Mixed Integer Linear Programming Research Articles

Cost-effective reliability level in 100% renewables-based standalone microgrids considering investment and expected energy not served costs

Loss of load probability (LOLP) and expected energy not served (EENS) are commonly used in electrical power systems to evaluate reliability. LOLP defined as the probability that available generation capacity will be inadequate to supply customer demand. EENS defined as the expected amount of energy not being served to consumers by the system during the period considered due to system capacity shortages or unexpected power outages. Loss of Load Frequency (LOLF) is referred to a number of loss of load (LOL) event happened in the operation life span of the SMG. Loss of Load Reduction (LOLR) is defined as the required reduction in LOLF to obtain a specific reliability level. While power systems are designed to minimize LOLP and EENS, this is constrained by the total cost: investment cost, operation and maintenance cost, and cost of customer interruption (CCI). This research considers Standalone Microgrid (SMG), also known as Autonomous Microgrid which only operates in off-grid mode and cannot be connected to wider electrical power system. When designing a 100 % renewable energy integrated SMGs, it is crucial to determine the cost-effective reliability level (CERL). The CERL occurs when the total cost is minimum. This research proposes an approach to calculate the CERL for a fully renewable SMG. An analytical formulation is proposed to represent the LOLR needed to obtain a specific reliability level as a function of the required size of reliability improvement alternatives. The CCI is evaluated using LOLF and EENS indices. Finally, the total cost of the SMG system is evaluated for each reliability level. Consequently, the total cost of the SMG system is expressed as a function of reliability levels, and the minimum value of total cost and the corresponding reliability level are evaluated. In this research, a Monte Carlo Simulation (MCS) approach is used to find hourly LOLF, considering 25 years (219,000 h) of SMG lifespan, regression analysis is used for an analytical formulation, and mixed integer linear programming (MILP) is used for the investment decision making based on a cost minimisation approach. The result demonstrates that the CERL of the SMG system evaluated in the case study is 98.71 %.

Distributionally Robust Portfolio Optimization under Marginal and Copula Ambiguity

AbstractWe investigate a new family of distributionally robust optimization problem under marginal and copula ambiguity with applications to portfolio optimization problems. The proposed model considers the ambiguity set of portfolio returns in which the marginal distributions and their copula are close—in terms of the Wasserstein distance—to their nominal counterparts. We develop a cutting-surface method to solve the proposed problem, in which the distribution separation subproblem is nonconvex and includes bilinear terms. We propose three approaches to solve the bilinear formulation, namely (1) linear relaxation via McCormick inequalities, (2) exact mixed-integer linear program reformulation via disjunctive inequalities, and (3) inner approximation method via a novel iterative procedure that exploits the structural properties of the bilinear optimization problem. We further carry out a comprehensive set of computational experiments with distributionally robust portfolios featuring Conditional Value-at-Risk (CVaR) measures. These tests aim to compare the accuracy of the proposed algorithms, analyze the impact of the radius of the Wasserstein ambiguity ball on the portfolio, and assess portfolio performance. We use a rolling-horizon approach to conduct the out-of-sample tests, which show the superior performance of the portfolios under marginal and copula ambiguity over the equally weighted and ambiguity-free Mean-CVaR benchmark portfolios.

Bad data identification method considering the on-load tap changer model

With the connection and integration of renewable energy, the on-load tap-changer (OLTC) has become an important device for regulating voltage in distribution networks. However, due to non-smooth and non-linear characteristics of OLTC, traditional bad data identification and state estimation methods for transmission network are limited when applied to the distribution network. Therefore, the nonlinearity and droop control constraints of the OLTC model are considered in this paper. At the same time, the quadratic penalty function is introduced to realize the fast normalization of the tap position. It proposes a two-stage bad data identification method based on mixed-integer linear programming. In the first stage, suspicious measurements are identified using projection statistics and maximum normalized residual methods for preprocessing the measurement data. In the second stage, a linearization approach utilizing hyhrid data-physical driven is applied to handle nonlinear constraints, leading to the development of a bad data identification model based on mixed-integer linear programming. Finally, the proposed methodology is validated using a modified IEEE-33 node test feeder example, demonstrating the accuracy and efficiency of bad data identification.

Optimization of the electric vehicle charging strategy problem for sustainable intercity travels with multiple refueling stops

Electric vehicle (EV) drivers considering long-distance trips still face range anxiety due to the limited range of EVs and the scarcity of charging stations. Thus, it becomes important to ensure the feasibility of the selected route and determine an optimal charging strategy. As a crucial aspect of decision support for EV drivers, this study proposes a mixed integer linear programming (MILP) approach for the EV charging strategy problem (EVCSP), incorporating a piecewise linear approximation technique to address the challenges posed by nonlinear charging times. The proposed optimization model, namely CSPM determines where, when, and how much to charge an EV for a specified route to minimize travel time and cost. The solution time of large-scale test problems and a case study on Türkiye reveal the robustness and reliability of the CSPM. Furthermore, two multi-objective optimization methods (the weighted sum and the lexicographic method) are applied to the case study, and the results are analyzed. The results indicate that the travel cost is more sensitive to the selected charging strategy, with a range of 46.09 % across the applied charging strategies, whereas travel time remains more resilient, with a maximum fluctuation of 19.77 %. A comparative analysis with a full charging strategy reveals that the CSPM reduces the travel time by 60.1 % and improves the cost efficiency by 105.72 %.

Robust Optimization Models for Planning Drone Swarm Missions

This article presents methods of planning unmanned aerial vehicle (UAV) missions in which individual platforms work together during the reconnaissance of objects located within a terrain. The planning problem concerns determining the flight routes of a swarm, where each UAV has the ability to recognize an object using a specific type of sensor. The experiments described in this article were carried out for drone formation; one drone works as a swarm information hub and exchanges information with the ground control station (GCS). Numerical models for mission planning are presented, which take into account the important constraints, simplifying the description of the mission without too much risk of losing the platforms. Several types of objective functions were used to optimize swarm flight paths. The mission models are presented in the form of mixed integer linear programming problems (MILPs). The experiments were carried out on a terrain model built on the basis of graph and network theory. The method of building a network on which the route plan of a drone swarm is determined is precisely presented. Particular attention was paid to the description of ways to minimize the size of the network on which the swarm mission is planned. The presented methods for building a terrain model allow for solving the optimization problem using integer programming tasks.

Robust day-ahead scheduling of cooperative energy communities considering multiple aggregators

Future cities must play a vital role in reducing energy consumption and decarbonizing the electricity sector, thus evolving from passive structures towards more efficient smart cities. This transition can be facilitated by energy communities. This emerging paradigm consists of collectivizing a set of residential installations equipped with onsite renewable generators and storage assets (i.e., prosumers), which can eventually share resources to pursue collective welfare. This paper focuses on cooperative communities, where prosumers share resources without seeking selfish monetary counterparts. Despite their apparent advantages, energy management and scheduling of energy communities suppose a challenge for conventional tools due to the high level of uncertainty (especially due to intermittent renewable generation and random demand), and privacy concerns among prosumers. This paper addresses these issues. Specifically, a novel management structure based on multiple aggregators is proposed. This paradigm preserves users' confidential features while allowing them to extract the full potential of their assets. To efficiently manage the variety of assets available under uncertainty, an adaptive robust day-ahead scheduling model is developed, which casts as a solvable and portable Mixed Integer Linear Programming framework, which eases its implementation in real-world cases. The new proposal concerns uncertain generation and demand using a polyhedral representation of the uncertainty set. A case study is conducted to validate the developed model, showing promising results. Moreover, different results are obtained and analysed. Finally, it is worth remarking on how the level of robustness impacts the collective bill, incrementing it by 75 % when risk-averse conditions are assumed. In addition, the role of storage assets under pessimistic conditions is remarked, pointing out that these assets rule the scheduling plan of the community instead of renewable generators.

Optimisation of Ammonia Production and Supply Chain from Sugarcane Ethanol and Biomethane: A Robust Mixed-Integer Linear Programming Approach

Optimisation of Ammonia Production and Supply Chain from Sugarcane Ethanol and Biomethane: A Robust Mixed-Integer Linear Programming Approach

Deep reinforcement learning driven trajectory-based meta-heuristic for distributed heterogeneous flexible job shop scheduling problem

As the production environment evolves, distributed manufacturing exhibits heterogeneous characteristics, including diverse machines, workers, and production processes. This paper examines a distributed heterogeneous flexible job shop scheduling problem (DHFJSP) with varying processing times. A mixed integer linear programming (MILP) model of the DHFJSP is formulated with the objective of minimizing the makespan. To solve the DHFJSP, we propose a deep Q network-aided automatic design of a variable neighborhood search algorithm (DQN-VNS). By analyzing schedules, sixty-one types of scheduling features are extracted. These features, along with six shaking strategies, are used as states and actions. A DHFJSP environment simulator is developed to train the deep Q network. The well-trained DQN then generates the shaking procedure for VNS. Additionally, a greedy initialization method is proposed to enhance the quality of the initial solution. Seven efficient critical path-based neighborhood structures with three-vector encoding scheme are introduced to improve local search. Numerical experiments on various scales of instances validate the effectiveness of the MILP model and the DQN-VNS algorithm. The results show that the DQN-VNS algorithm achieves an average relative percentage deviation (ARPD) of 3.2%, which represents an approximately 88.45% reduction compared to the best-performing algorithm among the six compared, with an ARPD of 27.7%. This significant reduction in ARPD highlights the superior stability and performance of the proposed DQN-VNS algorithm.

Practical solutions for microgrid energy management: Integrating solar forecasting and correction algorithms

Nowadays, the shift towards renewable energies is happening at an exponential pace. Microgrids are an integral part in integrating renewable energies into the global energy mix. Due to the volatility of renewables, such as solar or wind, proper energy management is needed to avoid generation or load curtailment. This issue is addressed throughout the literature using a wide variety of strategies, ranging from classical linear to nature inspired metaheuristic optimization algorithms. This paper goes beyond the simulation phase of different optimization algorithms and addresses the optimal energy management problem by proposing a novel framework to integrate optimization strategies into the daily operation of microgrids. The proposed framework showcases 3 practical methods for controlling the distributed generators, as well as a communication scheme which facilitates the data transfer between the machine running the optimization strategy and the energy management system of the microgrid. The framework is demonstrated on a small-scale islanded microgrid setup, located at the Technical University of Cluj-Napoca. The microgrid setup consists of a photovoltaic array, a battery energy storage system and two dispatchable generators. A two-stage optimization strategy is used, consisting of a day-ahead stage, which uses the solar forecast provided by the Global Forecast System, prior to each day, to issue a working plan for the dispatchable generators, and an intra-day stage, which uses hourly solar forecasts, issued during the day, to adjust the operation of the microgrid and minimize the error between the day-ahead solar forecast and the actual weather conditions. Mixed integer linear programming is used to solve the optimization problem and validate the correct operation of the proposed framework over two case studies. The results show that the microgrid works as intended, in both scenarios, as well as the benefits of using an intra-day correction stage.

Augmented ɛ‐constraint‐based matheuristic methodology for Bi‐objective production scheduling problems

AbstractMatheuristic is an optimisation methodology that integrates mathematical approaches and heuristics to address intractable combinatorial optimisation problems, where a common framework is to insert mixed integer linear programming (MILP) models as local search functions for evolutionary algorithms. However, since a mathematical programming formulation only tries to find the solution with the best objective value, matheuristics are rarely adopted to multi‐objective scenarios asking for a set of Pareto optimal solutions, for example, vehicle routing problems and production scheduling problems. In this situation, the ɛ‐constraint, which transforms multi‐objective problems into single‐objective formulations by considering selected objectives as constraints, seems to be a promising approach. First, an augmented ɛ‐constraint‐based matheuristic methodology (ɛ‐MH) is proposed to apply the idea of ɛ‐constraint to embedded MILP models, so that Pareto fronts obtained by meta‐heuristics can be further improved by solving a set of MILP models. Afterwards, four speed‐up strategies are developed to alleviate the computational burden resulting from repeatedly solving mathematical formulations, which also imply preferable scenarios for taking advantages of the ɛ‐MH. Finally, several real‐world bi‐objective scheduling problems are discussed to present potential applications for the proposed methodology.

Game theoretic operation optimization of photovoltaic storage charging station considering uncertainty and carbon trading

With the advancement of energy conservation and emission reduction efforts, the orderly charging of electric vehicles and the operation of photovoltaic-storage-charging stations associated with electric vehicles have become increasingly important topics. This study constructs an optimization model for the operation of stations under the synergy of electricity and carbon markets from a game theory perspective. Firstly, Latin hypercube sampling and Monte Carlo sampling are employed to handle the uncertainties in photovoltaic output and the stochastic nature of electric vehicles charging. Secondly, a ladder-type carbon trading mechanism is introduced, and a charging optimization model based on Stackelberg game theory is developed to describe the benefit interaction between charging stations and electric vehicles users. In this model, the upper level represents the charging station operator aiming to maximize joint electricity and carbon revenue while minimizing load fluctuations, whereas the lower level represents electric vehicles users aiming to maximize consumer surplus. Finally, a genetic algorithm nested with mixed-integer linear programming is used to solve the optimization model. The simulation results validate the model's effectiveness and superiority. The results indicate that, compared to centralized optimization methods, the Stackelberg game mechanism can increase consumer surplus by 119.40 %, reduce carbon emissions by 217.92 %, and achieve a win-win situation for both parties. Compared to a fixed carbon trading value, the ladder-type carbon trading mechanism can reduce carbon emissions by 23.84 % and smooth load fluctuations.

An Improved Variable Neighborhood Search for the Reconfigurable Assembly Line Reconfiguring Problem

Throughout the past several decades, the manufacturing industry has been confronted with rapidly evolving market demands. The Reconfigurable Assembly Line (RAL) changes the quantity and variety of products produced through reconfiguration at different stages. The reconfiguration stages of RAL include the reallocation of operations and resources. However, existing research has neglected the resources. In the paper, a mixed integer linear program for the Reconfigurable Assembly Line Reconfiguring Problem is first formulated. Subsequently, an efficient data structure is developed to model the process precedence constraint and alternative resources. And an improved Variable Neighborhood Search which includes new problem-specific neighborhood structures and local search for resource selection is proposed to solve the problem. Computational results demonstrate the contribution of the proposed data structure and the improved procedure. Furthermore, the experiments verify the superiority and stability of the proposed method compared with other heuristic algorithms.

Stochastic Risk-driven Bidding of a Solar and Storage Aggregator in Primary Frequency and Energy Markets: A Performance-based Capacity Allocation Approach

The ever-increasing impact of deploying renewable energy resources reducing power system inertia, requires distributed energy resource (DER) aggregators to secure high-performance, fast-responding primary frequency reserve (PFR) in ancillary service markets. However, enabling immediate and local regulation of the operating point upon detecting frequency deviations imposes the necessity of allocating specific headroom for DERs. The individual headroom susceptibility to uncertainty and response speed of a certain DER not only compromises aggregated headroom performance in the PFR market but also leads to incurring unnecessary opportunity costs in the energy market and inadequate frequency control in the power system. To address this challenge, this paper introduces an integrated probabilistic performance index for DERs that considers their individual interactive heterogeneous response speed and uncertainty. Using this index, aggregators overseeing photovoltaic and smart buildings equipped with energy storage can evaluate, allocate, and remunerate optimal headroom capacity for individual DERs based on their performance. This approach ensures maximum aggregated profits and minimum opportunity costs in the PFR and energy markets while improving the performance of the provided PFR capacity, respectively. These tasks are aimed by a novel optimal capacity allocation strategy for an aggregator in both markets. This strategy employs a mixed-integer linear programming model to find the optimal solution and the conditional value-at-risk measure to tackle the uncertainties faced with the problem. The efficiency of the proposed model is assessed within a system reflecting the ERCOT market structure, demonstrating improved aggregated headroom performance, reduced energy opportunity costs, diminished risk of capacity shortage, and improved load frequency control (LFC) from the power system perspective.

A sustainable network design for municipal solid waste management considering waste-to-energy conversion under uncertainty

This research proposes a sustainable municipal solid waste management (MSWM) system to address population growth, consumerism, and resource scarcity. It introduces a two-phase decision structure combining multi-attribute decision-making (MADM) tools. Phase one develops a hybrid risk-oriented fuzzy MADM tool for selecting optimal waste-to-energy technologies, considering environmental and technological factors. Phase two uses a fuzzy bi-objective multi-period mixed-integer linear programming model to design the MSWM supply chain efficiently under uncertainty. A case study in North Tehran demonstrates the framework’s practical applicability and effectiveness in real-world scenarios, highlighting that effective source separation boosts recycling, reduces costs, and provides social benefits. Sensitivity analyses offer insights to enhance waste segregation practices. The study emphasizes integrating economic, environmental, and social sustainability in waste management decision-making. By offering a novel, holistic approach, this research addresses existing gaps in systematic decision-making processes and provides a robust tool for municipalities to optimize strategies under uncertainty.

Sustainable optimization of balancing valve settings in urban heating systems with an enhanced Jaya algorithm

With the continuous growth of global energy demand, energy conservation in heating has become a research hotspot for urban sustainable development. However, the research on energy conservation by adjusting the balance valve in the secondary network of heating system is relatively few. To address this gap, this paper solves the balancing valve opening adjustment (BVOA) problem with the goal of minimizing heat consumption and the energy usage of circulation pumps. A mixed-integer linear programming model is designed for this problem, and an effective EJaya algorithm is proposed. Initially, a random initialization method with three scrambling strategies is employed to generate high-quality solutions. Moreover, a segmented operator is proposed to enhance the algorithm’s search efficiency. Subsequently, Q-learning is utilized to select from three neighborhood operators, enhancing the algorithm’s local search capabilities. Additionally, an adaptive search balance strategy is designed to balance the methods of updating solutions, aiding in finding better solutions. Finally, extensive experiments have validated the effectiveness and efficiency of the proposed algorithm in solving the BVOA problem, contributing to the promotion of urban energy sustainability and social welfare.

Efficient Algorithm for Generating Optimal Inequality Candidates for MILP Modeling of Boolean Functions

Mixed-Integer Linear Programming (MILP) modeling has become an important tool for both the analysis and the design of symmetric cryptographic primitives. The bit-wise modeling of their nonlinear components, especially the S-boxes, is of particular interest since it allows more informative analysis compared to word-oriented models focusing on counting active S-boxes. At the same time, the size of these models, especially in terms of the number of required inequalities, tends to significantly influence and ultimately limit the applicability of this method to real-world ciphers, especially for larger number of rounds. It is therefore of great cryptographic significance to study optimal linear inequality descriptions for Boolean functions. The pioneering works of Abdelkhalek et al. (FSE 2017), Boura and Coggia (FSE 2020) and Li and Sun (FSE 2023) provided various heuristic techniques for this computationally hard problem, decomposing it into two algorithmic steps, coined Problem 1 and Problem 2, with the latter being identical to the well-known NP-hard set cover problem, for which there are many heuristic and exact algorithms in the literature. In this paper, we introduce a novel and efficient branch-and-bound algorithm for generating all minimal, non-redundant candidate inequalities that satisfy a given Boolean function, therefore solving Problem 1 in an optimal manner without relying on heuristics. We furthermore prove that our algorithm correctly computes optimal solutions. Using a number of dedicated optimizations, it provides significantly improved runtimes compared to previous approaches and allows the optimal modeling of the difference distribution tables (DDT) and linear approximation tables (LAT) of many practically used S-boxes. The source code for our algorithm is publicly available as a tool for researchers and practitioners in symmetric cryptography.

A fix-propagate-repair heuristic for mixed integer programming

AbstractWe describe a diving heuristic framework based on constraint propagation for mixed integer linear programs. The proposed approach is an extension of the common fix-and-propagate scheme, with the addition of solution repairing after each step. The repair logic is loosely based on the WalkSAT strategy for boolean satisfiability. Different strategies for variable ranking and value selection, as well as other options, yield different diving heuristics. The overall method is relatively inexpensive, as it is basically LP-free: the full linear programming relaxation is solved only at the beginning (and only for the ranking strategies that make use of it), while additional, typically much smaller, LPs are only used to compute values for the continuous variables (if any), once at the bottom of a dive. While individual strategies are not very robust in finding feasible solutions on a heterogeneous testbed, a portfolio approach proved quite effective. In particular, it could consistently find feasible solutions in 189 out of 240 instances from the public MIPLIB 2017 benchmark testbed, in a matter of a few seconds of runtime. The framework has also been implemented inside the commercial MIP solver Xpress and shown to give a small performance improvement in time to optimality on a large internal heterogeneous testbed.

Data-Driven Shelf Stocking: Robustness and Vertical-Location Effect

In this article, we study a shelf-stock allocation problem in a data-driven setting where the demand distribution is unknown. The retailer needs to decide which commodities to place on each level of the shelf (location decision), and determine the stock level for each selected commodity at the shelf (stock decision) that can be adjusted based on updated demand prediction, under constraints on allocation and capacity, so as to maximize total expected profit. Key issues of the problem include demand endogeneity on vertical-location effect and the demand ambiguity that may arise from the ever shorter life cycles of commodities and other complicated determinants. We address the problem in a rolling-horizon fashion and develop a two-phase data-driven robust optimization model leveraging a decision-dependent Wasserstein metric that incorporates time-series demand forecasts and captures vertical-location effect under demand ambiguity. We derive tractable solution schemes for the model, in which the phase-I model determines the optimal location decision by solving one instance of mixed-integer linear program, while the phase-II model admits an analytical solution for updating the optimal stock. Furthermore, we discuss the sensitivity implications on how the ambiguity aversion impacts the optimal stock, and prove the consistency of the model solution under regularity condition on the underlying demand process that well justifies the asymptotic performance of the proposed data-driven approach. Finally, we extend our model to incorporate substitution effect, and conduct sufficient numerical experiments with real-life data that demonstrate the performance of our model.

Optimizing customized bus services for multi‐trip urban passengers: A bi‐objective approach

AbstractCustomized bus services typically focus on single‐trip requests, which often struggle to accommodate the growing needs for varied multiple trips in urban daily travel. This paper addresses the customized bus routing problem for passengers with multiple trips. A bi‐objective mathematical model is established for maximizing the operational profit and minimizing the travel costs by considering the characteristics of the multi‐trip requests and time‐dependent travel time. Besides, a novel profit objective function is proposed considering the service's completion status and the starting price. Since the proposed mixed integer linear programming model is an NP‐hard problem, a non‐dominated sorting genetic algorithm II‐based method is proposed to handle different sizes of instances. Finally, the instances with multi‐trip requests are carried out to test the accuracy of the model and the effectiveness of our method compared with Gurobi and the local search‐based multi‐objective algorithm approach.

Applying the Simulated Annealing Algorithm to the Set Orienteering Problem with Mandatory Visits

This study addresses the set orienteering problem with mandatory visits (SOPMV), a variant of the team orienteering problem (SOP). In SOPMV, certain critical sets must be visited. The study began by formulating the mathematical model for SOPMV. To tackle the challenge of obtaining a feasible route within time constraints using the original MILP approach, a two-stage mixed-integer linear programming (MILP) model is proposed. Subsequently, a simulated annealing (SA) algorithm and a dynamic programming method were employed to identify the optimal route. The proposed SA algorithm was used to solve the SOP and was compared to other algorithms, demonstrating its effectiveness. The SA was then applied to solve the SOPMV problem. The results indicate that the solutions obtained using SA are superior and more efficient compared to those derived from the original MILP and the two-stage MILP. Additionally, the results reveal that the solution quality deteriorates as the ratio of the set of mandatory visits increases or the maximum allowable travel time decreases. This study represents the first attempt to integrate mandatory visits into SOP, thereby establishing a new research direction in this area. The potential impact of this research is significant, as it introduces new possibilities for addressing complex combinatorial optimization problems.