Mixed Integer Linear Programming Formulation Research Articles

Assessing and Improving Automated Viewpoint Planning for Static Laser Scanning Using Optimization Methods

Abstract. The preparation of laser scanning missions is important for efficiency and data quality. Furthermore, it is a prerequisite for automated data acquisition, which has numerous applications in the built environment, including autonomous inspections and monitoring of construction progress and quality criteria. The scene and potential scanning locations can be discretized to facilitate the analysis of visibility and quality aspects. The remaining mathematical problem to generate an economic scan strategy is the Viewpoint Planning Problem (VPP), which asks for a minimum number of scanning locations within the given scene to cover the scene under pre-defined requirements. Solutions for this problem are most commonly found using heuristics. While these efficient methods scale well, they cannot generally return globally optimal solutions. This paper investigates the VPP based on a problem description that considers quality-constrained visibility in 3D scenes and suitable overlaps between individual viewpoints for targetless registration of acquired point clouds. The methodology includes the introduction of a preprocessing method designed to simplify the input data without losing information about the problem. The paper details various solution methods for the VPP, encompassing conventional heuristics and a mixed-integer linear programming formulation, which is solved using Benders decomposition. Experiments are carried out on two case study datasets, varying in specifications and sizes, to evaluate these methods. The results show the actual quality of the obtained solutions and their deviation from optimality (in terms of the estimated optimality gap) for instances where exact solutions can not be achieved.

Mathematical programming models for order picking in warehouses with decoupling of picker and cart

As a special Travelling Salesman Problem (TSP), the Single-Picker Routing Problem (SPRP) in warehouses is of important theoretical and practical significance. In manual order picking, the items are usually picked up by a picker with a cart. To speed up the picking process, the picker is allowed to stop the cart to pick up the items individually and return to the location of the cart. This paper proposes a mixed integer linear programming formulation for an order picking problem with the decoupling of the picker and the cart in a multi-block warehouse, where the picker capacity is at most four. And based on this model, order batching problem is also analysed. Through numerical experiments, the cost of above decoupling problems is computed under different settings of the speed ratio and the capacity of the picker while he/she is travelling alone. The results indicate how the decoupling of picker and cart leads to cost reduction. Our model shows good performance for orders of small and medium scale and hence has great potential to improve the order picking operation which is executed by decoupling containers and robots/vehicles in modern digital warehouse systems.

A blockchain-based optimal peer-to-peer energy trading framework for decentralized energy management with in a virtual power plant: Lab scale studies and large scale proposal

The integration of distributed energy resources (DERs) such as wind, solar, batteries, etc. into the grid presents opportunities as well as challenges with regard to efficient coordination and trading. Virtual power plants (VPPs) have emerged as an important paradigm for aggregating and managing numerous DERs. This paper proposes a decentralized blockchain-based peer-to-peer (P2P) multilayer energy trading framework to facilitate optimized coordination of DERs within a VPP context. A control center, prosumers, and service operators constitute the model’s multi-layered architecture. Profits from feed-in tariffs, ancillary services, and P2P sales are optimized for revenue while expenses are minimized using a mixed-integer linear programming formulation. Smart contracts enable prosumers to trade energy units in the Ethereum blockchain with ease via a participatory web platform. The effectiveness of the framework in synchronizing power supply and demand within the VPP by coordinating the charging and discharging of storage systems in accordance with consumption patterns is demonstrated through performance evaluation. In contrast to conventional centralized architectures, the proposed framework utilized decentralized architecture efficiently consolidates and optimizes distributed energy resources (DERs) to improve grid stability and renewable energy utilization.

Management of a post-disaster emergency scenario through unmanned aerial vehicles: Multi-Depot Multi-Trip Vehicle Routing with Total Completion Time Minimization

One of the most valuable and promising applications for Unmanned aerial vehicles (UAVs) is in natural disaster management, where these aircraft can operate autonomously without any need for human intervention during their flights.In this paper, we foster the interface of Operational Research with computer science in general and sensor networking in particular by focusing on managing a post-disaster emergency scenario where the use of a fleet of UAVs helps rescue teams identify people needing help inside an affected area. We model this situation as an original graph theoretical problem called Multi-Depot Multi-Trip Vehicle Routing Problem with Total Completion Time minimization (MDMT-VRP-TCT). The main novelty of the MDMT-VRP-TCT is the combination of the following three features: multi-depot, multi-trip, and completion time minimization.We propose a mixed-integer linear programming (MILP) formulation, develop a matheuristic framework to address large instances, and present an extended set of experiments to test the performance of the proposed matheuristic: first, we compare the matheuristic with the MILP formulation on a set of small instances (up to 30 nodes); then, we compare our matheuristic with two heuristics from networking literature, showing that it outperforms the existing algorithms.

A Granular Approach to Optimal and Fair Patient Placement in Hospital Emergency Departments

Prolonged emergency department (ED) length of stay (LOS) is associated with detrimental effects on patient care quality and outcomes. There is evidence that certain groups of patients experience longer LOS based on their gender or race, especially with regard to the part of LOS that is attributable to waiting to be seen by a clinician. This work tackles the patient prioritization and placement aspects of ED operations with the goal of improving throughput and wait time in a fair, equitable way. We present a novel Mixed Integer Linear Programming (MILP) predictive-prescriptive formulation that incorporates a breakdown of predicted patient ED LOS into actionable pieces. We incorporate considerations for fairness and reformulate the MILP formulation into a compact and computationally tractable formulation that can be solved efficiently in real time. To deal with uncertainty, we propose a sampling-based solution, and provide provable guarantees regarding its convergence, stability and sample complexity. The proposed solution increases the throughput of the ED by [Formula: see text] and decreases the average wait time by [Formula: see text] compared to current hospital practice. In addition, the method is near-optimal in terms of throughput, and produces high-quality solutions in terms of average wait time compared to a clairvoyant oracle. Our proposed approach demonstrates desirable properties when it comes to fairness in patient prioritization, illustrating a path for addressing hidden biases in patient ED wait times and hospital operations as a whole. This work was conducted in collaboration with a large US academic medical center. Data from more than 40,000 patient visits were used to shape and evaluate the predictive-prescriptive models. An important practical contribution is translating a complex algorithm’s output into recommendations that can be operationalized in the context of existing processes in the ED. Specifically, we develop an interpretable metamodel that is trained to mimic the predictive-prescriptive algorithm’s decisions and provides a transparent set of rules for patient placement. The method will be used by the hospital to improve patient flow and quality of care as well as to support more fair and consistent bed allocation decisions.

A logistics provider’s profit maximization facility location problem with random utility maximizing followers

We introduce a strategic decision-making problem faced by logistics providers (LPs) seeking facility location decisions that lead to profitable operations. The profitability depends on the revenue generated through agreements with shippers, and the costs arising when satisfying these agreements. The latter depends, in turn, on the service levels and characteristics of the shippers’ customers. However, at a strategic level, the LP has imperfect information thereof.We propose a stochastic bilevel formulation where a given LP (leader) anticipates the decisions of shippers (followers) arising from a random utility maximization model. Using a sample average approximation and properties of the associated optimal solutions, we propose a heuristic that can compute high-quality solutions. It is based on a reduced single-level mixed integer linear programming formulation that can be solved by a general-purpose solver after a preprocessing phase. We can quickly identify situations that lead to zero expected profit for the LP. Experimental results show that expected profit highly depends on shippers’ price sensitivities. Underestimating price sensitivities can lead to an overestimation of the expected profit.

Resource Extraction Evaluation Using a Mathematical Programming Framework for Surface-Underground Mining Options and Transitions Optimization

A mathematical programming framework based on Mixed Integer Linear Programming (MILP) model for surfaceunderground mining options and transition optimization for resource extraction is presented in this paper. Existing models are mainly based on a stepwise optimization approach with limited constraints which produces localized optimal solutions and are often impractical. For mineral deposits amenable to both surface and underground mining options, the MILP framework determines the most suitable mining option and associated schedule to exploit the orebody. The MILP formulation is tested and implemented on a gold deposit case study. The NPV of the optimal mining option ($ 2.515 billion) is sensitive to the gold price, ore quantity delivered from the underground mine, and delay factor associated in supporting the operational development and stopes. Positive changes in the delay factors associated with operational development support and mining stope support have more impact on the NPV than negative changes. However, the NPV is highly sensitive to the mining stope support delay than the operational development support delay

The Continuous Time-Resource Trade-off Scheduling Problem with Time Windows

We introduce a variant of the cumulative scheduling problem (CuSP) characterized by continuous modes, time windows, and a criterion that involves safety margin maximization. The study of this variant is motivated by the Geospatial based Environment for Optimisation Systems Addressing Fire Emergencies Horizon 2020 Project, which is devoted to the design of evacuation plans in the face of natural disasters and more specifically, wildfire. People and goods have to be transferred from endangered places to safe places, and evacuation planning consists of scheduling evacuee moves along precomputed paths under arc capacities and deadlines. The resulting model is relevant in other contexts, such as project or industrial process scheduling. We consider here several formulations of the continuous time-resource trade-off scheduling problem (CTRTP-TW) with a safety maximization objective. We establish a complete complexity characterization distinguishing polynomial and NP-hard special cases depending on key parameters. We show that the problem with fixed sequencing (i.e., with predetermined overlap or precedence relations between activities) is convex. We then show that the preemptive variant is polynomial, and we propose lower and upper bounds based on this relaxation. A flow-based mixed-integer linear programming formulation is presented, from which a branch-and-cut exact method and an insertion heuristic are derived. An exact dedicated branch-and-bound algorithm is also designed. Extensive computational experiments are carried out to compare the different approaches on evacuation planning instances and on general CTRTP-TW instances. The experiments also show the interest of the continuous model compared with a previously proposed discrete approximation. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms—Discrete. Funding: This work was funded by the Horizon 2020 Marie Skłodowska-Curie Research and Innovation Staff Exchange European Project 691161 GEO-SAFE (Geospatial based Environment for Optimisation Systems Addressing Fire Emergencie). This work has also been supported by ANITI, the Artificial and Natural Intelligence Toulouse Institute. 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.2022.0142 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0142 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Optimizing service networks to support freight rail decarbonization: Flow selection, facility location, and energy sourcing

We present a framework to support decarbonization of energy intensive transportation systems offering periodic service on expansive networks (e.g., freight rail, trucking, and intercity bus services). The framework consists of two optimization problems that respectivelyaddress (i) flow selection and facility location, and (ii) energy sourcing/procurement at the service facilities to enable the selected flows. The framework generalizes mixed integer linear programming formulations for flow refueling facility location and flow-based set cover models appearing in the literature to situations where it is of interest to account for repositioning of assets along cyclical trajectories to allow for periodic service, and to account for intermediate flow capture (i.e., trip chaining). The framework also consists of a minimum cost network flow model to determine optimal energy sourcing and distribution strategies, which dictate capacity requirements at the service facilities. The energy demands are obtained from the solution to the flow selection and facility location model. To illustrate the framework, we analyze the deployment of charging stations to support battery-electric locomotive service on a subset of the US freight rail network (i.e., an aggregate network of 3 Class I Railroads). The results show that the deployment of 30 charging stations can support battery-electric locomotives (with 1600-km ranges) to serve 86% of distance-weighted flows (ton-km) and reduce emissions by approximately 50%.

Mixed-integer optimisation of graph neural networks for computer-aided molecular design

ReLU neural networks have been modelled as constraints in mixed integer linear programming (MILP), enabling surrogate-based optimisation in various domains and efficient solution of machine learning certification problems. However, previous works are mostly limited to MLPs. Graph neural networks (GNNs) can learn from non-euclidean data structures such as molecular structures efficiently and are thus highly relevant to computer-aided molecular design (CAMD). We propose a bilinear formulation for ReLU Graph Convolutional Neural Networks and a MILP formulation for ReLU GraphSAGE models. These formulations enable solving optimisation problems with trained GNNs embedded to global optimality. We apply our optimisation approach to an illustrative CAMD case study where the formulations of the trained GNNs are used to design molecules with optimal boiling points.

A Bi-Objective Home Health Care Routing and Scheduling Problem under Uncertainty.

Home health care companies provide health care services to patients in their homes. Due to increasing demand, the provision of home health care services requires effective management of operational costs while satisfying both patients and caregivers. In practice, uncertain service times might lead to considerable delays that adversely affect service quality. To this end, this paper proposes a new bi-objective optimization problem to model the routing and scheduling problems under uncertainty in home health care, considering the qualification and workload of caregivers. A mixed-integer linear programming formulation is developed. Motivated by the challenge of computational time, we propose the Adaptive Large Neighborhood Search embedded in an Enhanced Multi-Directional Local Search framework (ALNS-EMDLS). A stochastic ALNS-EMDLS is introduced to handle uncertain service times for patients. Three kinds of metrics for evaluating the Pareto fronts highlight the efficiency of our proposed method. The sensitivity analysis validates the robustness of the proposed model and method. Finally, we apply the method to a real-life case and provide managerial recommendations.

Performance Optimization Across the Edge-Cloud Continuum: A Multi-agent Rollout Approach for Cloud-Native Application Workload Placement

The advancements in virtualization technologies and distributed computing infrastructures have sparked the development of cloud-native applications. This is grounded in the breakdown of a monolithic application into smaller, loosely connected components, often referred to as microservices, enabling enhancements in the application’s performance, flexibility, and resilience, along with better resource utilization. When optimizing the performance of cloud-native applications, specific demands arise in terms of application latency and communication delays between microservices that are not taken into consideration by generic orchestration algorithms. In this work, we propose mechanisms for automating the allocation of computing resources to optimize the service delivery of cloud-native applications over the edge-cloud continuum. We initially introduce the problem’s Mixed Integer Linear Programming (MILP) formulation. Given the potentially overwhelming execution time for real-sized problems, we propose a greedy algorithm, which allocates resources sequentially in a best-fit manner. To further improve the performance, we introduce a multi-agent rollout mechanism that evaluates the immediate effect of decisions but also leverages the underlying greedy heuristic to simulate the decisions anticipated from other agents, encapsulating this in a Reinforcement Learning framework. This approach allows us to effectively manage the performance–execution time trade-off and enhance performance by controlling the exploration of the Rollout mechanism. This flexibility ensures that the system remains adaptive to varied scenarios, making the most of the available computational resources while still ensuring high-quality decisions.

An optimal solutions-guided deep reinforcement learning approach for online energy storage control

As renewable energy becomes more prevalent in the power grid, energy storage systems (ESSs) are playing an ever-increasingly crucial role in mitigating short-term supply–demand imbalances. However, the operation and control of ESS are not straightforward, given the ever-changing electricity prices in the market environment and the stochastic and intermittent nature of renewable energy generations, which respond to real-time load variations. In this paper, we propose a deep reinforcement learning (DRL) approach to address the electricity arbitrage problem associated with optimal ESS management. First, we analyze the structure of the optimal offline ESS control problem using the mixed-integer linear programming (MILP) formulation. This formulation identifies optimal control actions to absorb excess renewable energy and perform price arbitrage strategies. To tackle the uncertainties inherent in the prediction data, we then recast the online ESS control problem into a Markov Decision Process (MDP) framework and develop the DRL approach, which involves integrating the optimal offline control solution obtained from the training data into the training process and introducing noise to the state transitions. Unlike typical offline DRL training over a long time interval, we employ the Deep Deterministic Policy Gradient (DDPG) and Proximal Policy Optimization (PPO) algorithms with smaller neural networks training over a short time interval. Numerical studies demonstrate the promising potential of the proposed DRL-enabled approach for achieving better online control performance than the model predictive control (MPC) method under different price errors. This highlights the sample efficiency and robustness of our DRL approaches in managing ESS for electricity arbitrage.

Hub location with congestion and time-sensitive demand

This work studies the effect of hub congestion and time-sensitive demand on a hub-and-spoke location/allocation system. The Hub Location with Congestion and Time-sensitive Demand Problem is introduced, which combines these two main characteristics. On the one hand, hubs can be activated at several service levels, each of them characterized by a maximum capacity, expressed as the amount of flow that may circulate through the hub, which is associated with a hub transit time. On the other hand, alternative levels are available for served commodities, where each demand level is characterized by its amount of demand, unit revenue, and maximum service time. In this problem the efficiency of a hub-and-spoke system is given by the maximum net profit it may produce. To the best of our knowledge this is the first work where hub congestion and time-sensitive demand are jointly considered.Two alternative mixed-integer linear programming formulations are proposed. They include a new set of constraints, which are necessary to guarantee the consistency of the obtained solutions under the presence of the capacity-type constraints derived from hub service levels and served demand levels. The efficiency of the formulations is analyzed through a set of computational experiments. The results of the computational experiments allow to study the structure of the obtained solutions and to analyze how the different parameters affect them.

Mixed-integer linear programming formulations and column generation algorithms for the Minimum Normalized Cuts problem on networks

This paper deals with the k-way normalized cut problem in complex networks. It presents a methodology that uses mathematical optimization to provide mixed-integer linear programming formulations for the problem. The paper also develops a branch-and-price algorithm for the above-mentioned problem which scales better than the compact formulations. Additionally, a heuristic algorithm which is able to approximate large-scale image problems in those cases where the exact methods are not applicable is presented. Extensive computational experiments assess the usefulness of these methods to solve the k-way normalized cut problem. Finally, we have applied the minimum normalized cut objective function to the segmentation of actual images, showing the applicability of the introduced methodology.

A mixed-integer linear programming approach for collateral management

Collateral management involves the efficient monitoring and allocation of assets to mitigate credit risk in financial transactions and is crucial for institutions such as commercial banks, investments banks, and central counterparties. It helps to ensure financial stability and compliance with regulatory requirements. This study is centered on a fundamental objective of collateral management, namely to collateralize as much as possible a set of transactions given a limited asset portfolio. In practice, an allocation is dynamic, necessitating periodic adjustments due to fluctuations over time in the asset pool, the transactions set, and asset value. Consequently, the objective is not merely the identification of an optimal allocation from zero but the optimization of an obsolete allocation in response to these fluctuations. The size and complexity of real-world problems often preclude direct resolution through conventional MILP solvers. In addition, each collateral adjustment movement incurs associated costs. In this research, a two-step method based on linear and mixed-integer linear programming formulations with ℓ1 penalties is devised, with the principal aim of minimizing under-allocation while simultaneously controlling the number of adjustment movements required. The method is tested on 20 real-world collateral givers and invariably reaches a good trade-off between solution quality, computational time, and the number of movements.

Identifying Socially Optimal Equilibria Using Combinatorial Properties of Nash Equilibria in Bimatrix Games

Nash equilibrium is arguably the most fundamental concept in game theory, which is used to analyze and predict the behavior of the players. In many games, there exist multiple equilibria, with different expected payoffs for the players, which in turn raises the question of equilibrium selection. In this paper, we study the [Formula: see text]-hard problem of identifying a socially optimal Nash equilibrium in two-player normal-form games (called bimatrix games), which may be represented by a mixed integer linear program (MILP). We characterize the properties of the equilibria and develop several classes of valid inequalities accordingly. We use these theoretical results to provide a decomposition-based reformulation of the MILP, which we solve by a branch-and-cut algorithm. Our extensive computational experiments demonstrate superiority of our approach over solving the MILP formulation through feeding it into a commercial solver or through the “traditional” Benders’ decomposition. Of note, our proposed approach can find provably optimal solutions for many instances. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms. Funding: This work was supported by the National Institute of Dental and Craniofacial Research [Grant R01DE028283]. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health. The funding agreements ensured the authors’ independence in designing the study, interpreting the data, writing, and publishing the report. 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.2022.0072 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0072 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Markov decision process design: A framework for integrating strategic and operational decisions

We consider the problem of optimally designing a system for repeated use under uncertainty. We develop a modeling framework that integrates the design and operational phases, which are represented by a mixed-integer program and discounted-cost infinite-horizon Markov decision processes, respectively. We seek to simultaneously minimize the design costs and the subsequent expected operational costs. This problem setting arises naturally in several application areas, as we illustrate through examples. We derive a bilevel mixed-integer linear programming formulation for the problem and perform a computational study to demonstrate that realistic instances can be solved numerically.

Autonomous Ride-Sharing Service Using Graph Embedding and Dial-a-Ride Problem: Application to the Last-Mile Transit in Lyon City

Autonomous vehicles are anticipated to revolutionize ride-sharing services and subsequently enhance the public transportation systems through a first–last-mile transit service. Within this context, a fleet of autonomous vehicles can be modeled as a Dial-a-Ride Problem with certain features. In this study, we propose a holistic solving approach to this problem, which combines the mixed-integer linear programming formulation with a novel graph dimension reduction method based on the graph embedding framework. This latter method is effective since accounting for heterogeneous travel demands of the covered territory tends to increase the size of the routing graph drastically, thus rendering the exact solving of small instances computationally infeasible. An application is provided for the real transport demand of the industrial district of “Vallée de la Chimie” in Lyon city, France. Instances involving more than 50 transport requests and 10 vehicles could be easily solved. Results suggest that this method generates routes of reduced nodes with lower vehicle kilometers traveled compared to the constrained K-means-based reduction. Reductions in terms of GHG emissions are estimated to be around 75% less than the private vehicle mode in our applied service. A sensitivity analysis is also provided.

A Mixed‐Integer Linear Programming Framework for Optimization of Water Network Operations Problems

AbstractWater distribution systems (WDSs) are critical infrastructure used to convey water from sources to consumers. The mathematical framework governing the distribution of flows and heads in extended period simulations of WDSs lends itself to application in a wide range of optimization problems. Applying the classical mixed integer linear programming (MILP) approach to model WDSs hydraulics within an optimization framework can contribute to higher solution accuracy with lower computational effort. However, adapting WDSs models to conform to a MILP formulation has proven challenging because of the intrinsic non‐linearity of system hydraulics and the complexity associated with modeling hydraulic devices that influence the state of the WDS. This paper introduces MILPNet, an adjustable framework for WDSs that can be used to build and solve an extensive array of MILP optimization problems. MILPNet includes constraints that represent the mass balance and energy conservation equations, hydraulic devices, control rules, and status checks. To conform to MILP structure, MILPNet employs piece‐wise linear approximation and integer programming. MILPNet was implemented and tested using Gurobi Python API. Modeling accuracy was shown to be comparable to EPANET, a public domain software for hydraulic modeling, and sensitivity analyses were conducted to examine the impacts of the modeling assumptions on the performance of MILPNet. Additionally, application of the framework was demonstrated using pump scheduling optimization examples in single and rolling horizon scenarios. Our results show that MILPNet can facilitate the construction and solution of optimization problems for a range of applications in WDSs operations.