Mixed Integer Programming Solver Research Articles

Optimizing the design and operation of water networks: Two decomposition approaches

We consider the design and operation of water networks simultaneously. Water network problems can be divided into two categories: the design problem and the operation problem. The design problem involves determining the appropriate pipe sizing and placements of pump stations, while the operation problem involves scheduling pump stations over multiple time periods to account for changes in supply and demand. Our focus is on networks that involve water co-produced with oil and gas. While solving the optimization formulation for such networks, we found that obtaining a primal (feasible) solution is more challenging than obtaining dual bounds using off-the-shelf mixed-integer nonlinear programming solvers. Therefore, we propose two methods to obtain good primal solutions. One method involves a decomposition framework that utilizes a convex reformulation, while the other is based on time decomposition. To test our proposed methods, we conduct computational experiments on a network derived from the PARETO case study.

Configuring Mixed-Integer Programming Solvers for Large-Scale Instances

Algorithm configuration techniques automatically search for parameters of solvers and algorithms that provide minimal runtime or maximal solution quality on specified instance sets. Mixed-integer programming (MIP) solvers pose a particular challenge for algorithm configurators due to the difficulty of finding optimal, or even feasible, solutions on the large-scale problems commonly found in practice. We introduce the OPTANO Algorithm Tuner (OAT) to find configurations for MIP solvers and other optimization algorithms. We present and evaluate several critical components of OAT for solving MIPs in particular and show that OAT can find configurations that significantly improve the solution time of MIPs on two different datasets.

Loss-optimal classification trees: a generalized framework and the logistic case

Classification trees are one of the most common models in interpretable machine learning. Although such models are usually built with greedy strategies, in recent years, thanks to remarkable advances in mixed-integer programming (MIP) solvers, several exact formulations of the learning problem have been developed. In this paper, we argue that some of the most relevant ones among these training models can be encapsulated within a general framework, whose instances are shaped by the specification of loss functions and regularizers. Next, we introduce a novel realization of this framework: specifically, we consider the logistic loss, handled in the MIP setting by a piece-wise linear approximation, and couple it with ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-regularization terms. The resulting optimal logistic classification tree model numerically proves to be able to induce trees with enhanced interpretability properties and competitive generalization capabilities, compared to the state-of-the-art MIP-based approaches.

Matheuristic for the lot-sizing and scheduling problem in integrated pulp and paper production

Vertical pulp and paper production is challenging from a process point of view. Managers must deal with floating bottlenecks, intermediate storage levels, and by-product production to control the whole process while reducing unexpected downtimes. Thus, this paper aims to address the integrated lot sizing and scheduling problem considering continuous digester production, multiple paper machines, and a chemical recovery line to treat by-products. The aim is to minimize the total production cost to meet customer demands, considering all productive resources and encouraging steam production (which can be used in power generation). Production planning should define the sizes of production lots, the sequence of paper types produced in each machine, and the digester working speed throughout the planning horizon. Furthermore, it should indicate the rate of by-product treatment at each stage of the recovery line and ensure the minimum and maximum storage limits. Due to the difficulty of exactly solving the mixed integer programming model representing this problem for real-world instances, mainly with planning horizons of over two weeks, constructive and improvement heuristics are proposed in this work. Different heuristic combinations are tested on hundreds of instances generated from data collected from the industry. Comparisons are made with a commercial Mixed-Integer and Linear Programming solver and a hybrid metaheuristic. The results show that combining the greedy constructive heuristic with the new variation of a fix-and-optimize improvement method delivers the best performance in both solution quality and computational time and effectively solves realistic size problems in practice. The proposed method achieved 69.41% of the best solutions for the generated set and 55.40% and 64.00% for the literature set for 1 and 2 machines, respectively, compared with the best solution method from the literature and a commercial solver.

Parallel Beam Search Algorithms for Domain-Independent Dynamic Programming

Domain-independent dynamic programming (DIDP), a model-based paradigm based on dynamic programming, has shown promising performance on multiple combinatorial optimization problems compared with mixed integer programming (MIP) and constraint programming (CP). The current DIDP solvers are based on heuristic search, and the state-of-the-art solver, complete anytime beam search (CABS), uses beam search. However, the current DIDP solvers cannot utilize multiple threads, unlike state-of-the-art MIP and CP solvers. In this paper, we propose three parallel beam search algorithms and develop multi-thread implementations of CABS. With 32 threads, our multi-thread DIDP solvers achieve 9 to 39 times speedup on average and significant performance improvement over the sequential solver, finding the new best solutions for two instances of the traveling salesperson problem with time windows. In addition, our solvers outperform multi-thread MIP and CP solvers in four of the six combinatorial optimization problems evaluated.

Solution algorithms for dock scheduling and truck sequencing in cross-docks: A neural branch-and-price and a metaheuristic

In this study, we present a novel mathematical model for the integrated challenge of dock scheduling and truck sequencing in cross-docking facilities. Many existing models for variations of this problem in the literature rely on big-M constraints, known for their poor performance when applied to general-purpose mixed-integer programming solvers. This introduces significant limitations, especially when tackling smaller instances. Consequently, most solution methods in the literature gravitate towards metaheuristic approaches. Our proposed model offers a compact formulation without any big-M constraints, utilizing 4-index variables. It is strategically designed to lend itself well to a dual decomposition approach (Dantzig–Wolfe) and can be effectively solved using a branch-and-price methodology. We address the pricing problem through a branch-and-cut scheme and illustrate its efficiency in handling large instances. Recognizing the pricing problem as the primary bottleneck, we employ a deep neural network trained on a comprehensive set of instances with the same distribution to predict promising duals. Extensive computational experiments demonstrate a notable reduction of over 50% in overall computational times. Additionally, we introduce a heuristic sharing similarities with certain Variable Neighborhood Search (VNS) approaches. Our proposed heuristic has proven highly efficient in extensive computational experiments.

Exact Solution of the Single-Picker Routing Problem with Scattered Storage

We present a new modeling approach for the single-picker routing problem with scattered storage (SPRP-SS) for order picking in warehouses. The SPRP-SS assumes that an article is, in general, stored at more than one pick position. The task is then the simultaneous selection of pick positions for requested articles and the determination of a minimum-length picker tour collecting the articles. It is a classical result of Ratliff and Rosenthal that, for given pick positions, an optimal picker tour is a shortest path in the state space of a dynamic program with a linear number of states and transitions. We extend the state space of Ratliff and Rosenthal to include scattered storage so that every feasible picker tour is still a path. The additional requirement to make consistent decisions regarding articles to collect is not modeled in the state space so that dynamic programming can no longer be used for the solution. Instead, demand covering can be modeled as additional constraints in shortest-path problems. Therefore, we solve the resulting model with a mixed-integer (linear) programming (MIP) solver. The paper shows that this approach is not only convenient and elegant for single-block parallel-aisle warehouse SPRP-SSs but also generalizable: the solution principle can be applied to different warehouse layouts (we present additional results for a two-block parallel-aisle warehouse) and can incorporate further extensions. Computational experiments with other approaches for the SPRP-SS show that the new modeling approach outperforms the available exact algorithms regarding computational speed. History: Accepted by Andrea Lodi, Area Editor for Design and Analysis of Algorithms—Discrete. Supplemental Material: The e-companion is available at https://doi.org/10.1287/ijoc.2023.0075 .

A General Framework for Providing Interval Representations of Pareto Optimal Outcomes for Large-Scale Bi- and Tri-Criteria MIP Problems

The Multi-Objective Mixed-Integer Programming (MOMIP) problem is one of the most challenging. To derive its Pareto optimal solutions one can use the well-known Chebyshev scalarization and Mixed-Integer Programming (MIP) solvers. However, for a large-scale instance of the MOMIP problem, its scalarization may not be solved to optimality, even by state-of-the-art optimization packages, within the time limit imposed on optimization. If a MIP solver cannot derive the optimal solution within the assumed time limit, it provides the optimality gap, which gauges the quality of the approximate solution. However, for the MOMIP case, no information is provided on the lower and upper bounds of the components of the Pareto optimal outcome. For the MOMIP problem with two and three objective functions, an algorithm is proposed to provide the so-called interval representation of the Pareto optimal outcome designated by the weighting vector when there is a time limit on solving the Chebyshev scalarization. Such interval representations can be used to navigate on the Pareto front. The results of several numerical experiments on selected large-scale instances of the multi-objective multidimensional 0–1 knapsack problem illustrate the proposed approach. The limitations and possible enhancements of the proposed method are also discussed.

Maximizing value yield in wood industry through flexible sawing and product grading based on wane and log shape

The optimization of sawing processes in the wood industry is critical for maximizing efficiency and profitability. The introduction of computerized tomography scanners provides sawmill operators with three-dimensional internal models of logs, which can be used to assess value and yield more accurately. We present a methodology for solving the sawing optimization problem employing a flexible sawing scheme that allows greater flexibility in cutting logs into products while considering product quality classes influenced by wane defects. The methodology has two phases: preprocessing and optimization. In the preprocessing phase, two alternative algorithms are given that generate and evaluate the potential sawing positions of products by considering the 3D surface of the log, product size requirements, and product quality classes. In the optimization phase, a maximum set-packing problem is solved for the preprocessed data using mixed-integer programming (MIP), aiming to obtain a feasible cut pattern that maximizes value yield. This is implemented in a system named FlexSaw, which takes advantage of parallel computation during the preprocessing phase and utilizes a MIP solver during the optimization phase. The proposed sawing methods are evaluated on the Swedish Pine Stem Bank. Additionally, FlexSaw is compared with an existing tool that utilizes cant sawing. Results demonstrate the superiority of flexible sawing. While the practical feasibility of implementing a flexible way of sawing logs is constrained by the limitations of current sawmill machinery, the potential increase in yield promotes the exploration of alternative machinery in the wood industry.

Instance space analysis for 2D bin packing mathematical models

In this paper, we apply Instance Space Analysis (ISA) to study the two-dimensional bin-packing problem. We consider classical and newly-generated instances to test the performance of four mixed-integer programming (MIP) models from the literature. This is the first time ISA is used to compare MIP models. We set as a performance metric the time taken by the black-box MIP solver CPLEX to obtain a proven optimal solution when running each model. Our results provide a new perspective on the different models’ performance according to each instance’s features.

Optimal scheduling of BESS for congestion management considering reliability and OTS

The present research explores transmission congestion control in Battery based Energy Storage Systems (BESS) integrated grids using Optimal Transmission Switching (OTS). The oversimplification of underlying assumptions in prior OTS-based congestion management strategies leads to excessive line switching which is undesirable. Hence, a novel reliability constrained alternating current optimal power flow (AC-OPF) based OTS is presented to mitigate the line congestion in the BESS integrated grids. The suggested approach is solved to acquire the OTS solution for 24 h time frame with the primary goal of minimising generating costs subjected to ac power flow equality, BESS, reliability, and transmission congestion constraints. The proposed approach makes use of the integer variable to represent the transmission switching operations and is solved by using a mixed-integer non-linear programming (MINLP) solver. The inclusion of reliability constraints ensures the system reliability by limiting the value of load failure probability to a pre-specified limit. The feasibility of this approach is validated by testing it on a standard IEEE test system for various loading scenarios. The results exhibit that coordinated charging/discharging of BESS supports reducing the number of lines switchings required for congestion management compared to the existing approaches, substantially.

GAMMA: Graph Attention Model for Multiple Agents to Solve Team Orienteering Problem With Multiple Depots.

In this work, we present an attention-based encoder-decoder model to approximately solve the team orienteering problem with multiple depots (TOPMD). The TOPMD instance is an NP-hard combinatorial optimization problem that involves multiple agents (or autonomous vehicles) and not purely Euclidean (straight line distance) graph edge weights. In addition, to avoid tedious computations on dataset creation, we provide an approach to generate synthetic data on the fly for effectively training the model. Furthermore, to evaluate our proposed model, we conduct two experimental studies on the multi-agent reconnaissance mission planning problem formulated as TOPMD. First, we characterize the model based on the training configurations to understand the scalability of the proposed approach to unseen configurations. Second, we evaluate the solution quality of the model against several baselines-heuristics, competing machine learning (ML), and exact approaches, on several reconnaissance scenarios. The experimental results indicate that training the model with a maximum number of agents, a moderate number of targets (or nodes to visit), and moderate travel length, performs well across a variety of conditions. Furthermore, the results also reveal that the proposed approach offers a more tractable and higher quality (or competitive) solution in comparison with existing attention-based models, stochastic heuristic approach, and standard mixed-integer programming solver under the given experimental conditions. Finally, the different experimental evaluations reveal that the proposed data generation approach for training the model is highly effective.

Matheuristic for synchronized vehicle routing problem with multiple constraints and variable service time: Managing a fleet of sprayers and a tender tanker

This paper considers an extension of the vehicle routing problem with synchronization constraints and introduces the vehicle routing problem with multiple synchronization constraints and variable service time. This important problem is motivated by a real-world problem faced by one of the largest agricultural companies in the world providing precision agriculture services to their clients who are farmers and growers. The solution to this problem impacts the performance of farm spraying operations and can help design policies to improve spraying operations in large-scale farming. We propose a Mixed Integer Programming (MIP) model for this challenging problem, along with problem-specific valid inequalities. A three-phase powerful matheuristic is proposed to solve large instances enhanced with a novel local search method. We conduct extensive numerical analysis using realistic data. Results show that our matheuristic is fast and efficient in terms of solution quality and computational time compared to the state-of-the-art MIP solver. Using real-world data, we demonstrate the importance of considering an optimization approach to solve the problem, showing that the policy implemented in practice overestimates the costs by 15%–20%. Finally, we compare and contrast the impact of various decision-maker preferences on several key performance metrics by comparing different mathematical models.

A mixed-integer approximation of robust optimization problems with mixed-integer adjustments

In the present article we propose a mixed-integer approximation of adjustable-robust optimization problems, that have both, continuous and discrete variables on the lowest level. As these trilevel problems are notoriously hard to solve, we restrict ourselves to weakly-connected instances. Our approach allows us to approximate, and in some cases exactly represent, the trilevel problem as a single-level mixed-integer problem. This allows us to leverage the computational efficiency of state-of-the-art mixed-integer programming solvers. We demonstrate the value of this approach by applying it to the optimization of power systems, particularly to the control of smart converters.

Prioriactions: Multi‐action management planning in R

AbstractDesigning effective conservation strategies requires deciding not only where to locate conservation actions (i.e. which territorial units should be priortized), but also which type actions should be deployed. For most of conservation planning contexts, deciding where and what to do usually yields a complex and computationally challenging decision‐making setting. Although the resulting optimization problems have typically been tackled using heuristic approaches, recent advances in mixed integer programming (MIP) solver technology have turned MIP‐based approaches into a practical alternative for solving complex conservation planning problems.We introduce theRpackageprioriactions, which allows solving complex conservation planning problems comprising prioritization and action deployment decisions.prioriactionsfeatures a MIP approach that allows formulating and solving optimally (or nearly optimally) a wide class of conservation planning problems (characterized by different spatial and functional constraints and requirements). Furthermore, the package allows using a variety of commercial and open‐source exact solvers enhancing its usability as well as its practical effectiveness.Here, we present a comprehensive description of the main functions available inprioriactions. This package has a workflow of three straightforward steps: (a) validation of the input data, using theinputData()function that prepares input; (b) the creation of a prioritization model, using theproblem()function, allows the creation of two types of common models: theminimization of coststo achieve a recovery target andmaximizing the recovery benefitsgiven a limited budget; and (c) to solve of the model, using thesolve()function.Theprioriactionspackage provides a user‐friendly platform for addressing different multi‐actions management problems, allowing to identify more rigorously, transparently and in a reproducible way the spatial deployment of management actions.

Fast Globally Optimal Catalog Matching using MIQCP

We propose a novel exact method to solve the probabilistic catalog matching problem faster than previously possible. Our new approach uses mixed integer programming and introduces quadratic constraints to shrink the problem by multiple orders of magnitude. We also provide a method to use a feasible solution to dramatically speed up our algorithm. This gain in performance is dependent on how close to optimal the feasible solution is. Also, we are able to provide good solutions by stopping our mixed integer programming solver early. Using simulated catalogs, we empirically show that our new mixed integer program with quadratic constraints is able to be set up and solved much faster than previous large linear formulations. We also demonstrate our new approach on real-world data from the Hubble Source Catalog. This paper is accompanied by publicly available software to demonstrate the proposed method.

Mediating governance goals with patients and nurses satisfaction: a multi-actor multi-objective problem including fairness

ABSTRACT Balancing conflicting goals among different stakeholders is a challenging problem in various application domains. In this paper, we analyse it in the context of home healthcare. Fairness objective functions for nurses and patients are combined with system-level governance goals of the territorial centers in charge of the assistance service. From the solution of several multi-objective problems including one objective for each actor hierarchically ordered, the best goal for each stakeholder is identified and interesting managerial conclusions are drawn. To efficiently solve large-size instances, we introduce a parallel Adaptive Large Neighborhood Search algorithm with destroy and repair operators customised to solve multi-objective multi-actor problems. The algorithm proves to be highly efficient and effective when compared to a commercial Mixed Integer Programming solver, either in its plain form or enforced by a mild start and a primal heuristic. Additionally, we devise a metaheuristic method to generate the Pareto frontier of an instance.

Mixed-integer programming models for irregular strip packing based on vertical slices and feasibility cuts

The irregular strip-packing problem, also known as nesting or marker making, is defined as the automatic computation of a non-overlapping placement of a set of non-convex polygons onto a rectangular strip of fixed width and unbounded length, such that the strip length is minimized. Nesting methods based on heuristics are a mature technology, and currently, the only practical solution to this problem. However, recent performance gains of the Mixed-Integer Programming (MIP) solvers, together with the known limitations of the heuristics methods, have encouraged the exploration of exact optimization models for nesting during the last decade. Despite the research effort, there is room to improve the efficiency of the current family of exact MIP models for nesting. In order to bridge this gap, this work introduces a new family of continuous MIP models based on a novel formulation of the NoFit-Polygon Covering Model (NFP-CM), called NFP-CM based on Vertical Slices (NFP-CM-VS). Our new family of MIP models is based on a new convex decomposition of the feasible space of relative placements between pieces into vertical slices, together with a new family of valid inequalities, symmetry breakings, and variable eliminations derived from the former convex decomposition. Our experiments show that our new NFP-CM-VS models outperform the current state-of-the-art MIP models. Ten instances are solved up to optimality within one hour for the first time, including one with 27 pieces. Finally, we provide a detailed reproducibility protocol and dataset as supplementary material to allow the exact replication of our models, experiments, and results.

The min-Knapsack problem with compactness constraints and applications in statistics

In the min-Knapsack problem, one is given a set of items, each having a certain cost and weight. The objective is to select a subset with minimum cost, such that the sum of the weights is not smaller than a given constant. In this paper, we introduce an extension of the min-Knapsack problem with additional “compactness constraints” (mKPC), stating that selected items cannot lie too far apart. This extension has applications in statistics, including in algorithms for change-point detection in time series. We propose three solution methods for the mKPC. The first two methods use the same Mixed-Integer Programming (MIP) formulation but with two different approaches: passing the complete model with a quadratic number of constraints to a black-box MIP solver or dynamically separating the constraints using a branch-and-cut algorithm. Numerical experiments highlight the advantages of this dynamic separation. The third approach is a dynamic programming labelling algorithm. Finally, we focus on the particular case of the unit-cost mKPC (1c-mKPC), which has a specific interpretation in the context of the statistical applications mentioned above. We prove that the 1c-mKPC is solvable in polynomial time with a different ad-hoc dynamic programming algorithm. Experimental results show that this algorithm vastly outperforms both generic approaches for the mKPC and a simple greedy heuristic from the literature.

Instance-specific algorithm configuration via unsupervised deep graph clustering

Instance-specific Algorithm Configuration (AC) methods are effective in automatically generating high-quality algorithm parameters for heterogeneous NP-hard problems from multiple sources. However, existing works rely on manually designed features to describe training instances, which are simple numerical attributes and cannot fully capture structural differences. Targeting at Mixed-Integer Programming (MIP) solvers, this paper proposes a novel instances-specific AC method based on end-to-end deep graph clustering. By representing an MIP instance as a bipartite graph, a random walk algorithm is designed to extract raw features with both numerical and structural information from the instance graph. Then an auto-encoder is designed to learn dense instance embeddings unsupervisedly, which facilitates clustering heterogeneous instances into homogeneous clusters for training instance-specific configurations. Experimental results on multiple benchmarks show that the proposed method can improve the solving efficiency of CPLEX on highly heterogeneous instances, and outperform existing instance specific AC methods.