theoretical support can be obtained for the urban rail transit operator.

Uncertain programming model for designing multi-objective reverse logistics networks

Given the contradiction between the rapid growth of products and the modest recovery rate of end-of-life products, there is a pressing need to understand the societal significance of establishing a reverse logistics network for end-of-life products. This research constructs an open-loop five-tier reverse logistic network model encompassing customers, centres for collection, disassembly and inspection, remanufacturing, and disposal. A multi-objective mixed-integer nonlinear programming model under uncertainty has been developed. Unlike previous research, this model accounts for surrounding residents' disutility of facilities while simultaneously minimizing economic costs and environmental impact. Besides, uncertainty theory is introduced in solving the proposed model. More specifically, the formulated model converts all uncertain variables into uncertain distributions by implementing the uncertain multi-objective programming method. Furthermore, a customised non-dominated sorting genetic algorithm III (NSGA-III) is proposed and is employed for the first time to address facility selection and recycling volume distribution within the network. The model is then validated using a real-life case study focusing on end-of-life vehicles in Changchun, China. This research could assist decision-makers in both governmental and private sectors in achieving a balanced approach to the interests of the economy, environment, and local communities comprehensively when designing reverse supply chains.

Planning of electric vehicle charging stations: An integrated deep learning and queueing theory approach

This study presents a hybrid solution for the charging station location-capacity problem. The proposed approach simultaneously determines the location and capacity of charging stations (i.e., number of charging piles), and assigns piles to electric vehicles based on their level of charge. The problem is formulated as a bi-objective mixed-integer nonlinear programming model to minimize the total cost of establishing charging stations together with the average customers’ waiting time. The proposed solution combines queueing theory with mathematical modelling to estimate the average waiting time. A deep learning algorithm is then developed to enhance the precision of waiting time estimation. Another contribution is involving a deep neural network model in improving NSGA-II algorithm. Numerical experiments are conducted in Halifax, Canada to assess the performance of the proposed framework. The results demonstrate the strong predictive performance of the deep learning algorithm and highlight the limitations of traditional queueing models in estimating waiting times in charging stations (i.e., 99.8% improvement in computation time, as well as accuracy improvement of time estimations from 13% to 1.6% deviation). Several valuable insights are obtained to improve the operational performance of charging stations such as achieving a significant (i.e., 61.5%) drop in the average waiting time across the network by a modest (i.e., 29.2%) increase in the initial investments. Also, it reveals that the variability of service rate significantly impacts the average waiting time (i.e., a 50% increase in the variability of service rate causes a substantial 950.56% surge in the average waiting time). The findings underscore the need to control service rate fluctuations to reduce wait times and boost driver satisfaction. The improved NSGA-II algorithm shows 12.77% improvement in the Pareto front solutions. Finally, the proposed prioritization strategy based on the charging level of vehicles could reduce the average waiting time and cost compared to the first-come-first-served strategy.

A short-term wind-hydrothermal operational framework in the presence of pumped-hydro storage

Short-term operation is one of the most significant aspects of power systems operation and planning and it mainly includes resource scheduling including renewable and non-renewable energy sources. The mentioned problem is modeled as an optimization problem with one or more objectives depending upon the priorities of the decision maker (system operator) and several equality and inequality constraints. In this respect, this paper proposes a short-term operational model for the hydrothermal power systems in the presence of wind power generation and pumped-hydro storage (PHS) units. The problem is formulated as a single-objective optimization problem that aims to minimize the total operating cost including the fuel cost of thermal units. The problem has been comprehensively investigated through five case studies where the first case study is used for the sake of validation. Then, the other case studies assessed the impacts of wind power generation, PHS unit, and uncertainty on the problem. The five case studies are as follows: Case 1: Short-term hydrothermal (STH) scheduling which aims to minimize the total cost. Case 2: Deterministic short-term hydrothermal-wind (STHTW) scheduling which aims to minimize the total cost. Case 3: Stochastic wind-hydro-thermal scheduling that seeks to minimize the total cost taking into consideration the uncertainty of the wind power generation. Case 4: Deterministic wind-hydro-thermal scheduling in the presence of the PHS unit with the total cost minimization as the objective function. Case 5: Stochastic wind-hydro-thermal scheduling in the presence of the PHS unit that is used to evaluate the impact of uncertainty of wind power generation with the total cost minimization as the objective function. It is noteworthy that the first three case studies have been formulated using a non-linear programming (NLP) model and solved using the CONOPT solver in GAMS. Case studies 4 and 5 have been presented using a mixed-integer non-linear programming (MINLP) model and solved using the DICOPT solver in GAMS. A sensitivity analysis has also been carried out to assess the system's performance with different loading conditions. In this respect, the total cost and PHS operation with different amounts of load demand were evaluated and the results have been presented and discussed.

A novel approach to optimize an integrated network design and pricing of a healthcare supply chain

Adopting various pricing policies has been highly regarded in recent years for setting prices and increasing firms' profits. One of the most common steps in pricing is to identify costs. Since a significant part of costs is related to the corresponding supply chain, many researchers in different fields have used decision-making for simultaneous pricing and network design. However, there is no such approach in the field of healthcare. This paper tries to fill this gap by formulating a mixed-integer nonlinear bi-level programming model examining the interaction of hospitals and their medicines suppliers. At the upper level, there is a competitive market where a new firm (entrant) intends to enter the market and faces the challenge of pricing medicines and making network design decisions. At the lower level, there is a private hospital competing with a public hospital, and it also struggles with healthcare services pricing and supplier selection. A comprehensive utility function that considers healthcare services prices, quality, waiting time, health insurance, readmission rate, and referral rate is extended at this level. Three novel meta-heuristic algorithms are recommended, including bi-level, improved fruit fly, jellyfish optimization, and forensic-based investigation optimization algorithms to solve the presented complex mathematical problem.

Using Simplified Distflow-Based Mixed-Integer Quadratically Constrained Programming Formulation for Optimum Selection of Conductor Size in Electrical Distribution Networks

Determining conductor size is a subproblem that plays a vital role in the planning process of distribution systems. The problem of selecting the conductor cross-sectional area is usually made a model of mixed-integer non-linear programming (MINLP). It is important to note, however, that the MINLP model is not always capable of ensuring convergence to a solution that is globally optimum. In this research, a mixed-integer quadratically constrained programming (MIQCP) formulation is developed as a method for determining the conductor cross-sectional area in power distribution networks in an optimal manner. The goal function aims at minimizing the lifetime cost of lines, including initial capital cost together with operational expenses. The suggested optimization model’s constraints consist of power balance equations, thermal loading capacity of branches, nodal voltage restrictions, budget limitations for investment, and the need to have the same wire size in the main feeder. By using the simplified DistFlow approach for electrical distribution networks and precisely linearizing the product of a binary variable and a continuous variable, the MIQCP formulation is derived from the MINLP model. It is likely possible to solve the MIQCP formulation efficiently in the GAMS environment by using available commercial solvers like CPLEX. The developed MIQCP model is validated using three medium voltage distribution grids of IEEE 33 buses, IEEE 85 nodes, and the Vietnamese real 102 buses. The calculation results revealed the accuracy along with the efficacy of the proposed optimization procedure.

Fair and effective vaccine allocation during a pandemic

This paper presents a novel model for the Vaccine Allocation Problem (VAP), which aims to allocate the available vaccines to population locations over multiple periods during a pandemic. We model the disease progression and the impact of vaccination on the spread of the disease and mortality to minimise total expected mortality and location inequity in terms of mortality ratios under total vaccine supply and hospital and vaccination centre capacity limitations at the locations. The spread of the disease is modelled through an extension of the well-established Susceptible–Infected–Recovered (SIR) epidemiological model that accounts for multiple vaccine doses. The VAP is modelled as a nonlinear mixed-integer programming model and solved to optimality using the Gurobi solver. A set of scenarios with parameters regarding the COVID-19 pandemic in the UK over 12 weeks are constructed using a hypercube experimental design on varying disease spread, vaccine availability, hospital capacity, and vaccination capacity factors. The results indicate the statistical significance of vaccine availability and the parameters regarding the spread of the disease.

Influence of Load Models and DG Operational Power Factor on Optimization of Shunt Capacitor Placement in Distribution Grids Considering Voltage Stability

The paper proposes a mathematical model of mixed-integer nonlinear programming (MINLP) to optimize the location and size of shunt capacitors in power distribution grids with distributed generation (DG), considering voltage stability constraints. At the same time, the paper also analyzes the influence of the load models, such as voltage-dependence load (ZIP load) and constant power load, together with DG’s operating power factor on the optimal solution. The objective function of the proposed optimization formulation is minimizing the total expenses, including the capital investment of capacitors and the expense incurred by network energy loss. This MINLP model includes constraints in the current and critical loading conditions, such as the system of power flow equations, nodal voltage limits, branch thermal limits, capacitor-related constraints, and restrictions of minimum security level. The proposed optimization model was evaluated on a modified IEEE 33-node distribution grid using the KNITRO commercial solver with the GAMS programming language. The calculation results show that the voltage stability constraints, load models, and operational power factor of the DG units have an essential influence on the optimal position and capacity of the shunt capacitors.