Mixed-integer Nonlinear Optimization Model Research Articles

Second‐order cone programming models for the unitary weighted Weber problem and for the minimum sum of the squares clustering problem

AbstractIn this work, new mixed integer nonlinear optimization models are proposed for two clustering problems: the unitary weighted Weber problem and the minimum sum of squares clustering. The proposed formulations are convex quadratic models with linear and second‐order cone constraints that can be efficiently solved by interior point algorithms. Their continuous relaxation is convex and differentiable. The numerical experiments show the proposed models are more efficient than some classical models for these problems known in the literature.

Optimizing tensile strength and energy consumption for FDM through Mixed-Integer Nonlinear Multi-objective optimization and design of experiments

This study presents a methodology for optimizing key parameters of a fused deposition modeling (FDM) printer to minimize energy consumption (EC) while exceeding a specified tensile strength (TS) threshold. Employing Design of Experiments (DoE) with Taguchi and Response Surface analysis, we identify influential parameters affecting TS and EC. A Mixed-Integer Nonlinear Multi-Objective Optimization model is then utilized to balance TS and EC, resulting in optimal parameter values. Validation using fabricated specimens demonstrates less than 5 % error in Tensile Strength and less than 2 % error in Energy Consumption, confirming the efficacy of the proposed methodology.

Optimizing large-scale hydrogen storage: A novel hybrid genetic algorithm approach for efficient pipeline network design

As hydrogen production technologies continue to advance, efficient and high-capacity hydrogen storage solutions become increasingly crucial. To address the complex optimization challenges in the design of large-scale hydrogen storage pipeline networks, the study introduces a mixed integer nonlinear optimization model inspired by hydrogen pipeline design optimization theory. The model not only focuses on optimizing network design parameters but also aims to minimize investment costs. To achieve this, the study proposes an innovative hybrid genetic algorithm that combines the Modified Feasible Directions Method (MFDM) and Genetic Algorithm Theory (GAT), specifically targeting the optimization of pipeline diameter's discrete distribution challenges. Comparative analyses with SUMT and GA algorithms demonstrate the superiority of the approach. In continuous space, the algorithm achieves a lower convergence cost of 232.4437 × 104 Yuan, while in discrete space, the cost further decreases to 212.1636 × 104 Yuan. Additionally, the study delves into the sensitivity of the HGA-MFDM algorithm to ambient temperature and wellhead temperature in hydrogen storage facilities. The analysis reveals that wellhead temperature has a more significant impact on pipeline network investment, whereas ambient temperature exhibits a more pronounced effect on iteration curves. The findings provide valuable insights for the precise adjustment and optimization of hydrogen storage pipeline networks in practical applications.

Exploring optimal pathways of the high-CO2 content natural gas source to chemicals and fuels using superstructure multi-objective optimization

This study aims to develop a mathematical formulation of superstructure multi-objective optimization to find optimal pathways of high-CO2 natural gas and PV-hydrogen to chemicals and fuels integrated with CO2 sequestration. A mixed-integer nonlinear programming optimization model is used to compete along 56 pathways in the superstructure optimization network. The model was developed with the general algebraic modeling system and computed with the standard branch and bound (SBB) solver to maximize annual net profit and minimize annual net CO2 emissions. The result shows the selected products in 2020 are liquefied natural gas, methanol, dimethyl ether, formic acid, and acetic acid, with annual net profits and net CO2 emissions of 38.17 million $/year and −8.47 million tons of CO2/year, respectively. In the projection analysis, the annual net profit growth results at a compounding annual growth rate of 7.85% per year between 2020 and 2060, while annual net CO2 emissions decrease until −16.37 million tons of CO2/year in 2060. In 2030, blue acetic acid will be selected via all separation processes. For 2050, the synfuel via partial membrane separation is selected. The other products, formic acid, methanol, and acetic acid via membrane separation (full and partial), will always be selected between 2020 and 2060. Therefore, this could be a future strategy for monetizing high–CO2–content natural gas sources with low-carbon process development.

Learning and forgetting interactions within a collaborative human-centric manufacturing network

Learning and forgetting (LaF) phenomena are characteristic of labor-intensive production and service industries. To mitigate the effects of LaF in a human-centric manufacturing system integrated with outsourcing, managers need to coordinate their decisions with partners for assigning operations and scheduling processes following a hierarchy. A model that addresses this should consider the expected latency of various tasks across assignments and production sequences and similarities among jobs as that affects learning. This paper develops a novel bi-level LaF model to help determine the leader-follower decisions in a decentralized network. It models the learning concept as a factor of task execution order and task variety. The mixed-integer non-linear optimization model determines the best order coordination and scheduling scheme by minimizing the processing, operating, and holding costs and penalties for missing deadlines. This study also develops an efficient column-and-constraint generation algorithm based on the duplication method, which enables solving bi-level models in which the lower-level model includes integer variables. This study also provides an illustrative real-sized example to validate the model and prove the efficiency of our resolution method. The results indicate that adopting compromise solutions enables preoccupied workers to be released earlier than expected, reducing the costs associated with learning and forgetting (due to latency). Despite the effects of LaF and the decentralized structure of the supply chain, which includes rising network costs, the schedules become more precise, and the cost balance among actors effectively increases.

A deterministic bounding algorithm vs. a hybrid meta-heuristic to deal with a bilevel mixed-integer nonlinear optimization model for electricity dynamic pricing

In the electricity retail market, the retailer company aims to determine the optimal time-of-use (ToU) prices to maximize profits resulting from buying energy in organized (long-term, day-ahead, balancing) markets and selling it to consumers. Therefore, the retailer should take into account the consumer’s demand response actions to minimize the electricity bill in face of time varying prices. In this paper, this problem is formulated as a bilevel mixed-integer nonlinear programming model in which the retailer is the leader and the consumer is the follower. The consumer’s problem encompasses the integrated optimization of all home energy resources, considering re-scheduling appliance operation, charging/discharging of electric vehicle and stationary batteries, local microgeneration and (buying and selling) exchanges with the grid. The accurate physical modelling of appliance operation to generate effective load scheduling solutions imposes a high computational burden. Two algorithms are proposed to address this problem: a deterministic bounding algorithm (DBA) using an optimal-value-function approach for bilevel optimization, and a hybrid meta-heuristic using a particle swarm optimization algorithm to tackle the upper-level problem that calls an exact mixed-integer linear programming solver to deal with the lower-level problem. In the framework of the DBA, three different techniques were implemented to deal with the nonlinearities arising from the products of integer and continuous variables (bilinear terms): 1) solving the (non-convex) subproblems of DBA using a mixed-integer nonlinear solver, 2) using the McCormick envelopes to approximate the bilinear terms by linear ones, and 3) expressing the integer variables by binary ones and linearizing the bilinear terms using an exact form. Computational experiments are presented and discussed for real data settings of the problem under study considering a computational budget to compare the different algorithms and techniques employed. The results showed that the DBA with the approximate linearization technique (2) leads to the best solutions.

Coordinated Control of Electric Vehicles and PV Resources in an Unbalanced Power Distribution System

Improving air quality, reducing greenhouse gas emissions, and achieving independence from fossil fuels have led most countries towards deploying solar photovoltaics (PV) in the power distribution grid and electrifying the transportation fleet. Internal combustion engine (ICE) vehicles are, in particular, one of the main culprits of injecting greenhouse gas emissions into the atmosphere, making electric vehicles (EVs) an important tool in combating climate change. Despite their considerable environmental and economic benefits, the integration of PVs and EVs can introduce unique operational challenges for the power distribution grid. If not coordinated, high penetration of PVs and EVs can result in variety of power quality issues, such as instances of overvoltage and undervoltage, frequency fluctuations, and/or increased losses. This paper proposes a mixed-integer multi-objective nonlinear optimization model for optimal energy dispatch in a power distribution grid with high penetration of PV and EV resources. The model proposed here is an extension of the traditional voltage and var optimization (VVO) into a comprehensive and coordinated control of voltage, active power, and reactive power. A modified version of the IEEE 123-bus test distribution system is used to demonstrate the effectiveness of the proposed solution.

A versatile optimization framework for sustainable post-disaster building reconstruction

This paper proposes an optimization framework for sustainable post-disaster building reconstruction. Based on mathematical optimization, it is intended to provide decision makers with a versatile tool to optimize building designs and to explore the trade-off between costs and environmental impact (represented by embodied energy) of alternative building materials. The mixed-integer nonlinear optimization model includes an analytical building model that considers structural and safety constraints and incorporates regional building codes. Using multi-objective optimization concepts, Pareto-optimal designs are computed that represent the best trade-off designs from which a decision maker can choose when they take additional criteria into consideration. As a case study, we consider the design of a multi-room one-story masonry building in Nepal. We demonstrate how the framework can be employed to address a variety of questions, such as the optimal building design and material selection, the sensitivity of the decision to material prices, and the impact of regional safety regulation thresholds.

Simultaneous identification of the number, location and release intensity of groundwater contamination sources based on simulation optimization and ensemble surrogate model

Abstract In most instances, the number, location and release intensity of groundwater contamination sources (GCSs) are all unknown. The 0-1 mixed integer nonlinear programming optimization model (0-1 MINPOM) used previously could only identify the location and release intensity for GCSs. Thus a 0-1 MINPOM was improved and applied to simultaneously identify the number, location and release history of GCSs. In addition, to reduce the time of calling the simulation model for iterative calculation, the deep belief neural network (DBNN), long short-term memory network (LSTM) and Kriging method were applied to establish an ensemble surrogate model of the simulation model to replace the simulation model for iterative calculation. The results show that compared with the three single surrogate models, the ensemble surrogate model has the highest approximation accuracy to the simulation model, and could save 99% of the calculation time when iterative calculation was performed in place of the simulation model. The improved 0-1 MINPOM could identify the number, location, and release history of GCSs simultaneously. The accuracy of identifying the number of GCSs was 100%, and the accuracies of identifying the locations and release history were above 95.12% and 83.51%, respectively.

Cost-based site and capacity optimization of multi-energy storage system in the regional integrated energy networks

The unbalance between the renewable energy sources and user loads reduces the performance improvement of regional integrated energy systems (RIES), in which the multi-energy storage system with battery and heat tank is necessarily integrated. This paper aims to optimize the sites and capacities of multi-energy storage systems in the RIES. A RIES model including renewable wind power, power distribution network, district heating network, multi-energy storage system, and heat pump to convert electricity to heat is constructed. An optimization method combining a mixed-integer nonlinear programming optimization model is proposed to minimize the comprehensive cost of RIES. The second-order cone relaxation method performs convex relaxation of the power flow constraints, and the district heat network is operated under a constant-flow-variable-temperature strategy. The original optimization model is transformed into a mixed-integer second-order cone programming problem to solve. Three solution methods, including enumeration method, improved enumeration method and genetic algorithm, are employed to determine the optimum installation locations, and they are compared in computation time and efficiency. The results demonstrate that the distributed hybrid integration of electricity and thermal energy storages improves the RIES economy by collaborating the power and heat balances and reducing the purchased electricity from the grid. By integrating the hybrid storage system, the curtailed wind is reduced by 53.9%, and the economy is improved by 13.4%.

A hybrid optimization method to design a sustainable resilient supply chain in a perishable food industry.

To integrate the location, inventory, and routing (LIR) problems arising in designing a resilient sustainable perishable food supply network (RSPFSN), a bi-objective optimization model is developed. To improve the resiliency and sustainability of the RSPFSN, a dynamic pricing strategy is used to cope with the disrupting events, along with minimizing the total cost and CO2 emission of the whole network. One of the important features of the proposed model is taking into account the effects of route disruptions and traffic conditions on the deterioration of products. To solve the mixed-integer nonlinear bi-objective optimization model, a novel hybrid method is developed using the Heuristic Multi-Choice Goal Programming and Utility Function Genetics Algorithm (HMCGP-UFGA). To improve resiliency, the dynamic pricing strategy, considering the traffic condition, can lead to around a 20% improvement in both cost and CO2 emission, based on the results of our case study in a dairy supply chain. Besides, the results of sensitivity analysis display the high flexibility of the proposed approach for various problems.

Emergency resources scheduling in distribution system: From cyber-physical-social system perspective

With the frequently happened aging electrical grid and extreme weather events, outage management and service restoration face the challenges of flexible scheduling multiple resources for different kinds of objectives. Previous research mainly focuses on the single restoration objective or single emergency resource management. This paper proposes a new emergency scheduling framework under the cyber-physical-social system (CPSS) perspective, which includes the scheduling of multiple components from CPSS, i.e., crew, mobile distributed generators, and resources. Under the framework, a novel co-scheduling model with the goal of minimizing time-consuming, cost, and shedding load are also built as a mixed-integer non-linear optimization model. Then we formulate this problem as advanced multiple salesmen traveling problem with a routing stage and a scheduling stage, and design a solution algorithm based on the genetic algorithm to solve this problem. In the solving algorithm, the optimal routing solutions are got at first, then the emergency scheduling solutions are optimized in the second stage based on the routing results. Finally, a numerical case study on the IEEE 123-bus network shows the effectiveness of the co-optimization process, which can provide a routing solution for each depot, and the corresponding optimization objective results will increase 25.75% and 73.92% when compared with time-consuming optimization and load-shedding optimization.

Joint Optimization of Ticket Pricing Strategy and Train Stop Plan for High-Speed Railway: A Case Study

In this study, we examined ticket pricing and train stop planning for the high-speed railway (HSR), which integrates two key aspects of railway operation and organization. We considered that passenger demand is sensitive to the generalized travel cost (depending on the ticket price and the travel time) and that the train stop plan can affect the travel time and passenger distribution. Then, a mixed-integer non-linear optimization model was proposed for the joint problem of ticket pricing and train stop planning to maximize HSR’s transport revenue and minimize passengers’ travel time. Based on the high similarity between combinatorial optimization problems and the solid annealing principle, we designed a combined simulated annealing (CSA) algorithm to solve practical problems. The results of a numerical example in the real HSR network showed that the proposed method can improve transport revenue by 5.1% and reduce passengers’ travel time loss by 11.15% without increasing transport capacity.

The parallel mobile charging service for free-floating shared electric vehicle clusters

Due to the simple rental procedures and unrestricted parking spots, free-floating shared electric vehicles (SEVs) are widely selected by travelers. A large number of SEVs gather together and even form SEV clusters at popular destinations. In most studies, the low-charge SEVs are usually relocated to charging stations to recharge their batteries. This paper proposes the parallel mobile charging service to route mobile charging vehicles (MCVs) to charge SEVs at their parking spots. The MCV routes are optimized through a mixed integer nonlinear optimization model where MCVs with limited battery capacity could recharge their batteries at available charging stations and sequentially or simultaneously charge the filtered SEVs at the same SEV cluster node. The parallel mobile charging service proposed without relocating SEVs fully takes advantage of SEV clusters (i.e., cluster-based overlapping idle time windows and small inter-vehicle distance) and MCVs (i.e., charging multiple SEVs simultaneously). An adaptive large neighborhood search algorithm is developed to solve large-scale instances where a generalized cost function is constructed to balance the effects of battery capacity violations and time window violations on solutions. Comparing with the optimization solver CPLEX, the adaptive large neighborhood search algorithm developed demonstrates high performance. In addition, the cluster-based parallel acceleration time ratio is introduced to demonstrate the advantages of parallel services in utilizing the cluster-based overlapping time windows.

Olive oil supply chain design with organic and conventional market segments and consumers’ preference to local products

Recent market studies showed that the demand for organic and local agrifood products is increasing despite their higher prices. The agribusiness actors should therefore rethink the supply chain configuration to cope with new market trends characterized by the rise of the organic segment and the increase of consumers' preference to more local products. This study focuses on the olive oil sector and proposes a mixed-integer non-linear optimization model for the design of olive oil supply chains while incorporating organic and conventional market segments and considering, for each segment, a supply chain proximity- and price-sensitive demand. The model is developed with the collaboration of olive oil producers in the Mediterranean area. Thanks to this industrial collaboration, we account for real-world practices and constraints and apply the model to a realistic case study. We first linearize the model and show that it can be efficiently solved with commercial optimization softwares. Based on numerical experiments, we derive a series of managerial insights that are applicable to the considered case study, some of them are not intuitive. For instance, we show that an increase in consumers’ preference to more local products may lead the producer to offer products with a more global supply chain. The conventional product variety may be produced with a more local supply chain than the organic (premium) variety. Finally, offering a mix of organic and conventional varieties instead of only one variety would lead to implementing a more local supply chain.

Globally optimal distillation tray design using a mathematical programming approach

This article presents the optimization of the design of distillation column trays. We develop a mixed-integer nonlinear optimization model, which we solve using mathematical programming. The objective of this formulation is to size all geometric variables of the tray. Two possible alternative objective functions are tested: column cost and column mass. The utilization of the proposed approach is illustrated through a design example from the literature. In this example, the optimal result obtained through the proposed approach is compared with the one obtained following the spirit of a traditional heuristic design procedure employed in process equipment design textbooks. The comparison indicates a reduction of both objective functions tested as compared to the result of the heuristic procedure.

A data-driven matching algorithm for ride pooling problem

This paper proposes a data-driven matching algorithm for the problem of ride pooling, which is a transportation mode enabling people to share a vehicle for a trip. The problem is considered as a variant of matching problem, since it aims to find a matching between drivers and riders. Proposed algorithm is a machine learning algorithm based on rank aggregation idea, where every feature in a multi-feature dataset provides a ranking of candidate drivers and weight for each feature is learned from past data through an optimization model. Once weight learning and candidate ranking problems are considered simultaneously, resulting optimization model becomes a nonlinear bilevel optimization model, which is reformulated as a single level mixed-integer nonlinear optimization model. To demonstrate the performance of the proposed algorithm, a real-life dataset from a mobile application of a ride pooling start-up company is used and company’s current approach is considered as benchmark. Results reveal that proposed algorithm correctly predicts the first choice of riders 17% to 28% better compared to the benchmark in different scenarios. Similarly, proposed algorithm offers recommendation lists in which the preferred driver is ranked 0.38 to 1.12 person closer (to the rider’s actual choice) compared to the benchmark.

Development of a multiple response-based mixed-integer nonlinear optimization model with both controllable and uncontrollable design factors

Response surface methodology (RSM) is an appropriate tool for modeling and analyzing existing or new products. In the literature, considerable attention has been paid to develop RSM models while using controllable design factors. However, there are some situations where uncontrollable design factors are required to conduct an experiment. Therefore, this paper is four-fold. One, an A-optimal design is selected as the appropriate design to generate the design matrix. Two, an exchange algorithm is proposed to construct A-optimal design points while dealing with uncontrollable design factors. In addition, fitted mean and standard deviation response functions are obtained with both controllable and uncontrollable design factors. Three, a multiple response-based mixed-integer nonlinear model and its solution procedure are proposed to find optimum operating conditions of both controllable and uncontrollable design factors. Next, a case study is presented to show the effectiveness of the proposed methodology. It is also reported from the case study that the proposed optimization model may achieve a variance reduction of up to 73% compared to the traditional counterpart. Finally, comparison and validation studies are conducted to verify the optimum operating conditions.

A Dispatch Optimization Model for Hybrid Renewable and Battery Systems Incorporating a Battery Degradation Model

Abstract A recent increase in the integration of renewable energy systems in existing power grids along with a lack of integrated dispatch models has led to waste in power produced. This paper presents a mixed-integer nonlinear optimization model for hybrid renewable-generator-plus-battery systems, with the objective of maximizing long-term profit. Prior studies have revealed that both high and low state of charge (SOC) of the battery is detrimental to its lifetime and results in reduced battery capacity over time. In addition, increased number of cycles of charge and discharge also causes capacity reduction. This paper models these two factors with a constraint relating capacity loss to the SOC and number of cycles completed by the battery. Loss in capacity is penalized in the objective function of the optimization model, thereby disincentivizing high and low SOCs and frequent cycling. A rolling time horizon optimization approach is used to overcome the computational difficulties of achieving global optimality within a long-term time horizon. By incorporating battery degradation, the model is capable of maximizing the profits from the power dispatch to the grid while also maximizing the life of the battery. This paper exercises the model within a case study using a sample photovoltaic system generation time series that considers multiple battery capacities. The results indicate that the optimal battery lifetime is extended in comparison to conventional models that ignore battery degradation in dispatch decisions. Finally, we analyze the relationship between battery operational decisions and the resultant capacity fade.

Proposal and Solution of a Mixed-Integer Nonlinear Optimization Model That Incorporates Future Preparedness for Project Portfolio Selection

In the context of project management, the attention given to project portfolio management has increased in recent years. The use of mathematical programming for portfolio management is also on the rise, because it integrates the project interactions with the multiple objectives of portfolio management into a single model. Among the possible objectives, recent studies have paid special attention to the emerging objective of future preparedness, which has not yet been incorporated into the existing mathematical models. This paper presents a mixed-integer nonlinear optimization model for portfolio selection that considers four main performance measures for project management, namely, value maximization, strategic alignment, balance, and future preparedness. Given the importance of the last measure, the purpose of this paper is to provide a more complete model that provides the marginal contribution and the best combination of projects according to the needs of the company. The model is tested using real data from two companies, one in Brazil and one in Canada, and the results obtained are coherent with their respective practices.