Mixed Integer Nonlinear Programming Approach Research Articles

Efficient continuous piecewise linear regression for linearising univariate non-linear functions

Due to their flexibility and ability to incorporate non-linear relationships, Mixed-Integer Non-Linear Programming (MINLP) approaches for optimization are commonly presented as a solution tool for real-world problems. Within this context, piecewise linear (PWL) approximations of non-linear continuous functions are useful, as opposed to non-linear machine learning-based approaches, since they enable the application of Mixed-Integer Linear Programming techniques in the MINLP framework, as well as retaining important features of the approximated non-linear functions, such as convexity. In this work, we extend upon fast algorithmic approaches for modeling discrete data using PWL regression by tuning them to allow the modeling of continuous functions. We show that if the input function is convex, then the convexity of the resulting PWL function is guaranteed. An analysis of the runtime of the presented algorithm shows which function characteristics affect the efficiency of the model, and which classes of functions can be modeled very quickly. Experimental results show that the presented approach is significantly faster than five existing approaches for modeling non-linear functions from the literature, at least 11 times faster on the tested functions, and up to a maximum speedup of more than 328,000. The presented approach also solves six benchmark problems for the first time.

Efficient Microgrid Management with Meerkat Optimization for Energy Storage, Renewables, Hydrogen Storage, Demand Response, and EV Charging

Within microgrids (MGs), the integration of renewable energy resources (RERs), plug-in hybrid electric vehicles (PHEVs), combined heat and power (CHP) systems, demand response (DR) initiatives, and energy storage solutions poses intricate scheduling challenges. Coordinating these diverse components is pivotal for optimizing MG performance. This study presents an innovative stochastic framework to streamline energy management in MGs, covering proton exchange membrane fuel cell–CHP (PEMFC-CHP) units, RERs, PHEVs, and various storage methods. To tackle uncertainties in PHEV and RER models, we employ the robust Monte Carlo Simulation (MCS) technique. Challenges related to hydrogen storage strategies in PEMFC-CHP units are addressed through a customized mixed-integer nonlinear programming (MINLP) approach. The integration of intelligent charging protocols governing PHEV charging dynamics is emphasized. Our primary goal centers on maximizing market profits, serving as the foundation for our optimization endeavors. At the heart of our approach is the Meerkat Optimization Algorithm (MOA), unraveling optimal MG operation amidst the intermittent nature of uncertain parameters. To amplify its exploratory capabilities and expedite global optima discovery, we enhance the MOA algorithm. The revised summary commences by outlining the overall goal and core algorithm, followed by a detailed explanation of optimization points for each MG component. Rigorous validation is executed using a conventional test system across diverse planning horizons. A comprehensive comparative analysis spanning varied scenarios establishes our proposed method as a benchmark against existing alternatives.

Sizing and energy management of standalone hybrid renewable energy systems based on economic predictive control

This paper aims to design and run a standalone hybrid renewable energy System (HRES) feeding residential building load. The suggested HRES encompasses photovoltaic panels and wind turbines as energy sources, combined with a hydrogen subsystem and a bank of batteries for energy storage. The meteorological input data includes irradiance, temperature, and wind speed for four seasonal weeks. These data sets are collected for a rural site situated in the town of Ras Munif, northern Jordan. The load profiles characterize a selected benchmark residential building. Consequently, a bi-level mixed integer nonlinear programming (BMINLP) problem is formulated to incorporate the designing problem with the energy management process. The upper level represents the sizing problem, while the lower level problem denotes the energy management strategy (EMS). Each problem has its independent objective function, constraints, and optimization variables. The EMS is an economic-based single objective optimization problem expressed as a mixed integer linear programming (MILP) problem. On the contrary, a mixed integer nonlinear programming (MINLP) approach frames the sizing problem to optimize a techno-economic-based multi-objective function. The interaction between the two problems is that the lower problem is an embedded constraint of the upper problem. Moreover, the candidate solutions from the upper problem are treated as fixed parameters in the lower problem. A multi-objective genetic algorithm is used to execute the overall problem by the MATLAB toolbox. The technical and economic performance parameters, including the investment cost, maintenance, operating costs, and others, are assessed based on the best findings. The key finding demonstrates that the summer week has the lowest total cost and energy loss factor, with around 132,000 $ and 0, respectively. Furthermore, the findings reveal that the entire system's energy loss factor for each sample period is less than 0.0009. This value is substantially less than the permitted limit of 0.01.

Optimization under uncertainty of a hybrid waste tire and natural gas feedstock flexible polygeneration system using a decomposition algorithm

Market uncertainties motivate the development of flexible polygeneration systems that are able to adjust operating conditions to favor production of the most profitable product portfolio. However, this operational flexibility comes at the cost of higher capital expenditure. A scenario-based two-stage stochastic nonconvex Mixed-Integer Nonlinear Programming (MINLP) approach lends itself naturally to optimizing these trade-offs. This work studies the optimal design and operation under uncertainty of a hybrid feedstock flexible polygeneration system producing electricity, methanol, dimethyl ether, olefins or liquefied (synthetic) natural gas. A recently developed C++ based software framework (named GOSSIP) is used for modeling the optimization problem as well as its efficient solution using the Nonconvex Generalized Benders Decomposition (NGBD) algorithm. Two different cases are studied: The first uses estimates of the means and variances of the uncertain parameters from historical data, whereas the second assesses the impact of increased uncertain parameter volatility. The value of implementing flexible designs characterized by the value of the stochastic solution (VSS) is in the range of 260–405 M$ for a scale of approximately 893 MW of thermal input. Increased price volatility around the same mean results in higher expected net present value and VSS as operational flexibility allows for asymmetric exploitation of price peaks.

An inexact multi-objective mixed-integer nonlinear programming approach for water-soil-fertilizer management under uncertainty considering “footprint family-planetary boundary” assessment

Water, energy, and food (WEF) are interlinked and create a dynamic system that impacts both human well-being and ecology. Given the importance of ecology within the WEF system, water resources serve as the core issue. The allocation of water resources in irrigation districts is a challenging problem for the coordinated development of agricultural production, water resources, and the ecological environment. The integration of stochastic multi-objective programming, fuzzy credibility-constrained programming, and mixed integer programming offers a solution to this issue, with the construction of a fuzzy credibility-constrained stochastic multi-objective mixed-integer nonlinear programming model. The applicability and validity of this model were verified by applying it to the Kaikong Irrigation District (KID) of the Tarim River Basin in northwest China, with notable findings indicating that the optimized system reduces agricultural costs by 5.82%, increases irrigation water use efficiency by 1.80%, and reduces global warming potential by 6.45%. This study investigates the effects of diverse allocation strategies of water and land resources on the social, economic, and ecological subsystems and their interactions by downscaling four subprocesses PBs to the KID scale. The optimization model reveals that only the nitrogen footprint of Kuerle City surpasses the nitrogen boundary of the KID. The proposed solutions based on the model can encourage the green and ecologically-friendly development of agricultural production and can be applied to agricultural systems in arid regions with similar conditions.

A Sensitivity-based Heuristic for Vehicle Priority Assignment at Intersections

This paper presents a sensitivity-based heuristic to address the dynamic priority assignment problem of connected and autonomous vehicle (CAV) and human-driven vehicle (HDV) at traffic intersections. We exploit sensitivity analysis tools to approximatively predict the CAV's performance violation as a function of the HDV states. Such predictions are then used to decide on a crossing order that preserves optimality and feasibility despite the behavior of the HDV. The proposed algorithm is compared with the baseline first-come, first-serve (FCFS) and mixed-integer nonlinear programming (MINLP) approaches. In the closed-loop simulation, we show that the heuristic is computationally much faster than MINLP and able to retain a close-to-optimal solution, which is far better than FCFS.

Data-driven simultaneous process optimization and adsorbent selection for vacuum pressure swing adsorption

Technologies for post-combustion carbon capture are essential for the reduction of greenhouse gas emissions to the atmosphere. However, they are still associated with high costs and energy consumption. Intensified processes for carbon capture have the potential to overcome these challenges due to their higher efficiency, lower capital cost, and increased operational flexibility. This work investigates simultaneous optimization of process conditions and adsorbent selection for a modular Vacuum Pressure-Swing Adsorption system designed for CO2 capture. Both surrogate-based Nonlinear Programming and Mixed-Integer Nonlinear Programming approaches are applied and compared in terms of computational efficiency and solution accuracy. Moreover, process performance results are examined by applying several data analytics techniques to gain insights into the material-process correlations. Data-driven classifiers and neural networks can accurately predict whether a material is likely to satisfy purity, recovery, and energy constraints when operated at optimal process conditions.

Uncertainty Propagation Based MINLP Approach for Artificial Neural Network Structure Reduction

The performance of artificial neural networks (ANNs) is highly influenced by the selection of input variables and the architecture defined by hyper parameters such as the number of neurons in the hidden layer and connections between network variables. Although there are some black-box and trial and error based studies in the literature to deal with these issues, it is fair to state that a rigorous and systematic method providing global and unique solution is still missing. Accordingly, in this study, a mixed integer nonlinear programming (MINLP) formulation is proposed to detect the best features and connections among the neural network elements while propagating parameter and output uncertainties for regression problems. The objective of the formulation is to minimize the covariance of the estimated parameters while by (i) detecting the ideal number of neurons, (ii) synthesizing the connection configuration between those neurons, inputs and outputs, and (iii) selecting optimum input variables in a multi variable data set to design and ensure identifiable ANN architectures. As a result, suggested approach provides a robust and optimal ANN architecture with tighter prediction bounds obtained from propagation of parameter uncertainty, and higher prediction accuracy compared to the traditional fully connected approach and other benchmarks. Furthermore, such a performance is obtained after elimination of approximately 85% and 90% of the connections, for two case studies respectively, compared to traditional ANN in addition to significant reduction in the input subset.

A sustainable supply chain network considering lot sizing with quantity discounts under disruption risks: centralized and decentralized models.

This study proposes a framework for the main parties of a sustainable supply chain network considering lot-sizing impact with quantity discounts under disruption risk among the first studies. The proposed problem differs from most studies considering supplier selection and order allocation in this area. First, regarding the concept of the triple bottom line, total cost, environmental emissions, and job opportunities are considered to cover the criteria of sustainability. Second, the application of this supply chain network is transformer production. Third, applying an economic order quantity model lets our model have a smart inventory plan to control the uncertainties. Most significantly, we present both centralized and decentralized optimization models to cope with the considered problem. The proposed centralized model focuses on pricing and inventory decisions of a supply chain network with a focus on supplier selection and order allocation parts. This model is formulated by a scenario-based stochastic mixed-integer non-linear programming approach. Our second model focuses on the competition of suppliers based on the price of products with regard to sustainability. In this regard, a Stackelberg game model is developed. Based on this comparison, we can see that the sum of the costs for both levels is lower than the cost without the bi-level approach. However, the computational time for the bi-level approach is more than for the centralized model. This means that the proposed optimization model can better solve our problem to achieve a better solution than the centralized optimization model. However, obtaining this better answer also requires more processing time. To address both optimization models, a hybrid bio-inspired metaheuristic as the hybrid of imperialist competitive algorithm (ICA) and particle swarm optimization (PSO) is utilized. The proposed algorithm is compared with its individuals. All employed optimizers have been tuned by the Taguchi method and validated by an exact solver in small sizes. Numerical results show that striking similarities are observed between the results of the algorithms, but the standard deviations of PSO and ICA–PSO show better behavior. Furthermore, while PSO consumes less time among the metaheuristics, the proposed hybrid metaheuristic named ICA–PSO shows more time computations in all small instances. Finally, the provided results confirm the efficiency and the performance of the proposed framework and the proposed hybrid metaheuristic algorithm.

The MINLP Approach to Topology, Shape and Discrete Sizing Optimization of Trusses

The paper presents the Mixed-Integer Non-linear Programming (MINLP) approach to the synthesis of trusses. The solution of continuous/discrete non-convex and non-linear optimization problems is discussed with respect to the simultaneous topology, shape and discrete sizing optimization of trusses. A truss MINLP superstructure of different topology and design alternatives has been generated, and a special MINLP model formulation for trusses has been developed. In the optimization model, a mass objective function of the structure has been defined and subjected to design, load and dimensioning constraints. The MINLP problems are solved using the Modified Outer-Approximation/Equality-Relaxation (OA/ER) algorithm. Multi-level MINLP strategies are introduced to accelerate the convergence of the algorithm. The Modified Two-Phase and the Sequential Two-Phase MINLP strategies are proposed in order to solve highly combinatorial topology, shape and discrete sizing optimization problems. The importance of local buckling constraints on topology optimization is also discussed. Some simple numerical examples are shown at the end of the paper to demonstrate the suitability and efficiency of the proposed method.

An Agri-Fresh Food Supply Chain Network Design with Routing Optimization: A Case Study of ETKA Company

The Supply Chain Network Design (SCND) with perishability is an active research topic. The Agri-fresh Food Supply Chain (AFSC) is a relevant topic to SCND and this study aims to model a new AFSC for a real-world case study. Regarding the traditional AFSC, the geographically dispersed small farmers transport their product individually to market for selling. This leads to a higher transportation cost, which is the major cause of farmers’ low profitability. This paper formulates a traditional product movement model to represent the existing AFSC. The concept of sharing economic approach is employed by the aggregate and collaborative transportation of products to minimize transportation inefficiency. This paper proposes an aggregate product movement with the vehicle routing model to re-design an AFSC for a case study in Iran based on the data of ETKA Company-the largest domestic agri-fresh food supply chain. A four-echelon, multi-period, Mixed Integer Non-Linear Programming (MINLP) approach for the proposed location-inventory-routing model is formulated for perishable products via considering the clustering of farmers to minimize the total distribution cost.

A data-driven approach to support voltage profiles & loss reduction in wind generator integrated active distribution network considering solid-state transformers with twofold reactive power compensation

ABSTRACT The paper proposes an optimal planning strategy for an active distribution network (ADN) with wind energy-based distributed generation (WDG) by considering the twofold reactive power compensation (TRPC) feature of a three-stage type solid-state transformer (SST). A model for wind speed and load demand estimation for each hour of the day is formulated by using the partition around medoids (PAM) technique over an annual database of load demand and wind speed. A multi-objective mixed-integer non-linear programming (MINLP) approach for optimally locating and sizing the SST installations in the radial distribution network (RDN) is presented to address the dual objectives of voltage profile improvement (VPI) and energy loss reduction (ELR). Along with the SST parameters, the locations and number of wind turbine (WT) units to be integrated into the system are treated as optimization variables. Furthermore, the distribution power flow accounts for the distribution transformer’s (DT) operating losses as well as the approximate losses of the SST. The presented approach is tested on a 33-bus RDN and the results are reported for multiple case studies. The programming was developed in the MATLAB R2020a environment and the paretosearch algorithm (PSA) is used to address the MINLP problem. Other standard multi-objective optimization algorithms, including multi-objective multi-verse optimization (MOMVO), non-dominated sorting genetic algorithm (NSGA-II), and multi-objective salp swarm algorithm (MSSA) are used to compare the performance of the proposed method. The outcomes of the peak hour evaluation, with a 20% over-rating, show the active and reactive power demand on the MV side of the substation to have decreased by 56.3% and 30.1% respectively. The active line loss has been lowered by 76.95% and the reactive line loss has decreased by 76.14%. The absolute minimum voltage has increased by 7.314%. Furthermore, according to the annual technical evaluation, there was a 29.42% reduction in active energy served by the DTs and SSTs.

Two-Stage Metaheuristic Mixed Integer Nonlinear Programming Approach to Extract Optimum Hedging Rules for Multireservoir Systems

AbstractHedging policy is an applicable strategy for water resources systems with storage reservoirs to reduce the damage resulting from extended droughts. Optimizing the operation of parallel and ...

Sustainable Logistics Network Design for Delivery Operations with Time Horizons in B2B E-Commerce Platform

In the recent era, the rapidly increasing trend of e-commerce business creates opportunities for logistics service providers to grow globally. With this growth, the concern regarding the implementation of sustainability in logistic networks has received attention in recent years. Thus, in this work, we have focused on the vehicle routing problem (VRP) to deliver the products in a lesser time horizon with driver safety concern considerations in business (B2B) e-commerce platforms. We proposed a sustainable logistics network that captures the complexities of suppliers, retailers, and logistics service providers. A mixed-integer nonlinear programming (MINLP) approach is applied to formulate a model to minimize total time associated with order processing, handling, packaging, shipping, and vehicle maintenance. Branch-and-bound algorithms in the LINGO optimization tool and genetic algorithm (GA) are used to solve the formulated mathematical model. The computational experiments are performed in eight different case scenarios (small-sized problem to large-sized problem) to validate the model.

Optimal Selection of Conductors in Three-Phase Distribution Networks Using a Discrete Version of the Vortex Search Algorithm

In this study, a new methodology is proposed to perform optimal selection of conductors in three-phase distribution networks through a discrete version of the metaheuristic method of vortex search. To represent the problem, a single-objective mathematical model with a mixed-integer nonlinear programming (MINLP) structure is used. As an objective function, minimization of the investment costs in conductors together with the technical losses of the network for a study period of one year is considered. Additionally, the model will be implemented in balanced and unbalanced test systems and with variations in the connection of their loads, i.e., Δ- and Y-connections. To evaluate the costs of the energy losses, a classical backward/forward three-phase power-flow method is implemented. Two test systems used in the specialized literature were employed, which comprise 8 and 27 nodes with radial structures in medium voltage levels. All computational implementations were developed in the MATLAB programming environment, and all results were evaluated in DigSILENT software to verify the effectiveness and the proposed three-phase unbalanced power-flow method. Comparative analyses with classical and Chu & Beasley genetic algorithms, tabu search algorithm, and exact MINLP approaches demonstrate the efficiency of the proposed optimization approach regarding the final value of the objective function.

Enhancing the passive design of buildings: A mixed integer non-linear programming approach for the selection of building materials and construction building systems

Consumption of energy in buildings accounts for a considerable proportion of worldwide energy use. There is a dire need for enhancing the energy efficiency of building to limit their demand for operating energy as this leads to enhanced reductions in environmental impacts. Of particular relevance to the amount of energy utilised in a building during the operation phase is the nature of material and size of components utilised in the building. In this work, a mathematical programming framework is presented to optimise a number of building design objective functions, including heat gain, daylight and economic cost of material utilised. The variables that are focussed on in this study are the sizes of windows, type of material adopted for the building, embodied in the construction building systems used for various building components, and the type of lighting adopted. To validate the framework, two realistic case studies obtained from an industry partner are adopted and solved via the use of the proposed mathematical programming method. Results indicate that compared to the solutions proposed by an experienced engineer, the daylight, heating and cost of the building is enhanced by up to 39%, 43% and 23% respectively. The framework is hoped to help policy makers introduce more streamlined guidance for the building sector when it comes to optimised material choice and window sizing to result in energy-efficient and economical buildings.

Conceptual MINLP approach to the development of a CO2 supply chain network – Simultaneous consideration of capture and utilization process flowsheets

A large fraction of anthropogenic CO2 emissions comes from large point sources such as power plants, petroleum refineries, and large industrial facilities. A significant decrease of these CO2 emissions can be achieved with CO2 capture, utilization, and storage (CCUS) technologies. This study proposes a conceptually simplified model for the optimization of combined CO2 supply networks and capture and utilization technologies by the mixed-integer non-linear programming (MINLP) approach. The objective is to maximize the profit of CCUS technologies, considering chemisorption using methyl-diethanolamine (MDEA) as a capture technology and conversion of CO2 to CH3OH as a utilization technology. Additionally, avoided tax from reduced CO2 emissions is considered as a revenue. A hypothetical case study of five larger point sources of CO2 was investigated, namely coal power plants, biogas plant, aluminium production plant and two cement plants. Two scenarios were considered: i) Scenario A considering different values of the CO2 tax, and ii) Scenario B considering different flue gas flowrates at different values of the CO2 tax. The results show the potential of model-based optimization in reducing the amount of CO2 in the atmosphere by CCUS technology. Furthermore, the results in Scenario A show that CCUS technology is only profitable if the price of CO2 emissions is higher than 110 €/t emitted CO2. Moreover, the results in Scenario B show that both the profit and the production of CH3OH depend to a large extent on the flue gas flow.

A MINLP approach to improve heat transfer in flat plates through the optimal distribution of nanoparticles

In this work, the thermal effect of deploying nanoparticles on the steady-state two-dimensional heat transfer behavior on a square plate is presented. The aim is to strengthen process intensification through efficient energy use. The approach of this work is to present a general methodology to obtain the optimal configuration of the nanoparticles and the base fluid respectively in the system. Here, the objective function is maximizing the heat transfer of the system considering that the performance of the nanofluids depends on the distribution of the nanoparticles. A general model is presented that allows solving the two-dimensional heat equation for the nanoparticles and the base fluid independently. The mathematical formulation is presented as a problem of Mixed-Integer Non-linear Programming (MINLP) model. The system under consideration is a mixture of metallic nano-sized particles and conventional heat-transferring fluid. In addition, it is also considered that there is heat generation inside the base fluid and as well the thermal conductivity of the flat plate is assumed to vary as a linear function of the temperature. The results indicate that improved heat transfer can be achieved through optimal fluid location. However, in few cases this behavior was not observed and the explanation of this issue is linked to the the nonconvex MINLP problem rendering only optimal local solutions. In order to show the applicability of the proposed methodology, two parametric case studies are presented.

An Incentive-Based Optimization Approach for Load Scheduling Problem in Smart Building Communities

The impact of load growth on electricity peak demand is becoming a vital concern for utilities. To prevent the need to build new power plants or upgrade transmission lines, power companies are trying to design new demand response programs. These programs can reduce the peak demand and be beneficial for both energy consumers and suppliers. One of the most popular demand response programs is the building load scheduling for energy-saving and peak-shaving. This paper presents an autonomous incentive-based multi-objective nonlinear optimization approach for load scheduling problems (LSP) in smart building communities. This model’s objectives are three-fold: minimizing total electricity costs, maximizing assigned incentives for each customer, and minimizing inconvenience level. In this model, two groups of assets are considered: time-shiftable assets, including electronic appliances and plug-in electric vehicle (PEV) charging facilities, and thermal assets such as heating, ventilation, and air conditioning (HVAC) systems and electric water heaters. For each group, specific energy consumption and inconvenience level models were developed. The designed model assigned the incentives to the participants based on their willingness to reschedule their assets. The LSP is a discrete–continuous problem and is formulated based on a mixed-integer nonlinear programming approach. Zoutendijk’s method is used to solve the nonlinear optimization model. This formulation helps capture the building collaboration to achieve the objectives. Illustrative case studies are demonstrated to assess the proposed model’s effect on building communities consisting of residential and commercial buildings. The results show the efficiency of the proposed model in reducing the total energy cost as well as increasing the participants’ satisfaction. The findings also reveal that we can shave the peak demand by 53% and have a smooth aggregate load profile in a large-scale building community containing 500 residential and commercial buildings.

Optimized packing unequal spheres into a multiconnected domain: mixed-integer non-linear programming approach

The problem of packing unequal spheres into a multiconnected domain (container) is considered. Given a set of spheres, the objective is to maximize the packing factor. The problem is considered as a knapsack problem and modelled as a mixed-integer non-linear programming. Characteristics of the model are indicated. We propose a new solution method based on a combination of a branch-and-bound approach and the known local optimization method. The search procedure is represented by a tree which allows handling all possible subsets of spheres. We develop a set of truncation rules to reduce the number of variants under test. The local optimization algorithm proceeds from the assumption of spheres radii being variable. A number of numerical examples are given.