Nonlinear Programming Model Research Articles

Applying the Sine-Cosine Optimization Algorithm to the Parametric Estimation Problem in Three-Phase Induction Motors

The steady-state analysis of electrical machines requires a detailed characterization of their equivalent electrical circuit, which adequately represents the transformation and interaction between electrical and mechanical energy. This research aims to characterize the equivalent circuit of three-phase induction motors by minimizing the mean square error between the measured and calculated torque variables. These torques are obtained from data provided by the manufacturer, including starting, peak, and full-load torques. A metaheuristic optimization technique is applied to solve the resulting nonlinear programming model based on the interactions between the sine and cosine functions. The numerical results obtained with this algorithm demonstrate its efficiency in terms of response quality, reaching objective function values of less than \(1\times10^{-8}\) with regard to the measured and calculated variables. Simulation results in two test systems allow concluding that the parametric estimation problem in three-phase induction motors is a multimodal optimization problem. This implies a potentially infinite set of solutions that minimize the root mean square error and adequately represent the behavior of the motor's output torque under various probable operating conditions.

The role of storage in energy security performance based on diversification and concentration for distributed energy systems

This work addresses the use of diversification and deconcentration strategies of primary sources of energy supply for the multi-objective design of distributed generation systems based on renewable sources. The Shannon–Wiener and Herfindahl–Hirschman indices are proposed as optimal design objectives. The utopia-tracking method is used to identify trade-offs between the total annual cost of the system and the efficiency of the storage system. The storage system is included in the assessment of diversification and deconcentration due to its functions as a backup unit in the event of adverse system operating conditions. Cases of interconnected and off-grid systems are addressed. The results show that, by including the storage system, an over-diversification of supply sources is generated and that, in the absence of various sources, as occurs in off-grid systems, storage generates sufficient flexibility to reach a trade-off among economic performance, diversification of supply sources and energy efficiency. The robust nonlinear programming model is fed with real data from a residential building.

A mathematical model and solution algorithms for optimising system-level configurations of reconfigurable single part flow lines

This study addresses a system-level configuration selection problem for reconfigurable single part flow lines (R-SPFLs) with parallel identical flexible machines at each stage. The problem is to determine the number of stages, the number of machines of a certain type at each stage and the assignment of operations to each stage in order to satisfy the demand of a part type. Precedence relations among the operations and space limitations are also considered. For the objective of minimising the sum of machine purchase and operation processing costs, a nonlinear integer programming model is developed that gives the optimal solutions. Then, due to the problem's complexity, a basic variable neighbourhood search algorithm is proposed that constructs an initial solution using a greedy-type heuristic and improves it using a shaking and a local search improvement methods. Moreover, the basic algorithm is extended to a general one that uses a variable neighbourhood descent method. Computational results show that the two algorithms outperform the previous one significantly in both solution quality and computation times, and the general algorithm improves the basic one significantly. Finally, the applicability of the variable neighbourhood search algorithms is shown by reporting a brief case study.

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.

An Investigation into the Pricing and Replenishment Strategy of Vegetable Commodities in Fresh Food Shops

Fresh supermarkets face challenges in marketing and inventory management due to the short shelf life and high loss rate of fresh vegetables. This necessitates the immediate development of a practical replenishment plan and pricing strategy. The paper commences by analyzing the sales distribution patterns of various vegetable categories, identifying leaves and flowers as having notably higher sales compared to other categories. Subsequent examination of the sales trend over time reveals variations in the sales of different categories or individual vegetables across months or quarterly periods, indicating a degree of seasonality. This study calculates the Spearman correlation coefficient for each category and individual product sales, presenting the results in a heat map. Findings indicate a significant positive correlation among all vegetable categories except for solanaceae, implying bundled sales or a consumer tendency to purchase specific vegetable categories concurrently. Consequently, when formulating replenishment and pricing strategies, considering a rational sales mix is crucial. The paper constructs a linear regression model, solving the equation using the least squares approach, and identifies a negative association between sales volume and cost-plus pricing. To accommodate seasonal patterns, the Prophet time series model forecasts wholesale prices for various vegetable categories during July 1-7, 2023. The objective is to maximize profit by accurately predicting vegetable costs. A nonlinear programming model is formulated to achieve this objective, determining the maximum revenue from days 1 to 7 as ￥(108953.65 - K), where K represents the fixed indirect cost. The model optimizes replenishment and pricing strategies for different categories.

A Drone-Driven Delivery Network Design for an On-Demand O2O Platform Considering Hazard Risks and Customer Heterogeneity

Nowadays, the online-to-offline (O2O) retailers provide on-demand delivery service for online orders by their own fleets and riders. An intelligent delivery network lays an important foundation to support cost-effective delivery service in the long run. Drones have great potential to revolutionize the instant delivery industry regarding cost and timeliness, while the hazard risks to humans and the environment should be seriously considered through sophisticated network design. In this paper, we propose a framework for a drone-driven intelligent delivery network design problem with the consideration of the multi-dimensional risk map, which needs to determine store location, drone fleet size and allocation, customer assignment, customer delivery mode selection, and delivery routing. A bi-objective non-linear programming model is formulated to maximize profit and minimize integrated risks as well. To tackle large instances, a modified NSGA-III algorithm is developed, which is incorporated with problem-specific search operators and Pareto local search to obtain Pareto solutions efficiently. Real-world data-based numerical experiments are conducted to verify the performance of the modified NSGA-III algorithm compared to the modified NSGA-II. A case study based on the geographical information in Shanghai is analyzed to validate the effectiveness of the proposed model. Moreover, sensitivity analysis is presented to evaluate the effects of multiple parameters on the drone delivery service network design. Some managerial insights are obtained for the O2O retailer who offers on-demand delivery service through online platform.

Replenishment Pricing of Vegetable Commodities in a Supermarket as an Example in China

In this paper, for the automatic pricing and replenishment decision-making problem of vegetable commodities, it is found that there are certain seasonal characteristics in the sales of vegetables in each category, and wavelet analysis is carried out on the daily sales data of single products to get the sales cycle of each category. For the vegetable single product with large amount of data, all single products are divided into four categories of common household vegetables, packaged vegetables, seasonal special vegetables and special vegetables by systematic cluster analysis, and the time distribution characteristics of single products are described by the sales time series of typical members in each cluster. Based on the market demand theory, the cost-plus pricing and sales volume data of the categories were fitted, and the relationship was obtained to be negatively correlated. Taking the purchase quantity and pricing of each category as the decision variables, maximizing revenue as the objective function, and taking the historical maximum purchase quantity of each category and the historical maximum purchase quantity of the superstore as the constraints, a nonlinear programming model was established to formulate the pricing and replenishment strategies for vegetable products. By solving the model, the optimal strategy of replenishment volume and pricing for the future seven-day category is derived, corresponding to the maximum profit of ￥3124.75.

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.

Configuration and Operation Optimization for Industrial Steam Power System Sustainable Retrofit Considering Renewable Energy Integration

AbstractRenewable energy integration and operational optimization are crucial in energy sustainability and decarbonization, especially for industrial steam power systems (SPS). This study establishes an SPS superstructure that integrates wind, solar, and biomass energy. A mixed‐integer nonlinear programming (MINLP) model is developed to determine an optimal retrofit strategy that minimizes the life cycle cost, which includes carbon emission cost and energy cost. To solve this high‐dimensional complex optimization problem, a multistage strategy fusion differential evolution algorithm with dynamic partitioning of the SVM feasible domain (SVM‐MS‐DE) is developed. The results from the optimal strategy demonstrate a 9.02% reduction in the system's total cost and a 9.64% decrease in carbon emission through the incorporation of wind and solar energy. Additionally, the sensitivity analysis on biomass ratio, carbon emission price, energy demand, and carbon emission limits reveal that the region can contribute significantly to renewable energy initiatives. Several recommended policies are provided to encourage enterprises to move towards sustainable development.

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.

Optimization model for multibeam sounding line deployment based on genetic algorithm

Multibeam sounding technology is a cutting-edge method used for measuring underwater topography, widely applied across various fields. The actual seafloor terrain often exhibits significant fluctuations, leading to challenges such as insufficient coverage in shallow areas and excessive overlap in deep-water zones, resulting in inefficient operations. Rational selection of line spacing helps improve measurement efficiency and ensures data integrity. In this paper, taking the 2023 National College Student Mathematical Modeling Competition Problem B as a background, we conduct a thorough analysis of the variation patterns of the coverage width of multibeam sounding strips with respect to various parameters. We establish models for ocean depth and coverage width, determining how the strip coverage width changes with distance and the angle between survey lines in different scenarios. Finally, employing genetic algorithms and differential methods, we identify decision variables, objective functions, and constraints to formulate a nonlinear programming model. Through iterative searches, we obtain an optimal sounding line deployment plan that maximizes coverage of the target sea area while keeping the overlap rate below 20%. The calculated missed measurement error is approximately 32.774%, and the overlap rate is around 10.477%, indicating a high level of model accuracy.

Collaborative vehicle-crew scheduling for multiple routes with a mixed fleet of electric and fuel buses

Since some fuel buses have not yet reached their retirement age, many cities still have a mix of fuel buses and electric buses (EBs) operating. This study focuses on multiple bus routes with fuel-electric mixed fleet. To address operation problems arising from separately scheduling vehicles and crews (known as the two-stage method) of each route, we propose a collaborative approach to optimize the vehicle dispatching plan and crew scheduling plan. Initially, we establish a nonlinear programming model by minimizing the overall operating costs of bus companies, encompassing vehicle purchase costs, EB charging costs, fuel bus refueling costs, and driver labor costs. Next, we devise an improved simulated annealing algorithm to solve the model. The proposed method is validated through a case study involving three real-world routes. A comparison with the traditional two-stage method reveals promising outcomes. Results indicate that the proposed method can decrease the number of vehicle changes by 21.43 %, driver labor costs by 2.04 %, and daily operating costs for the bus company by 0.82 %.

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.

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.

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.

Optimal resource allocation in hierarchical organizations under uncertainty: An interval-valued n-person cooperative game approach

In many real production scenarios, departmental organizations often exhibit a hierarchical structure, where departments cooperate with subordinate departments to optimize resource allocation and maximize their respective benefits. However, due to a lack of information or data, many model parameters in the allocation process cannot be precisely defined. In response to this challenge, an interval n-person hierarchical resource allocation model is proposed to achieve maximum economic benefit in uncertain environments. Based on the concepts of satisfactory degrees of comparing intervals and interval-valued cores of interval-valued n-person cooperative games, an auxiliary nonlinear programming model and method are developed to solve the interval-valued cores of such cooperative games. The approach explicitly considers the inclusion and/or overlap relations between intervals, whereas the traditional interval ranking method may not guarantee the existence of interval-valued cores. The proposed method offers cooperative opportunities under uncertain conditions. Finally, the feasibility and applicability of the models and methods are demonstrated through a numerical example and comparison with other methods.

Enhancing Electric Bus Charging Scheduling: An Energy-Integrated Dynamic Bus Replacement Strategy with Time-of-Use Pricing

Existing studies on electric bus (EB) scheduling mainly focus on the arrangement of bus charging at the bus terminals, which may lead to inflexible charging plans, high scheduling costs, and low utilization of electricity energy. To address these challenges, this paper proposes a dynamic bus replacement strategy. When the power of an in-service EB is insufficient, a standby EB stationed at nearby charging stations is dispatched in advance to replace this in-service EB at a designated bus stop. Passengers then transfer to the standby bus to complete their journey. The replaced bus proceeds to the charging station and transitions into a “standby bus” status after recharging. A mixed-integer nonlinear programming (MINLP) model is established to determine the dispatching plan for both standby and in-service EBs while also designing optimal charging schemes (i.e., the charging time, location, and the amount of charged power) for electric bus systems. Additionally, this study also incorporates the strategy of time-of-use electricity prices to mitigate the adverse impact on the power grid. The proposed model is linearized to the mixed-integer linear programming (MILP) model and efficiently solved by commercial solvers (e.g., GUROBI). The case study demonstrates that EBs with different energy levels can be dynamically assigned to different bus lines using bus replacement strategies, resulting in reduced electricity costs for EB systems without compromising on scheduling efficiency.