Exact Mixed-integer Programming Research Articles

Exact Mixed-Integer Programming Approach for Chance-Constrained Multi-Area Reserve Sizing

Exact Mixed-Integer Programming Approach for Chance-Constrained Multi-Area Reserve Sizing

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 balanced maximally diverse grouping problem with attribute values

The paper investigates the assignment of items to groups such that all groups are as heterogeneous as possible and each pair of groups is as balanced as possible, i.e. all groups get the same number of items assigned, the total diversity is maximized, and all pairs of groups are as homogeneous as possible regarding the attribute values of their assigned items. We present a two-step exact mixed-integer programming approach, whereat the first step is a maximally diverse grouping problem (MDGP), i.e. our approach extends the MDGP by a balancing step. Moreover, we present a detailed complexity analysis for both steps and heuristic solution approaches based on 2-opt, very large-scale neighborhood search, and variable neighborhood search, which are, together with the mixed-integer program, evaluated in a computational study.

Enhanced solution capabilities for multiple patch land allocation

Multi-patch land allocation refers to the selection of management units to form individual patches, where each patch is contiguous and restricted in spatial extent. The purpose of a patch can vary, but often involves a coordinated activity or usage, such as species conservation, rehabilitation, treatment, waste management and others. The goal then is to identify a set of patches that provides the greatest total benefit, with strict limits on size combined with inter-patch connectivity. This paper focuses on exact mixed-integer programming solution approaches of a multi-patch problem. Methods for formulation strengthening are introduced and formally developed. Land allocation associated with wildfire fuels treatment for the USDA Forest Service is investigated. Application results highlight enhanced solution capabilities, and demonstrate planning support possible through explicit spatial optimization approaches.

Length-constrained cycle partition with an application to UAV routing*

This article discusses the Length-Constrained Cycle Partition Problem (LCCP), which constitutes a new generalization of the Travelling Salesperson Problem (TSP). Apart from nonnegative edge weights, the undirected graph in LCCP features a nonnegative critical length parameter for each vertex. A cycle partition, i.e. a vertex-disjoint cycle cover, is a feasible solution for LCCP if the length of each cycle is not greater than the critical length of each vertex contained in it. The goal is to find a feasible partition having a minimum number of cycles. Besides analyzing theoretical properties and developing preprocessing techniques, we propose an elaborate heuristic algorithm that produces solutions of good quality even for large-size instances. Moreover, we present two exact mixed-integer programming formulations (MIPs) for LCCP, which are inspired by well-known modeling approaches for TSP. Further, we introduce the concept of conflict hypergraphs, whose cliques yield valid constraints for the MIP models. We conclude with a discussion on computational experiments that we conducted using (A)TSPLIB-based problem instances. As a motivating example application, we describe a routing problem where a fleet of uncrewed aerial vehicles (UAVs) must patrol a given set of areas.

Optimal Allocation and Sizing of PV Generation Units in Distribution Networks via the Generalized Normal Distribution Optimization Approach

The problem of optimal siting and dimensioning of photovoltaic (PV) generators in medium-voltage distribution networks is addressed in this research from the perspective of combinatorial optimization. The exact mixed-integer programming (MINLP) model is solved using a master–slave (MS) optimization approach. In the master stage, the generalized normal distribution optimization (GNDO) with a discrete–continuous codification is used to represent the locations and sizes of the PV generators. In the slave stage, the generalization of the backward/forward power method, known as the successive approximation power flow method, is adopted. Numerical simulations in the IEEE 33-bus and 69-bus systems demonstrated that the GNDO approach is the most efficient method for solving the exact MINLP model, as it obtained better results than the genetic algorithm, vortex-search algorithm, Newton-metaheuristic optimizer, and exact solution using the General Algebraic Modeling System (GAMS) software with the BONMIN solver. Simulations showed that, on average, the proposed MS optimizer reduced the total annual operative costs by approximately 27% for both test feeders when compared with the reference case. In addition, variations in renewable generation availability showed that from 30% ahead, positive reductions with respect to the reference case were obtained.

Investigating the recoverable robust single machine scheduling problem under interval uncertainty

We investigate the recoverable robust single machine scheduling problem under interval uncertainty. In this setting, jobs have first-stage processing times p and second-stage processing times q and we aim to find a first-stage and second-stage schedule with a minimum combined sum of completion times, such that at least Δ jobs share the same position in both schedules.We provide positive complexity results for some important special cases of this problem, as well as derive a 2-approximation algorithm to the full problem. Computational experiments examine the performance of an exact mixed-integer programming formulation and the approximation algorithm, and demonstrate the strength of a proposed polynomial time greedy heuristic.

Minimizing Airplane Boarding Time

The time it takes passengers to board an airplane is known to influence the turnaround time of the aircraft and thus bears a significant cost-saving potential for airlines. Although minimizing boarding time therefore is the most important goal from an economic perspective, previous efforts to design efficient boarding strategies apparently never tackled this task directly. In this paper, we first rigorously define the problem and prove its NP-hardness. Although this generally justifies the development of inexact solution methods, we show that all commonly discussed boarding strategies may in fact give solutions that are far from optimal. We complement these theoretical findings by a simple time-aware boarding strategy with guaranteed approximation quality (under reasonable assumptions) as well as a local improvement heuristic and an exact mixed-integer programming (MIP) formulation. Our numerical experiments with simulation data show that for several airplane cabin layouts, provably high-quality or even optimal solutions can be obtained within reasonable time in practice by means of our MIP approach. We also empirically assess the sensitivity of boarding strategies with respect to disruptions of the prescribed boarding sequences and identify robustness against such disruptions as a bottleneck for further improvements.

Mixed-integer programming model and hybrid driving algorithm for multi-product partial disassembly line balancing problem with multi-robot workstations

To address the problem of considerable waste electromechanical product generation, a partial disassembly line balancing problem with multi-robot workstations that can synchronously disassemble multiple products (MPR-PDLBP) is investigated to improve the product capacity and efficiency of existing disassembly lines. First, an exact mixed-integer programming model is established to accurately obtain the minimum disassembly objectives: cycle time, energy consumption, and improved hazardous index. Second, compared with the conventional disassembly line balancing problem (DLBP), the solution space and optimization difficulty of MPR-PDLBP increase significantly. Thus, a multi-objective hybrid driving algorithm (HDA) based on a three-layer encoding method with a heuristic rule is proposed to effectively address MPR-PDLBP, and a driving strategy is proposed to improve the exploitation ability and convergence speed of HDA. Finally, validity of the proposed model and algorithm are verified by comparing the calculation results of GUROBI and HDA for two small-scale cases. The superiority of HDA is proved by comparing the optimization results of a large-scale case with three other classic algorithms.

Models and Algorithms for the Bin-Packing Problem with Minimum Color Fragmentation

In the bin-packing problem with minimum color fragmentation (BPPMCF), we are given a fixed number of bins and a collection of items, each associated with a size and a color, and the goal is to avoid color fragmentation by packing items with the same color within as few bins as possible. This problem emerges in areas as diverse as surgical scheduling and group event seating. We present several optimization models for the BPPMCF, including baseline integer programming formulations, alternative integer programming formulations based on two recursive decomposition strategies that utilize decision diagrams, and a branch-and-price algorithm. Using the results from an extensive computational evaluation on synthetic instances, we train a decision tree model that predicts which algorithm should be chosen to solve a given instance of the problem based on a collection of derived features. Our insights are validated through experiments on the aforementioned applications on real-world data. Summary of Contribution: In this paper, we investigate a colored variant of the bin-packing problem. We present and evaluate several exact mixed-integer programming formulations to solve the problem, including models that explore recursive decomposition strategies based on decision diagrams and a set partitioning model that we solve using branch and price. Our results show that the computational performance of the algorithms depends on features of the input data, such as the average number of items per bin. Our algorithms and featured applications suggest that the problem is of practical relevance and that instances of reasonable size can be solved efficiently.

Sparse regression at scale: branch-and-bound rooted in first-order optimization

We consider the least squares regression problem, penalized with a combination of the \(\ell _{0}\) and squared \(\ell _{2}\) penalty functions (a.k.a. \(\ell _0 \ell _2\) regularization). Recent work shows that the resulting estimators enjoy appealing statistical properties in many high-dimensional settings. However, exact computation of these estimators remains a major challenge. Indeed, modern exact methods, based on mixed integer programming (MIP), face difficulties for problems where the number of features \(p \sim 10^4\). In this work, we present a new exact MIP framework for \(\ell _0\ell _2\)-regularized regression that can scale to \(p \sim 10^7\), achieving speedups of at least 5000x, compared to state-of-the-art exact methods. Unlike recent work, which relies on commercial MIP solvers, we design a specialized nonlinear branch-and-bound (BnB) framework, by critically exploiting the problem structure. A key distinguishing component in our framework lies in efficiently solving the node relaxations using a specialized first-order method, based on coordinate descent (CD). Our CD-based method effectively leverages information across the BnB nodes through using warm starts, active sets, and gradient screening. In addition, we design a novel method for obtaining dual bounds from primal CD solutions, which certifiably works in high dimensions. Experiments on synthetic and real high-dimensional datasets demonstrate that our framework is not only significantly faster than the state of the art, but can also deliver certifiably optimal solutions to statistically challenging instances that cannot be solved with existing methods. We open source the implementation through our toolkit L0BnB.

Exact and Evolutionary Algorithms for Synchronization of Public Transportation Timetables Considering Extended Transfer Zones

This article addresses timetable synchronization in public transportation, an important problem in modern smart cities, in order to guarantee a proper quality of service to citizens. Two variants of the bus timetabling synchronization problem considering extended transfer zones are studied: optimizing offsets and optimizing offsets and headways for each line. An exact mixed integer programming and an evolutionary algorithm are developed to solve both problem variants. The algorithms are evaluated on 45 instances of a real case study, the intelligent transportation system of Montevideo, Uruguay. Experimental results reported significant improvements over the current timetable implemented by the city administration. The number of successful synchronizations improved up to 66.6% and 179.9% for the first and second problem variant, respectively. The average waiting times for transfers improved, especially in tight problem instances (up to 57.8% and 158.3% for the first and second problem variant, respectively). The proposed planning methods are useful to help decision makers to configure public transportation systems.

Optimization of Station-Skip in a Cyclic Express Subway Service

With rapid population growth and increasing demand for urban mobility, metropolitan areas such as Singapore, Tokyo, and Shanghai are increasingly dependent on public transport systems. Various strategies are proposed to improve the service quality and capacity of bus and subway systems. Express trains—i.e., trains that skip certain stations—are commonly used because they can travel at higher speeds, potentially reduce travel time, and serve more passengers. In this paper, we study cyclic express subway service (CESS), in which express trains provide routine transport service with cyclic (periodic) station-skip patterns that can be used in daily service. We propose an exact Mixed Integer Programming (MIP) model to optimize cyclic station-skip patterns for express trains operating in a single-track subway system. The objective is to reduce passengers’ total travel time—i.e., the sum of waiting time and riding time—while considering demand intensity and distribution and train headway, frequency, and capacity. We implement the model in a set of numerical experiments using real data from Singapore. To solve the optimization problem more efficiently, we also propose a heuristic to solve large-scale problems. We observe that the exact MIP model for CESS provides optimal cyclic express service patterns within a reasonable computational time, and the heuristic method can significantly reduce the computational time and provide a good solution. The case study demonstrates that passengers’ average travel time could be significantly reduced compared to local train service. We also discuss the potential transfer of passengers between express trains and evaluate its effects using numerical experiments.

Scheduling spatially distributed jobs with degradation: Application to pothole repair

This paper considers scheduling spatially distributed jobs with degradation. A mixed integer programming (MIP) model is developed for the linear degradation case in which no new jobs arrive. Properties of the model are analyzed, following which three heuristics are developed, enhanced greedy, chronological decomposition and simulated annealing. Numerical tests are conducted to: (i) establish limits of the exact MIP solution, (ii) identify the best heuristic based on an analysis of performance on small problem instances for which exact solutions are known, (iii) solve large problem instances and obtain lower bounds to establish solution quality, and (iv) study the effect of three key model parameters. Findings from our computational experiments indicate that: (i) exact solutions are limited to instances with less than 14 jobs; (ii) the enhanced greedy heuristic followed by the application of the simulated annealing heuristic yields high quality solutions for large problem instances in reasonable computation time; and (iii) the factors “degradation rate” and “work hours” have a significant effect on the objective function. To demonstrate applicability of the model, a case study is presented based on a pothole repair scenario from Buffalo, New York, USA. Findings from the case study indicate that scheduling spatially dispersed jobs with degradation such as potholes requires: (i) careful consideration of the number of servers assigned, degradation rate and depot location; (ii) appropriate modeling of continuously arriving jobs; and (iii) appropriate incorporation of equity consideration.

Designing time-of-use tariffs in electricity retail markets using a bi-level model – Estimating bounds when the lower level problem cannot be exactly solved

Time-of-use tariffs are a pricing strategy for a product or service in which the supplier establishes time-differentiated prices. Dynamic (e.g., day-ahead) time-differentiated electricity prices can contribute to increase the retailer's profit, allow end-users to reduce the consumption costs and enhance grid efficiency.The electricity retailer and the consumer are hierarchically related. The interaction between them can be modeled by a bi-level (BL) optimization model – the retailer is the upper level decision maker and the consumer is the lower level decision maker. The retailer and the consumer have different and conflicting goals: the retailer establishes the pricing scheme to sell electricity to consumers to maximize his profit; the consumer reacts to these prices by determining the operation of the controllable loads in order to minimize the discomfort and the electricity bill.In this work, a BL optimization model incorporating shiftable, interruptible and thermostatic loads is proposed. The upper level problem is tackled by a particle swarm optimization algorithm while the lower level problem is solved by an exact mixed-integer programming solver. The inclusion of the thermostatic load in the lower level problem imposes a much higher computational burden. Therefore, it may not be possible to find the optimal lower level solution, and a sub-optimal lower level solution is infeasible to the BL problem. Considering a computational budget, this work proposes an approach to compute good quality estimates of bounds for the upper level objective function, providing the leader further information and allowing him to make sounder decisions in an adequate time frame.

Models and algorithms for maximum flow problems having semicontinuous path flow constraints

ABSTRACTThis article considers a variation of the node- and arc-capacitated maximum flow problem having semicontinuous flow restrictions. A semicontinuous variable must either take a value of zero or belong in the interval [ℓ, u] for some 0 < ℓ ⩽ u. Of particular interest are problems in which the variables correspond to the amount of flow sent along an entire origin–destination path. We examine both static and dynamic variations of this problem. As opposed to solving a Mixed-Integer Programming (MIP) model, we propose a branch-and-price algorithm for the static version of this problem, including a specialized branching strategy that leverages the existence of cut-sets in a non-feasible solution. For the dynamic version of the problem, we present an exact MIP formulation along with a set of symmetry-breaking and subtour-elimination constraints to improve its solvability. We demonstrate the efficacy of our algorithms on a set of randomly generated test instances.

Routing and Scheduling for a Last-Mile Transportation System

The last-mile problem concerns the provision of travel services from the nearest public transportation node to a passenger’s home or other destination. We study the operation of an emerging last-mile transportation system (LMTS) with batch demands that result from the arrival of groups of passengers who desire last-mile service at urban metro stations or bus stops. Routes and schedules are determined for a multivehicle fleet of delivery vehicles, with the objective of minimizing passenger waiting time and riding time. An exact mixed-integer programming (MIP) model for LMTS operations is presented first, which is difficult to solve optimally within acceptable computational times. Computationally feasible heuristic approaches are then developed: a myopic operating strategy that uses only demand information from trains that have already arrived, a metaheuristic approach based on a tabu search that employs demand information over the entire service horizon, and a two-stage method that solves the MIP model approximately over the entire service horizon. These approaches are implemented in a number of computational experiments to evaluate the system’s performance, and demonstrate that LMTS is notably preferable to a conventional service system under certain conditions.

Minimizing total weighted tardiness and overtime costs for single machine preemptive scheduling

This paper studies the scheduling of a finite set of jobs on a single resource that operates under both regular and overtime capacity modes. Jobs, which can be preempted, have associated release and due dates. Limited overtime capacity can be utilized to reduce tardiness. However, since overtime is costly, justification of the overtime use depends on the trade-off between the tardiness and the overtime costs. The overall objective is to minimize the total cost of tardiness and overtime. To achieve this objective, we develop a holistic method composed of three-stages. We first provide a heuristic based on an effective priority rule for the base case where no overtime capacity is considered. This heuristic is later employed in the first-stage to produce a compact non-delay schedule built based on the assumption that overtime capacity incurs no additional cost. In the second stage, the overtime usage is reduced by shifting workload and generating a full-delay schedule without altering the tardiness of jobs produced in the first stage. The third stage improves the total costs by altering the tardiness of jobs in return for savings in overtime utilization. Using computational tests, we compare the performance of our heuristics to the upper bounds generated by the exact mixed-integer programming formulation. The results show that the proposed method is efficient in obtaining solutions that are considerably better than the generated upper bounds in significantly short times and as such, it can be quite useful as an effective solution approach especially for large size problems.

A MODEL-BASED HEURISTIC FOR THE IRREGULAR STRIP PACKING PROBLEM

ABSTRACT The irregular strip packing problem is a common variant of cutting and packing problems. Only a few exact methods have been proposed to solve this problem in the literature. However, several heuristics have been proposed to solve it. Despite the number of proposed heuristics, only a few methods that combine exact and heuristic approaches to solve the problem can be found in the literature. In this paper, a matheuristic is proposed to solve the irregular strip packing problem. The method has three phases in which exact mixed integer programming models from the literature are used to solve the sub-problems. The results show that the matheuristic is less dependent on the instance size and finds equal or better solutions in 87,5% of the cases in shorter computational times compared with the results of other models in the literature. Furthermore, the matheuristic is faster than other heuristics from the literature.

A model and optimization-based heuristic for the operational aircraft maintenance routing problem

This paper investigates the Operational Aircraft Maintenance Routing Problem (OAMRP). Given a set of flights for a specific homogeneous fleet type, this short-term planning problem requires building feasible aircraft routes that cover each flight exactly once and that satisfy maintenance requirements. Basically, these requirements enforce an aircraft to undergo a planned maintenance at a specified station before accumulating a maximum number of flying hours. This stage is significant to airline companies as it directly impacts the fleet availability, safety, and profitability. The contribution of this paper is twofold. First, we elucidate the complexity status of the OAMRP and we propose an exact mixed-integer programming model that includes a polynomial number of variables and constraints. Furthermore, we propose a graph reduction procedure and valid inequalities that aim at improving the model solvability. Second, we propose a very large-scale neighborhood search algorithm along with a procedure for computing tight lower bounds. We present the results of extensive computational experiments that were carried out on real-world flight networks and attest to the efficacy of the proposed exact and heuristic approaches. In particular, we provide evidence that the exact model delivers optimal solutions for instances with up to 354 flights and 8 aircraft, and that the heuristic approach consistently delivers high-quality solutions while requiring short CPU times.