Mixed-integer Programming Research Articles

Learning to Cut via Hierarchical Sequence/Set Model for Efficient Mixed-Integer Programming.

Cutting planes (cuts) play an important role in solving mixed-integer linear programs (MILPs), which formulate many important real-world applications. Cut selection heavily depends on (P1) which cuts to prefer and (P2) how many cuts to select. Although modern MILP solvers tackle (P1)-(P2) by human-designed heuristics, machine learning carries the potential to learn more effective heuristics. However, many existing learning-based methods learn which cuts to prefer, neglecting the importance of learning how many cuts to select. Moreover, we observe that (P3) what order of selected cuts to prefer significantly impacts the efficiency of MILP solvers as well. To address these challenges, we propose a novel hierarchical sequence/set model (HEM) to learn cut selection policies. Specifically, HEM is a bi-level model: (1) a higher-level module that learns how many cuts to select, (2) and a lower-level module-that formulates the cut selection as a sequence/set to sequence learning problem-to learn policies selecting an ordered subset with the cardinality determined by the higher-level module. To the best of our knowledge, HEM is the first data-driven methodology that well tackles (P1)-(P3) simultaneously. Experiments demonstrate that HEM significantly improves the efficiency of solving MILPs on eleven challenging MILP benchmarks, including two Huawei's real problems.

Optimal energy management in a grid-tied solar PV-battery microgrid for a public building under demand response

Commercial buildings consume a substantial amount of energy, underscoring the need for consumption optimization. Embedded microgrids combined with demand side management strategies have potential to help end-users and utilities to better manage the grid. This paper presents an integrated optimal control stratrgy for a grid-tied solar PV-battery microgrid powering a public building under demand response program. Current literature predominantly focuses on residential buildings and, most importantly, only addresses a limited range of energy management objectives. In this study, a mixed-integer non-linear programming (MINLP) mathematical model that minimizes grid energy costs, grid peak demand, end-user inconvenience associated of appliance scheduling, and battery degradation is presented. To test and validate the model, two case study scenarios are considered: Case I applies the proposed model to a grid-tied, solar-PV-battery powered public building without demand response, while Case II integrates demand response into the resulting hybrid energy system. Simulation results from Case I demonstrate the model’s potential to reduce the building’s energy costs by 49%, with an additional 4.4% reduction achieved in Case II. Compared to the baseline model, the Case II model achieves a 37.5% reduction in system peak demand (coincident peak) and a 21.8% decrease in the building’s non-coincident peak demand. The proposed microgrid retrofit model is economically viable, with a payback period of 9-10 years. The obtained results can be used as a decision-making framework for planning embedded microgrids in public buildings with demand response programs.

Machine Learning-based Dispatching for a Wet Clean Station in Semiconductor Manufacturing

The concept of cyber manufacturing has become a critical element in semiconductor fabrication environments, where automation and systemization are integral, for addressing the growing complexity of processes and facilitating predictive capabilities through data integration. This study deals with the dispatching problem to minimize makespan at a wet clean station in semiconductor fabrication using artificial intelligence-enabled manufacturing control techniques. The wet clean station is comprised of sequential chemical and rinsing baths for cleaning wafer lots and multiple robot arms for lot handling. In the station, wafer lots are sequentially immersed in several baths for cleaning to eliminate residual contaminants and stains that cause defects on wafer surfaces. The station can process various types of products, and the specific order of immersion differs depending on the product type. Unlike typical dispatching problems, the information required for dispatching, such as processing times and sequences inside the station, is not available. The only available data are historical logs that record when each lot enters and leaves the station. However, even when cleaning the same product type, the duration that lots spend in the station may vary based on the combination of product types being cleaned simultaneously and the settings of the station. Thus, using the time records, this study proposes a dispatching method based on machine learning models (multiple linear regression, deep neural network, and convolutional neural network). The proposed algorithms were evaluated and verified by comparing them with CPLEX solving a mixed integer programming and dispatching methods used in a semiconductor fab in Korea. Through this experiment, we observed that the proposed models can provide dispatching solutions that are practical and effective in a rapidly changing production setting. These models have the potential to enhance the capacity of a wet clean station and will contribute to artificial intelligence-based manufacturing system control.

An Improved Combinatorial Benders Decomposition Algorithm for the Human-Robot Collaborative Assembly Line Balancing Problem

As an emerging technology, human-robot collaboration (HRC) has been implemented to enhance the performance of assembly lines and improve the safety of human workers. By integrating the advantages of human workers and collaborative robots (cobots), HRC enables production systems to process tasks consecutively, concurrently, or collaboratively. However, the introduction of cobots also makes the corresponding human-robot collaborative assembly line balancing problem more complex and difficult to solve. To solve this problem, we first propose an enhanced mixed integer program (EMIP) with various enhancement techniques and tighter bounds, and then, we develop an improved combinatorial Benders decomposition algorithm (Algorithm ICBD) with new local search strategies, Benders cuts, and acceleration procedures. To verify the effectiveness of our proposed model and algorithms, we conduct extensive computational experiments, and the results show that our proposed EMIP model is significantly better than the existing mixed integer program model; the percentages of instances that can obtain feasible and optimal solutions are increased from 82.42% to 100% and from 29.17% to 43.5%, respectively, whereas the average gap is decreased from 19.81% to 5.64%. In addition, our proposed Algorithm ICBD can get 100% of feasible solutions and 65.92% of optimal solutions for all of the test instances, and the average gap is only 1.49%. Moreover, compared with existing Benders decomposition methods for this problem, our approach yields comparatively better solutions in notably shorter average computational time when run in the same computational environment. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: This research was supported by the National Natural Science Foundation Council of China [Grants 72401214, 92167206, 7221101377, 72471169, and 72231005], the Ministry of Education of China [Grant 24YJC630078], and Computation and Analytics of Complex Management Systems (Tianjin University). This research was also supported by the Tianjin Natural Science Foundation Project [Grant 23JCQNJC01900] and the Tianjin Philosophy and Social Science Planning Project [Grant TJGL21-016]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0279 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0279 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Optimal Coordination of Directional Overcurrent Relays in Microgrids Considering European and North American Curves

Protecting AC microgrids (MGs) is a challenging task due to their dual operating modes—grid-connected and islanded—which cause sudden variations in fault currents. Traditional protection methods may no longer ensure network security. This paper presents a novel approach to protection coordination in AC MGs using non-standard features of directional over-current relays (DOCRs). Three key optimization variables are considered: Time Multiplier Setting (TMS), the plug setting multiplier’s (PSM) maximum limit, and the standard characteristic curve (SCC). The proposed model is formulated as a mixed-integer nonlinear programming problem and solved using four metaheuristic techniques: the genetic algorithm (GA), Imperialist Competitive Algorithm (ICA), Harmonic Search (HS), and Firefly Algorithm (FA). Tests on a benchmark IEC MG with distributed generation and various operating modes demonstrate that this approach reduces coordination times compared to existing methods. This paper’s main contributions are threefold: (1) introducing a methodology for assessing the optimal performance of different standard curves in MG protection; (2) utilizing non-standard characteristics for optimal coordination of DOCRs; and (3) enabling the selection of curves from both North American and European standards. This approach improves trip time performance across multiple operating modes and topologies, enhancing the reliability and efficiency of MG protection systems.

An Optimization Approach for Biosurveillance in Wastewater Networks

Biosurveillance is the ongoing systematic collection, analysis, and interpretation of biospheric data essential to planning, implementing, and evaluating national security interests. Biosurveillance efforts often focus on disease activity and threats to human, animal, or plant health to improve situational awareness of and early warnings of disease emergence or activity. Wastewater-based epidemiology (WBE) is an important subdomain of biosurveillance that monitors the consumption of, or exposure to, chemicals and pathogens at the community/population level. However, ecological, environmental, economic, cultural, and political variability yields an asymmetric landscape of threats, subsequent exposure(s), and community burden. This work presents an exact approach based on mixed-integer programming to optimize sensor placement in a wastewater network. The primary goal is to maximize the amount of information captured from the network while narrowing down the potential geographical sources of chemical or biological markers (i.e., signals). We use the wastewater systems of Los Angeles (US) and Regina (CAN) to demonstrate the scalability of our approach in realistic scenarios and how it can enhance stakeholders’ ability to identify emerging threats or assess the effectiveness of strategic interventions post-deployment.

A Modified Simulated Annealing and Enhanced Harmony Search Algorithms for the Continuous Facility Layout Problem (CFLP)

This paper focuses on solving the Continuous Facility Layout Problem (CFLP), which aims to minimize material handling costs by strategically placing facilities in a known or unknown area. The objective is to find optimal or near-optimal facility arrangements while adhering to non-overlapping and spatial constraints, thus enhancing the efficiency of production systems. The study addresses various layout scenarios, including single-row facility layout (with and without clearance), continuous layout, and unequal-area facility layout problems. To achieve this, the authors propose a mixed-integer nonlinear programming (MINLP) model tailored for continuous layouts. Two meta-heuristic algorithms are developed to optimize this model: a hybrid Simulated Annealing-Genetic Algorithm (SA-GA), which leverages genetic crossover operations, and an Enhanced Harmony Search Algorithm (EHSA), featuring dynamic parameters and a novel improvisation technique. The proposed methods provide flexible and efficient solutions for both small and large-scale layout problems, offering decision-makers practical tools for real-world applications.

Incentive Determination for Demand Response Considering Internal Rate of Return

The rapid expansion of renewable energy sources has led to increased instability in the power grid of Jeju Island, leading to the implementation of the plus demand response (DR) system, which aims to boost electricity consumption during curtailment periods. However, the frequency of curtailment owing to the increased utilization of renewable energy is outpacing the implementation of plus DR, highlighting the need for additional resources, such as energy storage systems (ESS). High initial investment costs have been the primary hindrance to the adoption of ESS by DR-participating companies but have not been fully considered in earlier studies on DR incentive determination. Therefore, this study proposes an algorithm for calculating appropriate incentives for plus DR participation considering the investment costs required for ESS. Based on actual load data, incentives are determined using an iterative mixed-integer programming (MIP) optimization method that progressively adjusts the incentive level to address the overall nonlinearity arising from both the multiplication of variables and the nonlinear characteristics of the internal rate of return (IRR), ensuring that the target IRR is achieved. A case study on the impact of factors such as IRR, ESS costs, and fluctuations in electricity rates on incentive calculations demonstrated that plus DR incentives required to achieve IRR targets of 5%, 10%, and 15% have increased linearly from 142.2 KRW/kWh to 363.0 KRW/kWh, confirming that the appropriate incentive level can be effectively determined based on ESS investment costs and target IRR. This result could help promote ESS adoption among DR companies and plus DR participation, thereby enhancing power grid stability.

Optimizing Connected Components Graph Partitioning With Minimum Size Constraints Using Integer Programming and Spectral Clustering Techniques

ABSTRACTIn this work, a graph partitioning problem in a fixed number of connected components is considered. Given an undirected graph with costs on the edges, the problem consists of partitioning the set of nodes into a fixed number of subsets with minimum size, where each subset induces a connected subgraph with minimal edge cost. The problem naturally surges in applications where connectivity is essential, such as cluster detection in social networks, political districting, sports team realignment, and energy distribution. Mixed Integer Programming formulations together with a variety of valid inequalities are demonstrated and computationally tested. An assisted column generation approach by spectral clustering is also proposed for this problem with additional valid inequalities. Finally, the methods are tested for several simulated instances, and computational results are discussed. Overall, the proposed column generation technique enhanced by spectral clustering offers a promising approach to solve clustering and partitioning problems.

The team orienteering problem with variable time windows

AbstractThe orienteering problem is well‐established within the domain of operational research. This paper introduces the team orienteering problem with variable time windows (TOPVTW) as a new variant of the team orienteering problem with time windows. The distinguishing feature of the TOPVTW is the adaptive time windows, which evolve dynamically as a consequence of the solution. This problem arises within the context of spread processes that must be contained. This propagation dictates the time windows, which should be observed by the teams. In turn, these teams must take actions influencing the spread and, consequently, the time windows. The time windows can affect both the nodes and the paths. Thus, two distinct versions of the problem are delineated, each supported by dedicated mixed integer programming models. An application is illustrated optimizing resources for wildfire suppression. This work serves as an initial stride toward further exploration and analysis within this domain.

Configuration of liveness-enforcing initial marking with the minimum resources for resource allocation systems

The enforcement of liveness is crucial for Petri nets models of resource allocation systems (RASs). It is interesting yet very challenging to establish an initial marking for Petri net plants so that the net is live. Such an initial marking is referred to as liveness-enforcing initial marking (LIM). Despite existing literature presenting various LIMs, no studies have addressed the issue of minimizing the number of resources in an LIM. This work focuses on designing an LIM with the minimum resources (LIM-MR) for a class of Petri nets called systems of sequential systems with shared resources (S4PRs) by assigning a token capacity to each resource place, such that the sum of all involved resources is minimized. This work first establishes a kind of necessary and sufficient liveness condition for S4PR, which is then encoded into a series of variables and constraints in a mixed-integer programming (MIP) formulation. Although LIM-MR may not be unique, solving the proposed MIP formulation can obtain at least one LIM-MR for S4PR under consideration. The experimental results show the solvability of this approach for S4PRs.

A multi-objective energy-efficient scheduling algorithm for solving hybrid flow workshops with parallel heterospeed machines

Abstract. In hybrid flow assembly workshops, minimizing completion time and reducing energy consumption are two key indicators of efficient workshop management, which are often inversely related. This study investigates how parallel heterospeed machines with adjustable times can reduce energy costs and assembly times at the production system level. To address these issues, we propose the MO-HFSP-Batch algorithm based on the enhanced NSGA-II algorithm, aimed at improving the algorithm's optimization capability. Given the trade-offs between optimization objectives and the high computational complexity of the proposed multi-objective mixed integer programming, a three-tier chromosome encoding structure was introduced during the algorithm design phase. This structure meets the triple requirements of batching assembly tasks, matching operations with parallel machines, and sequencing operations in actual hybrid flow assembly workshops. Extensive data analysis proves that our proposed algorithm effectively solves the scheduling issues in hybrid flow workshops and performs better in solving completion time indicators and stability of non-dominated solution sets than other algorithms.

Optimum Planning of Carbon Capture and Storage Network Using Goal Programming

Carbon capture and storage (CCS) is a critical technology used for mitigating climate change by capturing carbon dioxide emissions from industrial sources and storing them underground to prevent their release into the atmosphere. Despite its potential, optimizing CCS systems for cost-effectiveness and efficiency improvement remains a significant challenge. In this paper, the optimization of CCS systems through the development and application of two mathematical optimization techniques is introduced. The first technique is based on using a superstructure optimization model, while the second technique relies on applying a goal programming optimization model. These models were solved using LINGO software version API 14.0.5099.166 to enhance the efficiency and cost-effectiveness of CCS systems. The first model, seeking to maximize the exchange of CO2 flowrate from sources to sinks, achieved a CO2 capture rate of 93.36% with an annual total cost of USD 1.175 billion. The second model introduced a novel mixed-integer non-linear programming (MINLP) approach for multi-objective optimization, targeting the minimization of total system cost, alternative storage, and unutilized storage while maximizing CO2 load exchange. The application of the second model, when prioritized to maximize CO2 flowrate exchange using the goal programming technique, resulted in a cost reduction of 36.46% and a CO2 capture rate of 75.87%. In contrast, when the second model prioritized minimizing the total annual cost, a 48% cost reduction was achieved, and the CO2 capture rate was decreased by 68.37%. A comparison of the two models’ results is presented. The results showed that the second model, with the priority of maximizing CO2 capture, provides the best economic–environmental objective balance, which offers notable cost reductions while keeping an efficient CO2 capture rate. This study highlights the potential of advanced mathematical modeling in increasing the feasibility of CCS as one of the very important strategies of mitigating climate change and reducing global warming.

Station-based, free-float, or hybrid: An operating mode analysis of a bike-sharing system

The profit-oriented bike-sharing industry, such as Mobike in China and Limebike in the USA, has widely adopted the free-float mode, which allows users to rent and return bikes without any restrictions with regard to stations. Compared to the traditional station-based mode, this new mode improves convenience and satisfies a variety of travel demands, but it also results in higher operating costs and lower number of available bikes (the number of bikes can be found). So far, no theoretical study has been done comparing the performance levels of the two different operating modes. Some bike-sharing firms are now attempting to design a hybrid operating mode that combines the advantages of the two modes. We design a mixed-integer program to model these three operating modes under different operating conditions. The model captures some of the new features of free-float bike-sharing systems. These new features include discounts for returning bikes to designated stations, different numbers of available bikes at stations and free-float areas, differences in station-based return levels as well as in convenience levels in the three operating modes. This study also considers the interactions between the free-float and station-based modes in meeting the demand and in managing relocation processes. Extensive numerical experiments are conducted and the results reveal that in more than half of the total cases under different parameter settings, the profits of the hybrid mode outperform the profits of the other two modes. The station-based mode is suitable for settings with high convenience levels, low number of available free-float bikes, and limited budgets for bike acquisitions, whereas the free-float mode can ensure high profits with low convenience levels and a large number of bikes being available. Our models can serve as a decision-making tool for bike-sharing firms in their selection of an optimal operating mode.

Modeling EV dynamic wireless charging loads and constructing risk constrained operating strategy for associated distribution systems

Dynamic wireless charging (DWC) is an emerging technology that enables the charging of electric vehicles (EVs) while they are in motion. However, previous load modeling methods have not thoroughly explored the detailed analysis of DWC load characteristics. Existing research only considers the single-node supply mode for dynamic wireless charging roads (DWCRs), and the assessment of operational risks arising from the uncertain DWC loads has not been addressed. This paper begins by conducting an equivalent circuit analysis of a typical EV DWC system with multiple segmented coils. We present a more accurate trapezoidal power model for a single EV. Subsequently, we model the aggregated EV DWC load, accounting for traffic flow and headway using Poisson and negative exponential distribution functions, respectively. In the operation process, we consider a multi-node supply mode for DWCRs. To address the inaccuracy of long-term predictions, we propose a rolling optimization model to coordinate DWC and renewables with heterogeneous uncertainties by introducing a risk metric to manage potential uncertain risks. The proposed optimization model is transformed into a mixed-integer second-order cone programming (MISOCP) problem after convex relaxation. Finally, we conduct case studies to validate the proposed methods.

Renewables usage maximization in automated distribution networks by coordinated operation of dynamic line rating and dynamic network reconfiguration

Incorporation of smart-grid technologies (SGTs) in today’s electric distribution networks (DNs) has enabled distribution network operators (DNOs) to have an on-line supervision over the network equipment for optimal operation. This paper proposes a new approach for optimal scheduling of active DNs aiming at minimizing the curtailment power of wind and photovoltaic (PV) units as renewable energy sources (RESs). This purpose is fulfilled by minimizing the imported power from the transmission network resulting in maximization of renewables usage and minimization of power loss. The conducted approach is based on dynamic line rating (DLR) and dynamic network reconfiguration (DNR) as flexibility options. A convex formulation is employed to incorporate the objective function and constraints into a mixed-integer quadratically-constrained programming (MIQCP) model which is solved by global optimum solvers in GAMS. The proposed model is applied to the IEEE 33-bus system under different case studies, and the simulation results are analyzed. The obtained results indicate that the maximum scheduling of wind and PV units’ is fulfilled with the minimum energy losses. By the aid of DNR and DLR in a coordinated manner, the renewables scheduling is increased by about 64% while the energy loss is reduced by 29% compared to the base case.

Enhancing offshore parcel delivery efficiency through vessel-unmanned aerial vehicle collaborative routing

Offshore oil and gas production is vital for global energy supply, but it faces logistical challenges due to the high costs and inefficiencies of traditional supply methods. This paper introduces the vessel-unmanned aerial vehicle (UAV) routing problem with multiple visits in a single flight (VURP-M), a novel logistical model that addresses aimed at enhancing the replenishment of offshore platforms with small, essential items. The VURP-M allows the UAV to perform multiple visits during a single flight, optimising the delivery process. To tackle the VURP-M, we propose two mixed-integer second-order cone programs that capture the problem's complexities. Given its NP-hard nature, we employ an adaptive large neighbourhood search (ALNS) method, featuring a segmented initialisation process and problem-specific operators guided by a rule-based mechanism to improve solution efficiency. The ALNS formulates an initial solution by solving a travelling salesman problem to create a giant tour, which is then used to group sequential targets into multiple UAV flights. The subsequent optimal resolution of a SOCP determines the take-off and landing points for each flight. Subsequently, the ALNS refines the initial solution through destroy and repair operators, enhancing the search for superior sequences and allocation schemes. The effectiveness of our approach is demonstrated through a real-world case study and numerical experiments on random instances featuring up to 100 platforms. The results offer implications of the collaborative vessel-UAV model for the offshore logistics industry.

Single-machine scheduling with setup time: a case study in the cutting of large-scale structural parts

This article considers a single-machine scheduling problem with setup time to optimize the cutting schedule for structural parts within a railway equipment manufacturing enterprise. When performing parts-cutting operations for various orders, factors such as priority, the material attribute of the steel plate and part specifications need to be considered. Notably, it is beneficial to ensure continuous cutting to minimize steel waste when these parts need the same steel plates. A mixed integer programming model and a two-stage heuristic are proposed to determine the cutting sequence of the parts. The two-stage heuristic involves preprocessing and local search. Finally, the performance of the proposed approaches is assessed using real-world cutting production data. This includes scenarios with more than 10,000 parts, allowing for a thorough evaluation. The results indicate that the proposed two-stage heuristic exhibits significant efficiency and quality advantages compared to the Gurobi tm solver and local search.

Data-driven prediction of relevant scenarios for robust combinatorial optimization

We study iterative constraint and variable generation methods for (two-stage) robust combinatorial optimization problems with discrete uncertainty. The goal of this work is to find a set of starting scenarios that provides strong lower bounds early in the process. To this end we define the Relevant Scenario Recognition Problem (RSRP) which finds the optimal choice of scenarios which maximizes the corresponding objective value. We show for classical and two-stage robust optimization that this problem can be solved in polynomial time if the number of selected scenarios is constant and NP-hard if it is part of the input. Furthermore, we derive a linear mixed-integer programming formulation for the problem in both cases.Since solving the RSRP is not possible in reasonable time, we propose a machine-learning-based heuristic to determine a good set of starting scenarios. To this end, we design a set of dimension-independent features, and train a Random Forest Classifier on already solved small-dimensional instances of the problem. Our experiments show that our method is able to improve the solution process even for larger instances than contained in the training set, and that predicting even a small number of good starting scenarios can considerably reduce the optimality gap. Additionally, our method provides a feature importance score which can give new insights into the role of scenario properties in robust optimization.

A GRASP-based multi-objective approach for the tuna purse seine fishing fleet routing problem

Nowadays, the world’s fishing fleet uses 20% more fuel to catch the same amount of fish compared to 30 years ago. Addressing this negative environmental and economic performance is crucial due to stricter emission regulations, rising fuel costs, and predicted declines in fish biomass and body sizes due to climate change. Investment in more efficient engines, larger ships and better fuel has been the main response, but this is only feasible in the long term at high infrastructure cost. An alternative is to optimize operations such as the routing of a fleet, which is an extremely complex problem due to its dynamic (time-dependent) moving target characteristics. To date, no other scientific work has approached this problem in its full complexity, i.e., as a dynamic vehicle routing problem with multiple time windows and moving targets. In this paper, two bi-objective mixed linear integer programming (MIP) models are presented, one for the static variant and another for the time-dependent variant. The bi-objective approaches allow to trade off the economic (e.g., probability of high catches) and environmental (e.g., fuel consumption) objectives. To overcome the limitations of exact solutions of the MIP models, a greedy randomized adaptive search procedure for the multi-objective problem (MO-GRASP) is proposed. The computational experiments demonstrate the good performance of the MO-GRASP algorithm with clearly different results when the importance of each objective is varied. In addition, computational experiments conducted on historical data prove the feasibility of applying the MO-GRASP algorithm in a real context and explore the benefits of joint planning (collaborative approach) compared to a non-collaborative strategy. Collaborative approaches enable the definition of better routes that may select slightly worse fishing and planting areas (2.9%), but in exchange for a significant reduction in fuel consumption (17.3%) and time at sea (10.1%) compared to non-collaborative strategies. The final experiment examines the importance of the collaborative approach when the number of available drifting fishing aggregation devices (dFADs) per vessel is reduced.