Mixed Integer Linear Programming Formulation Research Articles

Optimization for the minimum fuel consumption problem of a hybrid electric vehicle using mixed-integer linear programming

In this study, a mixed-integer linear programming (MILP) formulation for obtaining the operating points of hybrid electric vehicle (HEV) powertrains is proposed. This study focuses on linearizing the nonlinear terms, such as an engine fuel consumption map, or bilinear terms represented by the energy of the motor. To address this problem, a conventional piecewise linear (PWL) method and a multi-layer perceptron (MLP) regression approach are adopted. Although the optimal solution cannot be determined using a PWL approximation for the fuel consumption map, it can be obtained using an MLP regression. Furthermore, the PWL method achieves better results than the MLP approach in terms of the accuracy of its bilinear approximation. Obtaining the optimal solution using MILP helps in acquiring a Lagrange multiplication of the design variables by solving the dual problem, which allows an efficient design revision strategy to be obtained.

Bandgap optimization in combinatorial graphs with tailored ground states: application in quantum annealing

A mixed-integer linear programming (MILP) formulation is presented for parameter estimation of the Potts model. Two algorithms are developed; the first method estimates the parameters such that the set of ground states replicate the user-prescribed data set; the second method allows the user to prescribe the ground states multiplicity. In both instances, the optimization process ensures that the bandgap is maximized. Consequently, the model parameter efficiently describes the user data for a broad range of temperatures. This is useful in the development of energy-based graph models to be simulated on Quantum annealing hardware where the exact simulation temperature is unknown. Computationally, the memory requirement in this method grows exponentially with the graph size. Therefore, this method can only be practically applied to small graphs. Such applications include learning of small generative classifiers and spin-lattice model with energy described by Ising hamiltonian. Learning large data sets poses no extra cost to this method; however, applications involving the learning of high dimensional data are out of scope.

Optimising vehicle and on-foot porter routing in urban logistics

As modern cities expand, parcel carriers are obliged to deal with the various last-mile challenges of the urban setting, in order to meet the ever-increasing requirements of consumers while ensuring an eco-friendly distribution system. Some of the urban last-mile challenges, such as kerbside parking limitations and vehicle access restrictions, pose additional constraints to carriers that operate van-based deliveries. This paper describes a new distribution model and an associated optimisation problem, namely a vehicle routing problem with portering (VRP-P) with time windows, that integrates on-foot porters with cargo vans. In this distribution strategy, vans rendezvous with porters at handover points at selected customer locations where the transfer of parcels can take place, avoiding the need for designated parking spaces. The VRP-P is modelled as a mixed-integer linear programming formulation and solved on real delivery data in central London. The paper also describes a fast heuristic that uses several improvement operators. Computational results show that the use of portering yields time and carbon savings of up to 89%, depending on the number of customers served. The heuristic obtains high-quality feasible solutions in short computational times and outperforms a commercial optimiser on large scale instances. The findings also show high utilisation of portering even at limited parcel carrying capacities. • We describe an urban delivery system that integrates routing of vans and porters. • We introduce a new mixed-integer linear programming model to solve the problem. • An actual data set from London deliveries is used for analysis. • We propose an efficient heuristic which produces results in short computational time. • Portering achieves up to 50% and 89% time and carbon savings, respectively.

Distributionally Robust Fair Transit Resource Allocation During a Pandemic

This paper studies the distributionally robust fair transit resource allocation model (DrFRAM) under the Wasserstein ambiguity set to optimize the public transit resource allocation during a pandemic. We show that the proposed DrFRAM is highly nonconvex and nonlinear, and it is NP-hard in general. Fortunately, we show that DrFRAM can be reformulated as a mixed integer linear programming (MILP) by leveraging the equivalent representation of distributionally robust optimization and monotonicity properties, binarizing integer variables, and linearizing nonconvex terms. To improve the proposed MILP formulation, we derive stronger ones and develop valid inequalities by exploiting the model structures. Additionally, we develop scenario decomposition methods using different MILP formulations to solve the scenario subproblems and introduce a simple yet effective no one left-based approximation algorithm with a provable approximation guarantee to solve the model to near optimality. Finally, we numerically demonstrate the effectiveness of the proposed approaches and apply them to real-world data provided by the Blacksburg Transit. History: This paper has been accepted for the Transportation Science Special Issue on Emerging Topics in Transportation Science and Logistics. Funding: This work was supported by the Division of Computing and Communication Foundations [Grant 2153607] and the Division of Civil, Mechanical and Manufacturing Innovation [Grant 2046426]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2022.1159 .

Alternative Pickup Locations in Taxi-Sharing: A Feasibility Study

Taxi-sharing is a known solution for reducing congestion and is more beneficial when the traveled miles, imposed by detours while serving additional passengers, are minimized. This study proposes incorporation of alternative meeting points in taxi-sharing routes to boost the efficiency of the system by eliminating unnecessary detours and improving the chances of passengers being matched. Unlike most ridesharing systems, in the proposed approach, passengers are not necessarily picked up or dropped off at their original location, but at a walkable distance from their origin. The proposed framework deals with practical challenges of including meeting points in a real-world high-demand taxi-sharing system with hundreds of requests per minute. This includes efficient selection of alternative pick up locations and incorporation of these alternatives into a novel mixed integer linear programming (MILP) formulation to find the optimal schedule. Using the 2015 New York City (NYC) yellow cab dataset, first, the potential benefits of introducing meeting points in Manhattan road network are demonstrated. Given the Nondeterministic Polynomial-time Hard (NP-hard) nature of the associated optimization problem, the problem is then broken down into smaller-sized problems forming clusters of passengers with high potential of sharing a ride. Then, the proposed MILP model is used to find the optimal route for each cluster, while selecting the best pickup point for each passenger. Testing on a sample of the NYC dataset, it is shown that the proposed methodology improves the efficiency of the taxi-sharing system by reducing the wait times by about 50% while considerably reducing total travel times and the number of vehicles used.

A continuous-time formulation for optimal task scheduling and quality-of-service assurance in nanosatellites

Satellite scheduling concerns an operations research problem that aims at planning the execution of a given set of tasks, either in real-time onboard the spacecraft or offline in the ground station. In particular, the task scheduling problem for nanosatellites has been the focus of recent research, given the unique set of constraints and applications, and considering their limited power resources. This paper applies the continuous-time methodology to deliver a novel scheduling formulation for the nanosatellite scheduling problem with increased flexibility and versatility to the decision-maker. Here, we adopt the concept of priority slots and follow the Multi-Operation Sequencing with Synchronized Start Times (MOS-SST) representation to yield a Mixed-Integer Linear Programming (MILP) formulation that considers constraints on time synchronization, mission quality-of-service, and management of shared resources. By comparing our continuous-time strategy with existing discrete-time alternatives, we showed that the proposed method can solve representative problem instances in a considerably shorter time, while being able to deal with real-valued constraint functions.

Management strategy for V2X equipped EV parking lot considering uncertainties with LSTM Model

The integration of electric vehicles (EVs) and renewable energy sources (RES), which are important key elements of green and sustainable energy, into power systems has become a crucial research subject in recent years due to their uncertain nature. Demand response (DR) strategies are expected to play a crucial role in the penetration of EVs and RESs. However, in order to improve the participation of EV owners in DR programs, their comfort should also be considered. In this study, a model was devised in the form of mixed-integer linear programming (MILP) that aims to mitigate the comfort violation of EV owners during vehicle-to-grid (V2G) and peak load limitation (PLL) operations through vehicle-to-vehicle (V2V) transactions and photovoltaic (PV) production in an electric vehicle parking lot (EVPL). A Machine learning (ML) based structure has been implemented using Long Short-Term Memory (LSTM) cells, a special type of recurrent neural network (RNN), to cope with the uncertainty of PV production and the uncertainty of EVs arrival times at EVPL. The results have shown that V2V transaction and PV generation may play an important role in terms of minimizing comfort violation during DR operations. All applications are developed in Python programming language.

A mathematical programming model for optimal fleet management of electric car-sharing systems with Vehicle-to-Grid operations

Electric car-sharing systems have attracted large attention in recent years as a new business model for achieving both economic and environmental benefits in urban areas. Among different types, the one considered in this paper is the so-called one-way car-sharing system whereby a user can begin and end a trip at any station of the system. At the same time, the Vehicle-to-Grid (V2G) concept is emerging as a possible innovative solution for smart power grid control. A management system that combines car-sharing system operations and V2G technology is a recent challenge for academia and industry. In this work, a mixed integer linear programming formulation is proposed to find the optimal management of electric vehicles in a one-way car-sharing system integrated with V2G technology. The proposed mathematical model allows finding the optimal start-of-day electric vehicles distribution that maximizes the total revenue obtained from system users and V2G profits through daily electric vehicles charging/discharging schedules. These schedules are based on mean daily users' electric vehicles requests and electricity prices. The model can be applied to evaluate the possible average daily profitability of V2G operations. In order to test the model performance, we applied it to a small-size test network and a real-size test network (the Delft network in the Netherlands). Under the model assumptions, the adoption of V2G technology allows to fully cover the daily charging costs due to users’ trips and to obtain V2G profits by taking advantage of electric vehicles unused time without significantly reducing the satisfied car-sharing system demand. Most of the energy purchased to charge the electric vehicles batteries is provided back to the grid during energy peak load demand, creating benefits also for energy providers.

Optimizing a complex multi-objective personnel scheduling problem jointly complying with requests from customers and staff

This paper deals with a complex multi-objective personnel scheduling problem motivated by a real case. A multi-objective mixed integer linear programming formulation of the problem is proposed. Constraints are classified into mandatory and optional. The work introduces a solution approach, dubbed PRIMP (Prioritize & Improve), that enforces constraint satisfaction by adopting additional objective functions. All the (given and additional) objective functions are lexicographically ordered. The method sequentially solves single-objective problems, according to their priority. Each problem is first processed by an exact solver; if no optimal solution is found within a given time limit, the problem is then addressed heuristically. The proposed multi-stage method is efficient (it takes just a few minutes to produce a daily schedule) and effective, compared both to the manual approach followed by the company and to the method that optimally tackles each single-objective problem by means of a competitive mixed-integer linear programming solver. Experimental results indicate that PRIMP can produce high quality schedules, where a larger number of optional constraints are satisfied and both the global idle time of employees and the waiting time of customers is reduced. The approach is modular and easily adaptable to manage different objective functions and/or constraints.

New exact and heuristic algorithms to solve the prize-collecting job sequencing problem with one common and multiple secondary resources

We study the prize-collecting job sequencing problem with one common and multiple secondary resources. In this problem, a set of jobs is given, each with a profit, multiple time windows for its execution, and a duration during which it requires the main resource. Each job also requires a preassigned secondary resource before, during, and after its use of the main resource. The goal is to select and schedule the subset of jobs that maximize the total profit. We present a new mixed integer linear programming formulation of the problem and a branch-cut-and-price algorithm as an exact solution method. We also introduce a heuristic algorithm to tackle larger instances. Extensive numerical experiments show that our exact algorithm can solve to optimality literature instances with up to 500 jobs for a particular dataset and up to 250 jobs for another dataset with different characteristics. Our heuristic builds high-quality solutions in a small computational time. It computes new best-known solutions for most of the larger instances.

Enhanced formulation for the Guillotine 2D Cutting Knapsack Problem

We advance the state of the art in Mixed-Integer Linear Programming formulations for Guillotine 2D Cutting Problems by (i) adapting a previously-known reduction to our preprocessing phase (plate-size normalization) and by (ii) enhancing a previous formulation (PP-G2KP from Furini et alli) by cutting down its size and symmetries. Our focus is the Guillotine 2D Knapsack Problem with orthogonal and unrestricted cuts, constrained demand, unlimited stages, and no rotation – however, the formulation may be adapted to many related problems. The code is available. Concerning the set of 59 instances used to benchmark the original formulation, the enhanced formulation takes about 4 hours to solve all instances while the original formulation takes 12 hours to solve 53 of them (the other six runs hit a three-hour time limit each). We integrate, to both formulations, a pricing framework proposed for the original formulation; the enhanced formulation keeps a significant advantage in this situation. Finally, in a recently proposed set of 80 harder instances, the enhanced formulation (with and without the pricing framework) found: 22 optimal solutions (5 already known, 17 new); better lower bounds for 25 instances; better upper bounds for 58 instances.

Clustering-based iterative heuristic framework for a non-emergency patients transportation problem

IntroductionNon-emergency patient transportation (NEPT) services are particularly important nowadays due to the aging population and contagious disease outbreaks (e.g., Covid-19 and SARS). In this work, we study a NEPT problem with a case study of patient transportation services in Hong Kong. The purpose of this work is to study the discomfort and inconvenience measures (e.g., waiting time and extra ride time) associated with the transportation of non-emergency patients while optimizing the operational costs and utilization of NEPT ambulances. MethodsA mixed-integer linear programming (MILP) formulation is developed to model the NEPT problem. This MILP model contributes to the existing literature by not only including the patient inconvenience measures in the objective function but also illustrating a better trade-off among different performance measures through its specially customized formulation and real-life characteristics. CPLEX is used to find the optimal solutions for the test instances. To overcome the computational complexity of the problem, a clustering-based iterative heuristic framework is designed to solve problems of practical sizes. The proposed framework distinctively exploits the problem-specific structure of the considered NEPT problem in a novel way to enhance and improve the clustering mechanism by repeatedly updating cluster centers. ResultsThe computational experiments on 19 realistic problem instances show the effective execution of the solution method and demonstrate the applicability of our approach. Our heuristic framework observes an optimality gap of less than 5% for all those instances where CPLEX delivered the result. The weighted objective function of the proposed model supports the analysis of different performance measures by setting different preferences for these measures. An extensive sensitivity analysis performed to observe the behavior of the MILP model shows that when operating costs are given a weightage of 0.05 in the objective function, the penalty value for user inconvenience measures is the lowest; when the weightage value for operating costs varies between 0.8 and 1.0, the penalty value for the same measures is the highest. ConclusionsThis research can assist decision-makers in improving service quality by balancing operational costs and patient discomfort during transportation.

Reducing blocking probability and QoT violation in dynamic elastic optical networks via load-aware margin selection

In this paper, we study the effect of load-aware margin selection on the performance of elastic optical networks (EONs) in dynamic service provisioning. First, a mixed-integer linear programming (MILP) formulation is proposed as a benchmark for dynamic routing, modulation level, spectrum, and power assignment (RMSPA). In this formulation, nonlinear interferences (NLIs) in optical fibers and the noise of amplifiers are considered. In addition, a margin which is calculated based on the network load is considered to estimate the additional noise which may degrade the signal quality of transmission (QoT) during its life time. The objective of this formulation is to minimize the spectrum usage and impairments of/from the signal which is optimized by choosing the optimum channel among a number of pre-calculated channels based on the QoT constraints. In order to reduce the RMSPA run time, heuristic algorithms are also proposed. The algorithms employ K-least weighted paths for routing. The defined weights are distance, the amount of traffic load, and a mixture scheme. Furthermore, the algorithms try to minimize the spectrum usage by choosing the highest possible modulation level while considering a specific criterion. The assumed criteria are: i) choosing the lowest possible power spectral density (PSD) in a discrete set, ii) selecting the optimum power among the set, iii) obtaining maximum SNR. We simulated the proposed algorithms and the MILP, and compared their performances on NSFNET and US Backbone topologies. The results reveal that the constrained best SNR scheme has the best performance among the heuristics and its performance is close to MILP, especially in higher loads. Moreover, results show that considering the links congestion leads to blocking reduction. We also simulated the first-fit based algorithms without considering NLIs which assign the modulation based on: i) considering only amplifier noise and self-channel interference (SCI), ii) experimental distance thresholds, and iii) overestimating the NLI. The results imply that the proposed MILP and heuristic algorithms outperform the other algorithms in terms of the summation of outage (probability of QoT violation) and blocking probability.

Toward Data Collection and Transmission in 6G Space–Air–Ground Integrated Networks: Cooperative HAP and LEO Satellite Schemes

The space–air–ground integrated network (SAGIN)-related issues are attractive in the sixth generation (6G) technologies, which facilitate the global coverage and seamless service. The cooperation of high altitude platforms (HAPs) and low-Earth orbit (LEO) satellites provides the remote area users with comprehensive coverage and service. In this work, we consider the cooperation of HAPs and LEO satellites in SAGIN to serve terrestrial users in the remote area for data collection and transmission. To deal with the periodical motion of LEO satellites, we employ the time expanding graph (TEG) to represent the multiple resources in SAGIN and depict task flow transmission processes. Based on TEG, we aim to maximize the total data received by the ground data processing center in a time horizon, considering multiple resource constraints and flow restrictions. The original problem is formulated in the form of mixed-integer linear programming, which is intractable to obtain the optimal solution by brute-force searching. To alleviate this intractability, we propose the Benders decomposition-based algorithm to obtain the optimal solution within an acceptable time complexity via iterations between the master problem and subproblem. Moreover, to further expedite the solution in large-scale systems, an acceleration algorithm is proposed by handling the master problem with an approximation algorithm and the subproblem with a unit-flow-based algorithm. Finally, simulations are conducted and the numerical results verify the efficiency of the proposed algorithms, and the effects of various network parameters are analyzed as well.

A Multi-Aspect Expanded Hypergraph Enabled Cross-Domain Resource Management in Satellite Networks

Satellite networks (SNs) are heterogeneous networks composed of typical functional domains, such as communication domains, observation domains, etc. Since the resources are independent among domains, and are highly dynamic with the movement of satellites, it is extremely difficult to capture the potential interaction relationship among various resources, even less to realize the coordinated scheduling of cross-domain resources in large-scale SNs. Motivated by these factors, we firstly propose a multi-aspect expanded hypergraph (MAEH) to accurately depict the Spatio-temporal features of resources in various domains as well as functional property, which is presented by different aspects. Particularly, the “aspect” can involve a group of resources with a similar feature in a hyperedge rather than a bilateral relationship between two individuals, thus the MAEH can dramatically reduce the redundant connections. By exploiting the MAEH, we model the multi-domain resource allocation problem in the form of mixed-integer linear programming to maximize the completed tasks. Through the topological nested characteristic of the MAEH, we propose a two-stage scheme to accomplish the resource allocation rapidly with lower computational complexity and to improve the resource utilization ratio efficiently. Simulation results validate that compared with the optimal performance, the executive time of the proposed scheme is average fivefold less under 4% extra communication resource cost. Besides, the resource utilization ratio improves over 15% by cross-domain collaboration.

Optimal Operation of Integrated Water–Power Systems Under Contingencies

With the sharply growing complexity and rapid deployment of smart technologies in our modern society, risk-aware management and coordination in day-to-day operation of the interlinked critical infrastructures is urgently needed. In particular, the interconnected water and power systems (WaPS) are in need of joint and cooperative operation to maximize the economic benefits during normal operating conditions and resilience services during emergencies. While contingency analysis is used to assist the system operators in gaining knowledge of the system’s static security, such understanding is more challenging to achieve in the case of integrated WaPS. This article proposes a novel optimization model for under-emergency operation of the integrated WaPS, considering contingencies in both networks. In order to ensure the delivery of water demand, the proposed formulation considers the hydraulic constraints of the water networks, which is naturally a nonlinear model. The proposed nonlinear model is approximated using a piece-wise linearization approach to convert the optimization model into a mixed-integer linear programming formulation. The proposed analytics are applied to a modified IEEE 24-bus reliability test system that is jointly operated with two and three commercial-scale water networks. The proposed model is evaluated using various disaster severity levels (i.e., <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N-k$</tex-math></inline-formula> contingency scenarios) and verify the promising performance of the proposed integrated WaPS model when facing failures and threatening high impact low probability emergencies.

Automating Bin Packing: A Layer Building Matheuristics for Cost Effective Logistics

In this paper, we address the problem of automating the definition of feasible pallets configurations. This issue is crucial for the competitiveness of logistic companies and is still one of the most difficult problems in internal logistics. In fact, it requires the fast solution of a three-dimensional Bin Packing Problem (3D-BPP) with additional logistic specifications that are fundamental in real applications. To this aim, we propose a matheuristics that, given a set of items, provides feasible pallets configurations that satisfy the practical requirements of items’ grouping by logistic features, load bearing, stability, height homogeneity, overhang as well as weight limits, and robotized layer picking. The proposed matheuristics combines a mixed integer linear programming (MILP) formulation of the 3D-Single Bin-Size BPP (3D-SBSBPP) and a layer building heuristics. In particular, the feasible pallets configurations are obtained by sequentially solving two MILP sub-problems: the first, given the set of items to be packed, aims at minimizing the unused space in each layer and thus the number of layers; the latter aims at minimizing the number of shipping bins given the set of layers obtained from the first problem. The approach is extensively tested and compared with existing approaches. For its validation we use both realistic data-sets drawn from the literature and real data-sets, obtained from an Italian logistics leader. The resulting outcomes show the effectiveness of the method in providing high-quality bin configurations in short computational times. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This work is motivated by the intention of facilitating the transition from Logistics 3.0 to Logistics 4.0 by providing an effective tool to automate bin packing, suitable for automated warehouses. On the one hand, the proposed technique provides stable and compact bin configurations in less than half a minute per bin on average, despite the high computational complexity of the 3D-SBSBPP. On the other hand, the approach allows to consider compatibility constraints for the items (e.g., final customer and category of the items), and the use of robotized layer picking in automated warehouses. In effect, layers composed by only one type of items (i.e., monoitem layers) can be directly picked and placed on the pallet by a robotic arm without the intervention of any operator. Consequently, the adoption of this approach in warehouses could drastically improve the efficiency of the packing process.

Stronger mixed-integer programming-formulations for order- and rack-sequencing in robotic mobile fulfillment systems

This paper addresses the order- and rack-sequencing problem at a single picking station in the context of robotic mobile fulfillment systems, a warehouse technology typically applied in large distribution centers. Following the parts-to-picker concept, items are stored on movable racks that are lifted and transported by automated guided vehicles from the storage area to picking stations for order-processing. The order-picking process involves two linked decisions: How to sequence the processing of orders and how to sequence the rack visits to supply the picking station with the requested items. We present a novel mixed-integer linear programming formulation achieving stronger linear programming bounds than a previous formulation. Including preprocessing techniques it quickly solves instances of medium-size to proven optimality for the first time in literature. For large real-world instances, we provide a three-stage heuristic solution procedure suitable in a dynamic environment, while providing competitive solutions within a short run time. Computational experiments on a broad set of benchmark instances and a comparative study with approaches from literature verify our results.

Scheduling on uniform machines with a conflict graph: complexity and resolution

AbstractThis paper deals with the problem of scheduling a set of unit‐time jobs on a set of uniform machines. The jobs are subject to conflict constraints modeled by a graph G called the conflict graph, in which adjacent jobs cannot be processed on a same machine. The objective considered herein is the minimization of maximum job completion time in the schedule, which is famous to be NP‐hard in the strong sense. The first part of this paper is an extensive study of the computational complexity of the problem restricted to several graph classes, namely: split graphs, interval graphs, forests, trees, paths and cycles. Afterward, we focus on the resolution of the problem with arbitrary conflict graphs. For this latter, a combination of a mixed integer linear programming (MILP) formulation, lower and upper bounds is proposed. A wild range of computational experiments proved the efficiency of this technique to tremendously reduce runtime and produce more optimal solutions (around 80% in average). Furthermore, a deep analysis of the resolution process based on both the density of the conflict graph as well as machine speeds (including identical machines) is thoroughly reported.

A Comprehensive Approach for an Approximative Integration of Nonlinear-Bivariate Functions in Mixed-Integer Linear Programming Models

As decentralized energy supply units, microgrids can make a decisive contribution to achieving climate targets. In this context, it is particularly important to determine the optimal size of the energy components contained in the microgrids and their optimal operating schedule. Hence, mathematical optimization methods are often used in association with such tasks. In particular, mixed-integer linear programming (MILP) has proven to be a useful tool. Due to the versatility of the different energetic components (e.g., storages, solar modules) and their special technical characteristics, linear relationships can often only inadequately describe the real processes. In order to take advantage of linear solution techniques but at the same time better represent these real-world processes, accurate and efficient approximation techniques need to be applied in system modeling. In particular, nonlinear-bivariate functions represent a major challenge, which is why this paper derives and implements a method that addresses this issue. The advantage of this method is that any bivariate mixed-integer nonlinear programming (MINLP) formulation can be transformed into a MILP formulation using this comprehensive method. For a performance comparison, a mixed-integer quadratic constrained programming (MIQCP) model—as an MINLP special case—is applied and transformed into a MILP, and the solution of the transformed problem is compared with the one of the MIQCP. Since there are good off-the-shelf solvers for MIQCP problems available, the comparison is conservative. The results for an exemplary microgrid sizing task show that the method delivers a strong performance, both in terms of approximation error (0.08%) and computation time. The method and its implementation can serve as a general user-tool but also as a basis for further methodological developments and research.