Mixed Integer Nonlinear Programming Model Research Articles

Efficient iterative optimization to real-time train regulation in urban rail transit networks combined with Benders decomposition method

Considering the rapid expansion of urban rail transit networks and increasing transfer demands, this paper investigates the design of real-time train regulation strategies for urban rail networks under disturbances. Specifically, a mixed-integer nonlinear programming (MINLP) model is constructed in a rolling horizon (RH) framework to reduce train deviations and passenger waiting times, which involves complex coupling relationships among network-wide non-transfer passengers, transfer passengers and train traffics of different lines. To effectively address the model complexity and adapt to the real-time nature of this issue, we proposed an efficient iterative optimization (IO) approach to split the original problem into a smaller-scale regulation issue and a computationally cheap passenger loading procedure, which can efficiently treat the introduced nonconvexity of the exact modelling scheme. Besides, tailored to the mixed-integer property of the model, the generalized Benders decomposition (GBD) technique is incorporated into our algorithm. Furthermore, to verify the effectiveness of our method, we describe the detailed implementation of a series of numerical experiments on a small-scale and a real-world case. Computational results show that our regulation method can effectively contribute to the reliability of train systems, improving operational efficiency and service level. Besides, it performs promising computational efficiency in dealing with large-scale real-world cases, applicable for real-time applications.

An optimal flow rate allocation model of the oilfield treated oil pipeline network

Oilfield treated oil pipeline network is the link connecting the upstream oilfields and the downstream refineries. Due to the differences in operating costs and transportation fee between different pipelines and the fluctuation in the demand and sales prices of the treated oil, there is an optimal flow allocation plan for the pipeline network to make the oilfield company obtain the highest social and economic benefit. In this study, a mixed integer nonlinear programming (MINLP) model is developed to determine the optimal flow rate allocation plan of the large-scale and complex treated oil pipeline network, and both the social and economic benefits are considered simultaneously. The optimization objective is the multi-objective which includes the largest user satisfaction and the highest economic benefit. The model constraints include the oilfield production capacity, refinery demand, pipeline transmission capacity, flow, pressure, and temperature of the node and station, and the pipeline hydraulic and thermal calculations. Python 3.7 is utilized for the programming of the off-line calculation procedure and the MINLP model, and GUROBI 9.0.2 is served as the MINLP solver. Moreover, the model is applied to a real treated oil pipeline network located in China, and three optimization scenarios are analyzed. For social benefit, the values of the user satisfaction of each refinery and the total network are 1 before and after optimization for scenarios 1, 2, and 3. For economic benefit, the annual revenue can be increased by 0.227, 0.293, and 0.548 billion yuan after the optimization in scenario 1, 2, and 3, respectively.

Optimising fleet sizing and management of shared automated vehicle (SAV) services: A mixed-integer programming approach integrating endogenous demand, congestion effects, and accept/reject mechanism impacts

Shared automated vehicles (SAV) are expected to benefit the sustainable development of urban regions and alleviate the negative impacts brought by the increasing number of private cars. In this paper, we envision a future scenario where non-pooled SAVs replace private cars and provide public on-demand mobility services to satisfy the mobility needs of a city’s residents. To help service providers make profitable fleet sizing and management decisions, we develop a mixed-integer non-linear programming model that considers the congestion effects and the mode choice of urban travellers in different income classes, between SAVs and bicycles. Our model optimises both strategic decisions (fleet size, initial fleet distribution, and service quality level) and operational decisions (trip assignment, vehicle routing, parking, and relocation). Travellers’ preference for both transport modes is described through a binary logit model and congestion effects are described by dynamically varying travel times with respect to traffic flow in a non-linear fashion. In addition, we investigate two types of accept/reject mechanisms (mandatory vs. non-mandatory acceptance) which lead to an endogenously determined acceptance rate that can affect travellers’ willingness to use SAV services. The computational challenge posed by the non-linear and non-convex nature of the model is addressed through reformulation and the use of outer-inner approximation methods combined with a breakpoint generation algorithm. We demonstrate the effectiveness of our proposed method in a case study of the city of Delft in The Netherlands, as well as a scaling analysis on three toy networks with various sizes and demand profiles. A sensitivity analysis of key parameters is carried out to assess system performance. Computational results indicate that fleet sizing decisions are influenced not only by the population’s geographical distribution and land use patterns but also by the pricing strategy, unit operating costs of the SAV fleet, network congestion level, and traveller behaviour. When the price rate of using SAVs is low, the fleet sizing decisions can also be influenced by the trip accept/reject mechanism and the travellers’ sensitivity to the service quality level. In addition, a low price of SAV service will attract more users but may not necessarily bring a higher profit because of the increased traffic congestion.

Multiperiod optimization of cooling water system with flexible topology network

Cooling water systems (CWSs) are extensively utilized in various industries to eliminate the excess heat and converse energy. Studies on CWSs mainly concentrated on finding the optimal cooler network structure. In addition, some works also considered the optimal design under varied operation conditions. However, in these works, once the optimal design of the cooler's network is determined, its arrangement remains fixed and cannot be adapted to accommodate diverse operating conditions. In this work, a flexible topology network concept is proposed to make the adjustment of network structure possible under different operation conditions. The CWS with integrated air cooler and flexible topology network has better overall performance, represented by a mixed integer nonlinear programming (MINLP) model that require advanced tools such as GAMS software. Case studies revealed that the proposed methodology can realize better energy-saving performance, and improve the economic performance under varied operation conditions. The impact of critical flexible nodes on system configuration and economy is achieved by sensitivity analysis.

Modeling and Optimization of Parallel Disassembly Line Balancing Problem With Parallel Workstations

To reasonably arrange disassembly facilities and plan enterprise space, we propose a parallel disassembly line balancing problem (PW-PDLBP) with parallel workstations. Additionally, a mixed-integer non-linear programming (MINLP) model that minimizes the line length, number of workstations, idle time balancing index, and energy consumption is established based on the problem characteristics and is solved using the GUROBI optimizer. Furthermore, a multi-objective enhanced differential evolution algorithm (MEDE) is developed to obtain high-quality disassembly schemes for PW-PDLBP. The correctness of encoding and decoding and the solving performance of MEDE are verified by comparing with the MINLP model and four existing algorithms. Then, an instance consisting of two different types of end-of-life TVs is optimized. Finally, the effectiveness of PW-PDLBP in improving enterprise space utilization is validated by comparing it with the parallel line layout without parallel workstations.

A two-stage optimization method for improving the load flexibility of existing district energy systems

Decarbonization of district energy systems is essential for China to meet its carbon neutrality goal by 2060. Most existing district energy systems are missing historical load data and have incomplete information, resulting in a lack of data support for the low-carbon transition. Moreover, demand-side load flexibility has not been fully exploited in the planning stage. In this paper, we developed a two-stage computational approach to optimize district loads. We first established an integrated Python framework, incorporating the TEASER simulation tool and AixLib model library, to efficiently calculate baseline loads through the bottom-up modeling and simulation of district buildings. Then, a price-based integrated demand response strategy was introduced. A mixed-integer nonlinear programming model was formulated to optimize the energy pricing strategy with the objective of minimum load fluctuations. Finally, a case study was employed to illustrate the feasibility of the calculation method, showing a normalized mean bias error of 7.17%. The results further demonstrated that the strategy could reduce the peak electric and heat loads by 3.55% and 9.57%, and increase load rates by 3.85% and 9.48%, respectively. The strategy could assist district energy service providers to optimize equipment capacity configuration and enhance the low-carbon planning potential of energy systems from the demand-side.

A Multi-Objective Mathematical Programming Model for Transit Network Design and Frequency Setting Problem

In this study, we propose a novel multi-objective nonlinear mixed-integer mathematical programming model for the transit network design and frequency setting problem that aims at designing the routes and determining the frequencies of the routes to satisfy passenger demand in a transit network. The proposed model incorporates the features of real-life transit network systems and reflects the views of both passengers and the transit agency by considering the in-vehicle travel time, transfers, waiting times at the boarding and transfer stops, overcrowding and under-utilization of vehicles, and vehicle fleet size. Unlike previous studies that simplify several aspects of the transit network design and frequency setting problem, the proposed model is the first to determine routes and their frequencies simultaneously from scratch, i.e., without using line and frequency pools while considering the aforementioned issues, such as transfers and waiting. We solve the proposed model using Gurobi. We provide the results of what-if analyses conducted using a real-world public bus transport network in the city of Kayseri in Türkiye. We also present the results of computational tests implemented to validate and verify the model using Mandl benchmark instances from the literature. The results indicate that the model produces better solutions than the state-of-the-art algorithms in the literature and that the model can be used by public transit planners as a decision aid.

A Multi-Period Model of Compressor Scheme Optimization for the Shale Gas Gathering and Transportation System

In the process of shale gas production, with the change in gas productive parameters, the pressurization demand for the shale gas gathering and transportation system (SGGTS) also changes, which affects the choice of pressurizing location and timing. Our purpose is to effectively respond to the impact of parameter changes during shale gas production and to better select the pressurization schemes. Therefore, we considered the modularization of the compressors and established a mixed-integer nonlinear programming (MINLP) model to minimize the total cost of the SGGTS. Taking an actual shale gas field as an example, by discretizing the time during a given production period to solve the model under multi-period and single-period conditions, the optimal pressurization scheme for the SGGTS in the specified production period is obtained. It indicates that the results obtained under a multi-period condition are more conducive to actual production. Compared with the results obtained under the single-period condition, the cumulative cost obtained in the multi-period condition is reduced by 17.19%. By deploying the MINLP model in the specified production period, the pressurization demand is met in each time period. This greatly improves the utilization rate of modular compressors, reduces the total cost, and improves the economic benefits of the SGGTS.

Long-term operation optimization of circulating cooling water systems under fouling conditions

Fouling caused by excess metal ions in hard water can negatively impact the performance of the circulating water system (CCWS) by depositing ions on the heat exchanger's surface. Currently, the operation optimization of CCWS often prioritizes short-term flow velocity optimization for minimizing power consumption, without considering fouling. However, low flow velocity promotes fouling. Therefore, it’s crucial to balance fouling and energy/water conservation for optimal CCWS long-term operation. This study proposes a mixed-integer nonlinear programming (MINLP) model to achieve this goal. The model considers fouling in the pipeline, dynamic concentration cycle, and variable frequency drive to optimize the synergy between heat transfer, pressure drop, and fouling. By optimizing the concentration cycle of the CCWS, water conservation and fouling control can be achieved. The model can obtain the optimal operating parameters for different operation intervals, including the number of pumps, frequency, and valve local resistance coefficient. Sensitivity experiments on cycle and environmental temperature reveal that as the cycle increases, the marginal benefits of energy/water conservation decrease. In periods with minimal impact on fouling rate, energy/water conservation can be achieved by increasing the cycle while maintaining a low fouling rate. Overall, the proposed model has significant energy/water saving effects and can comprehensively optimize the CCWS through its incorporation of fouling and cycle optimization.

Decomposition and approximate dynamic programming approach to optimization of train timetable and skip-stop plan for metro networks

Carefully coordinating train timetables of different operating lines can help reduce transfer delays, which in turn reduces station crowding and improves overall service quality. This paper explores the optimization to train timetable and skip-stop plans that aims to minimize the total passenger waiting time and station crowding. The problem is formulated as a mixed-integer non-linear programming model. To effectively address the complexity of the model, a decomposition and approximate dynamic programming approach is designed to convert the original network-level problem into a series of small-scale subproblems, one for each operating line, to be solved quickly in a distributed manner. The effectiveness and practicability of the model and algorithm are demonstrated on two case networks: a small-scale synthetic network of three metro lines and a real-world network based on Beijing metro. The computational results illustrate that the proposed strategy to generate train timetables and skip-stop plans can effectively reduce passenger waiting time and station crowing. The proposed decomposition and approximate dynamic programming approach is also shown to perform more efficiently than traditional heuristic algorithms, such as genetic algorithm and simulated annealing algorithm for large-scale networks.

Optimal and secure peer-to-peer carbon emission trading: A game theory informed framework on blockchain

We address two key challenges towards an effective and accountable emission trading system (ETS). The first is to determine the individual policies while ensuring a fair allocation of costs between the stakeholders to meet an overall environmental target for an industrial ecosystem. The second is to ensure privacy and traceability of shared data when participating in an ETS. Specifically, we propose a mixed-integer nonlinear program (MINLP) for peer-to-peer carbon trading in ETS under fixed carbon tax or cap-and-trade systems. To ensure data privacy and traceability, the MINLP model is integrated with smart contracts on the Ethereum blockchain. The model determines how much the participants buy/sell emission allocations, how much they invest in carbon capture, which capture technologies they select, and how they respond to changes in carbon prices. The planning requires simultaneous decision-making for all participants, which can be obtained as a Generalized Nash Equilibrium (GNE).

Real-time operational optimization for flexible multi-energy complementary integrated energy systems

Multi-energy complementary integrated energy system (MCIES) has gained widespread attentions due to its utilization of diverse energy sources, enhancing energy efficiency, and reducing carbon emissions. The scheduling optimization of MCIESs needs to address multiple factors, including solar/wind energy potential, fluctuating load demand, and prediction errors, to maintain its highly flexible and decarbonized operation. This study presents a scheduling optimization model based on a mixed integer nonlinear programming model (MINLP) with a novel flexibility index as the objective function. A case study of a MCIES in a swimming pool in Changsha is provided. The results demonstrate that real-time operational scheduling considering flexibility enhances the system flexibility of the MCIESs from 0.608 to 0.623 comparing with the day-ahead scheduling, maintaining superior performance across multiple dimensions. The coupling between the power grid and the natural gas network is also improved, significantly reducing the carbon dioxide emissions of the MCIES by 127.67kgCO2 per day. Furthermore, the study reveals that the flexibility-aware real-time operational scheduling for MCIES outperforms day-ahead scheduling in responding to weather and equipment failures, sustaining high flexibility operation, economic viability, and notably improved environmental performance. CO2 emissions are reduced by 21.088% and 14.638%, respectively. The proposed operational optimization approach in this study can enhance the flexibility and coordinated operation of MCIESs, leading to reduced carbon emissions.

Integration of hybrid renewable energy sources with the power system considering their economic complementarity

AbstractRenewable energy sources play an important role in providing clean energy for future electricity networks. As the penetration level of these resources grows, their integration with the grid will be more challenging. Each renewable energy source has different inherent characteristics that, if appropriately used in the generation mix, could complement each other and create many technical and economic benefits for the power system. Usually, renewable energy resource complementarity studies are carried out with the objectives of smooth effect, reducing the need for storage and load tracking. In this study, the economic complementarity approach is introduced with the help of a Mixed integer nonlinear programming (MINLP) model. This approach can integrate renewable and storage energy sources with the grid and determine the optimal capacity of these resources in complementary used mode. The results show that the proposed method always imposes the system's lowest annual operation and investment costs.

Designing a two-stage model for the resilient agri-food supply chain network under dynamic competition

PurposeSupply chain risk management can effectively reduce the loss of retailers. In this regard, retailers need to consider the competition risks of competitors in addition to the disruption risks. This paper designs a resilient retail supply chain network for perishable foods under the dynamic competition to maximize retailer's profits.Design/methodology/approachA two-stage mixed-integer non-linear model is presented for designing the supply chain network. In the first stage, an equilibrium model that considers the characteristics of perishable foods is developed. In the second stage, a mixed integer non-linear programming model is presented to deal with the strategic decisions. Finally, an efficient memetic algorithm is designed to deal with large-scale problems.FindingsThe optimal the selection of suppliers, distribution centers and the order allocation are found among the supply chain entities. Considering the perishability of agri-food products, the equilibrium retail price and selling quantity are determined. Through a numerical example, the optimal inventory period under different maximum shelf life and the impact of three resilient strategies on retailer's profit, selling price and selling quantity are analyzed.Research limitations/implicationsAs for future research, the research can be extended in a number of directions. First, this paper studies the retail supply chain network design problem under competition among retailers. It can be an interesting direction to consider retailers competing with suppliers. Second, the authors can try to linearize the non-linear model and solve the large-scale integer programming problem by exact algorithm. Finally, the freshness of perishable foods gradually declines linearly to zero as the maximum shelf life approaches, and it would be a meaningful attempt to consider the freshness of perishable foods declines exponentially.Originality/valueThis paper innovatively designs the resilient supply chain network for perishable foods under dynamic competition. The retailer's dynamic competition and resilient strategies are considered simultaneously when designing supply chain network for perishable foods. In addition, this paper gives insights into how to obtain the optimal inventory period and compare the retailer's resilient strategies.

Risk-based distributionally robust optimal dispatch for multiple cascading failures in regional integrated energy system using surrogate modeling

Regional integrated energy system (RIES) with the close coupling of multiple energies faces various cascading failure risks in its operation, and formulating the risk-based optimal dispatch scheme is important to ensure the secure and economic operation of the RIES. Considering that the occurrence probability of each subsequent failure path caused by the initial failure in the cascading failure chain varies with the change of dispatch scheme, the cooling/heating and gas pipeline dynamics and the uncertainty of renewable energy, a risk-based distributionally robust optimal dispatch model for multiple cascading failures in a RIES is proposed. A kernel ambiguity set is constructed by using the maximum mean discrepancy between the real and reference probability distributions and the kernel mean embedding technology, which can obtain the more accurate worst-case probability distribution. Moreover, a surrogate modeling method based on piecewise convex quadratic functions is proposed to learn the mapping between the cascading failure risks and the decision variables, and the original large-scale mixed integer nonlinear programming model is transformed into a small-scale mixed integer quadratic constrained programming model for efficient solution. Finally, case study on an actual RIES demonstrates the effectiveness and advantages of the proposed method. Compared to optimal dispatch without considering the cascading failure risks, the proposed method can reduce the total failure risk cost by nearly 60%.

Replacing DMF with ionic liquid in isoprene/n-pentane separation in C5 plants: Solvent design, process optimization, and industrial-scale assessment

In the petrochemical industries, separating isoprene and n-pentane in carbon-five plants has received significant interest because of relatively high energy consumption and carbon emissions. To address these issues, an energy-efficient isoprene/n-pentane extractive distillation (ED) separation process is designed, incorporating a tailor-made ionic liquid (IL) using the computer-aided ionic liquid design (CAILD) model. By addressing a mixed-integer nonlinear programming (MINLP) model based on CAILD, three optimal IL solvents, namely [MPy][TFA], [C1MPy][TFA], and [C2MPy][TFA], from 584,779 structurally potential IL solvents are identified. The performance of these ILs in isoprene/n-pentane separation in C5 plants is evaluated using rigorous process simulations in Aspen Plus. Subsequently, a comprehensive analysis is conducted, encompassing energy, environmental, and economic evaluations of the industrial-scale separation of isoprene/n-pentane. Notably, the [MPy][TFA]-based separation process exhibits the highest improvements in energy efficiency (69.8%), environmental sustainability (57.7%), and economic viability (30.0%) when compared to the conventional DMF-based separation process commonly used in industries. These findings underscore the significant potential of ILs for sustainable research in the field of isoprene and n-pentane separation.

A multi-product and multi-period supply chain network design problem with price-sensitive demand and incremental quantity discount

Demand is an important factor in supply chain network design (SCND). Among the many factors that affect demand, the impact of price cannot be underestimated. This study, for the first time, provides enterprises with direct multi-period pricing and demand decision solutions in SCND by simultaneously considering the impact of price-sensitive demand and incremental quantity discounts on the network. In this way, a mixed-integer nonlinear programming model is established based on the relationships between price and demand functions to maximize the overall profit of the supply chain. Due to the nonlinear characteristics of the model, this study proves the existence of the optimal solution of the problem and the feasibility of implementing the following two algorithms by analyzing the nature of the problem. The first algorithm is based on the second-order cone programming (SOCP) method, and the second algorithm is based on the outer-approximation method. The results of numerical experiments at different scales show that the SOCP method can obtain the optimal solution for small and medium-scale experiments. In contrast, the outer-approximation method can find approximate optimal solutions within a reasonable time for large-scale experiments. The results confirm that considering incremental quantity discount can significantly enhance the profitability of supply chain networks in long-term planning. Finally, some future research directions in the field of SCND are discussed.

Optimization Model and Application for Agricultural Machinery Systems Based on Timeliness Losses of Multiple Operations

Present agricultural practices confront issues such as mismatches between tractors and implements, imprecise machinery allocation, and excessive machinery investment. Optimization of agricultural machinery systems was a potent remedy for these concerns. To address inaccuracies in calculating objective functions and the incompleteness of constraints in existing models for agricultural machinery system optimization, a comprehensive mixed integer nonlinear programming (MINP) model for agricultural machinery system optimization was established. The model introduced timeliness loss costs for multiple key operations across various crops into the objective function, and constraints were enhanced by including operation sequence constraints and boundary constraints on initiation and completion dates of those key operations. Taking corn and soybeans as examples, timeliness loss functions of sowing and harvesting operations were derived through experiments. Solving the MINP model by Lingo (V.14.0) software, improvements in total power, workload per unit power, and total operational costs were shown when comparing the optimized machinery system through the MINP model against current systems. When the model omitted considerations for timeliness loss functions and operation sequence constraints, issues arose including an increase in total operational costs and an inversion of operation sequence. The model’s application in devising machinery allocation plans for production units of various operational scales revealed a gradual decrease in total power and costs per unit area with expanding scale, approaching stability when scale exceeded 1600 hm2. This study enriches theory and methodology for optimizing agricultural machinery systems, provides theoretical and technological underpinnings for rational machinery acquisition, and promotes the high-quality progression of comprehensive agricultural mechanization.

Towards efficient airline disruption recovery with reinforcement learning

Disruptions to airline schedules precipitate flight delays/cancellations and significant losses for airline operations. The goal of the integrated airline recovery problem is to develop an operational tool that provides the airline with an instant and cost-effective solution concerning aircraft, crew members and passengers in face of the emerging disruptions. In this paper, we formulate a decision recommendation framework which incorporates various recovery decisions including aircraft and crew rerouting, passenger reaccommodation, departure holding, flight cancellation and cruise speed control. Given the computational hardness of solving the mixed-integer nonlinear programming (MINP) model by the commercial solver (e.g., CPLEX), we establish a novel solution framework by incorporating Deep Reinforcement Learning (DRL) to the Variable Neighborhood Search (VNS) algorithm with well-designed neighborhood structures and state evaluator. We utilize Proximal Policy Optimization (PPO) to train the stochastic policy exploited to select neighborhood operations given the current state throughout the Markov Decision Process (MDP). Experimental results show that the objective value generated by our approach is within a 1.5% gap with respect to the optimal/close-to-optimal objective of the CPLEX solver for the small-scale instances, with significant improvement regarding runtime. The pre-trained DRL agent can leverage features/weights obtained from the training process to accelerate the arrival of objective convergence and further improve solution quality, which exhibits the potential of achieving Transfer Learning (TL). Given the inherent intractability of the problem on practical size instances, we propose a method to control the size of the DRL agent’s action space to allow for efficient training process. We believe our study contributes to the efforts of airlines in seeking efficient and cost-effective recovery solutions.

Optimization of system resilience in urban rail systems: Train rescheduling considering congestions of stations

In urban rail systems, unexpected events, due to equipment breakdown, extreme weather, etc., frequently cause the delay of trains and overcrowding of platforms, significantly affecting the service quality to passengers. Therefore, improving system resilience to recover to normal conditions more quickly has become a critical problem in the management of urban rail systems. In contrast to most previous studies that focused only on train rescheduling, our paper proposes optimizing the resilience of urban rail systems by considering the influence of delayed trains and overcrowded passenger flows. Specifically, we define a resilience-oriented quantitative evaluation index. Using the evaluation index as the objective function, we construct a mixed-integer nonlinear programming model. In our formulation, we consider a practical case in which the boarding and alighting of passengers at overcrowded stations may further delay trains. To solve the model, we develop a two-phase iterative optimization approach, in which phase 1 focuses on the adjustment of trains to enhance system resilience, and phase 2 evaluates the impact of passengers on the adjusted train schedule. Through the comparison of three different rescheduling methods, we prove the validity of the derived inequalities and our approach balances the computational time and system performance. Case studies are conducted on the Beijing Metro Line 1 to verify the effectiveness of the proposed approach. The results demonstrate that system resilience can be improved by as much as 39.7% using our optimization approach.