Column Generation Technique Research Articles

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.

Column generation approach for 1.5-dimensional cutting stock problem with technical constraints

In this study, the 1.5-dimensional cutting stock problem with technical constraints is considered. In the literature, this problem is also defined as a strip packing or open dimension problem. When given a strip of infinite length and bounded width, the problem is to define a packing of rectangular objects into a strip that minimizes its final length. Technical constraints, such as the order type and the number of strips, are indispensable in real life; however, they are often neglected in the literature because they make the problem difficult to solve. Only one study was reached in the literature that took into account technical constraints, but in that mentioned study, only a mathematical model was proposed for the problem. In this context, our aim is to solve the problem with a more effective approach. The research question in this study is the usability of the column generation technique to solve the 1.5-dimensional cutting stock problem. In this study, the column generation approach was proposed for the first time for the considered problem. To demonstrate the performance of the proposed solution method, randomly generated test problems were solved with GAMS/Cplex. As we report the results, proposed column generation approach (CG) reaches very close (such as %1 and %2 error) solutions to integrated mathematical model (IM) for small sized problems in a second. On the other hand, while CG solved all the problems in a reasonable time, IM could not produce a feasible solution to some problems. Numerical experiments showed that the column generation algorithm outperforms the integrated mathematical model for the problem.

Robust co-planning of transmission expansion and merchant energy storage investments considering long- and short-term uncertainties

The decreasing cost of energy storage technologies and the increasing utilization of renewable energy sources have made distributed transmission-scale energy storage economically viable. Merchant energy storage investors can collect significant profits through spatiotemporal energy arbitrage from congested transmission lines. This paper formulates a three-stage, four-level min–min–max–min problem from the viewpoint of power system operators and planners such that they will build new transmission lines and identify potential locations for profitable merchant energy storage investments. The first level determines transmission expansion decisions from the system operator’s perspective. The second level plans potential investments in merchant energy storage to ensure profitability at a desired rate of return. The third level realizes the worst-case situations of long-term uncertainties related to the expansion of wind farms and peak load growth in each regional zone, while the fourth level simulates market clearing for representative days, modeling short-term uncertainties. A novel iterative nested column and constraint generation (C&CG) technique is adopted to cope with numerical issues caused by binary variables of transmission lines and storage blocks and the enormous size of the problem. The case study of a modified IEEE 24-bus test system was used to assess the effectiveness of the proposed algorithm.

A framework for fair decision-making over time with time-invariant utilities

Fairness is a major concern in contemporary decision problems. In these situations, the objective is to maximize fairness while preserving the efficacy of the underlying decision-making problem. This paper examines repeated decisions on problems involving multiple stakeholders and a central decision maker. Repetition of the decision-making provides additional opportunities to promote fairness while increasing the complexity from symmetry to finding solutions. This paper presents a general mathematical programming framework for the proposed fairness-over-time (FOT) decision-making problem. The framework includes a natural abstraction of how a stakeholder’s acquired utilities can be aggregated over time. In contrast with a natural, descriptive formulation, we demonstrate that if the aggregation function possesses certain basic properties, a strong reformulation can be written to remove symmetry from the problem, making it amenable to branch-and-cut solvers. Finally, we propose a particular relaxation of this reformulation that can assist in the construction of high-quality approximate solutions to the original problem and can be solved using simultaneous row and column generation techniques.

Multi-route coordination for bus systems in response to road disruptions

Road disruptions frequently occur in practice, resulting in a productivity loss of bus services and widespread passenger delays. These negative impacts are significant, especially when multiple routes are influenced at the same time. In order to mitigate these impacts, this paper proposes a multi-route coordination approach that collaboratively adjusts multiple bus routes and optimizes bus timetables to provide effective alternative bus services for passengers. Three adjusting strategies are adopted for bus routes with varying passenger demand: detouring, short-running, and cancellation. To address the multi-route coordination problem, a column-generation-based two-stage framework is developed. Concretely, a column generation technique is utilized in the first stage to iteratively generate candidate adaptive bus routes and passenger paths. Following that, an integrated integer programming model is built in the second stage to simultaneously determine bus timetables and the combinations of those adaptive bus routes. After realizing the low computational efficiency for large-scale problems, this paper designs a customized decomposition algorithm based on set partitioning to solve the presented model and obtain near-optimal solutions efficiently. Finally, the proposed methodology is applied to an illustrative Sioux Falls network and a real-world bus network in Beijing to verify its validity and effectiveness. Three comparative analyses are conducted to discuss the advantages of the three adjusting strategies, to investigate the benefits of coordinating bus timetables, and to explore the applicability of different adjusting strategies, respectively.

PENYELESAIAN MASALAH CUTTING STOCK DENGAN PENGELASAN MENGGUNAKAN TEKNIK COLUMN GENERATION

The cutting stock problem with welding is a problem for cutting a number of raw materials into a number of cutting patterns to meet every demand such that the raw material needed is minimum. In this study, column generation techniques are used to solve the problem. The technique contains two models i.e the master model and the column generator submodel. The master model is a linear programming model that is used to determine the minimum number of raw materials needed to meet the needs according the cutting patterns. The submodel is constructed to generate a new cutting pattern that will improve the optimal solution of the master model. The column generation step is complete if there is no cutting pattern that can give the better solution. Since column generation technique does not always produce integer solutions, an additional mixed integer programming model is needed to meet unmet demands. The computationl results show that column generation techniques can solve cutting stock problems with welding, and it can give an optimal solution.

A Battery Electric Vehicle Transportation Network Design Model with Bounded Rational Travelers

With governments worldwide emphasizing environmental protection and the global focus on carbon reduction, the battery electric vehicle (BEV) industry has developed rapidly. An urban transportation network with BEVs as the main form of transportation will soon become mainstream. Motivated by the abovementioned background, a BEV transportation network design problem is investigated, and a network design model is established. The model aims to minimize the system travel time of BEV transportation networks and optimize the government’s lane expansion scheme (the location and number of lanes) under a limited budget. To consider the travel characteristics of BEV drivers, the charging time, range anxiety, and bounded rationality factors are simultaneously incorporated into the model. A heuristic algorithm is designed based on the active set algorithm to obtain the local optimal solution to the actual-scale problem. Moreover, a cutting-plane method is used to convert the original problem into a different form, and a column generation technique is embedded in the abovementioned algorithm to avoid the enumeration of paths. Sensitivity analyses of different levels of rationality of BEV drivers and government investment scales are performed. The experimental results demonstrate that the model and algorithm can effectively solve the problem and provide decision support for the government in formulating transportation infrastructure construction policies.

A three-phase heuristic for the Fairness-Oriented Crew Rostering Problem

The Fairness-Oriented Crew Rostering Problem (FCRP) considers the joint optimization of attractiveness and fairness in cyclic crew rostering. Like many problems in scheduling and logistics, the combinatorial complexity of cyclic rostering causes exact methods to fail for large-scale practical instances. In case of the FCRP, this is accentuated by the additionally imposed fairness requirements. Hence, heuristic methods are necessary. We present a three-phase heuristic for the FCRP combining column generation techniques with variable-depth neighborhood search. The heuristic exploits different mathematical formulations to find feasible solutions and to search for improvements. We apply our methodology to practical instances from Netherlands Railways (NS), the main passenger railway operator in the Netherlands Our results show the three-phase heuristic finds good solutions for most instances and outperforms a state-of-the-art commercial solver.

Integrated optimization of electric bus scheduling and charging planning incorporating flexible charging and timetable shifting strategies

In a battery electric bus (BEB) network, buses are scheduled to perform timetabled trips while satisfying time, energy consumption, charging, and operational constraints. Increasing research efforts have been dedicated to the integrated optimization of multiple planning tasks to reduce system costs. At a high integration level, this study determines the BEB scheduling and charging planning with flexible charging and timetable shifting strategies. We first formulate an integrated arc-based model to minimize the total costs considering the power grid pressure cost and subsequently reformulate it into a two-stage model, for which we develop an effective solution method. The first stage minimizes the total operational costs including the fleet, charging, and battery degradation costs based on the column generation technique, and the second stage minimizes the peak power demand through two timetable shifting strategies. It is found through numerical experiments that the proposed integrated optimization model and solution method can achieve significant improvement in the utilization rate and reductions in the fleet size, operational costs, and peak power demand compared to the two baseline models.

Joint column generation and Lagrangian relaxation technique for incident respondent location and allocation

AbstractIncident response operations require effective planning of resources to ensure timely clearance of roadways and avoidance of secondary incidents. This study formulates a mixed‐integer linear program to minimize the total expected travel time and maximize the demand covered. The model accounts for the location, severity, frequency of incidents, dispatching locations, and availability of incident respondents. An integrated methodology that includes column generation and Lagrangian relaxation with a density‐based clustering technique that defines incident hot spots is proposed. The hybrid approach is applied to an empirical case study in Raleigh, NC. A network instance with 10,672 incident sites, clustered with a search distance (ε) of 5 min, is solved efficiently with an optimality gap of 1.37% in 2 min. A Benders decomposition technique is implemented to conduct benchmark analyses. The numerical results suggest that the proposed algorithm can solve the problem efficiently and outperform the benchmark solutions.

Efficient MIP techniques for computing the relaxation complexity

The relaxation complexity {{,textrm{rc},}}(X) of the set of integer points X contained in a polyhedron is the minimal number of inequalities needed to formulate a linear optimization problem over X without using auxiliary variables. Besides its relevance in integer programming, this concept has interpretations in aspects of social choice, symmetric cryptanalysis, and machine learning. We employ efficient mixed-integer programming techniques to compute a robust and numerically more practical variant of the relaxation complexity. Our proposed models require row or column generation techniques and can be enhanced by symmetry handling and suitable propagation algorithms. Theoretically, we compare the quality of our models in terms of their LP relaxation values. The performance of those models is investigated on a broad test set and is underlined by their ability to solve challenging instances that could not be solved previously.

A branch-and-price algorithm to perform single-machine scheduling for additive manufacturing

Additive manufacturing (AM) has attracted significant attention in recent years based on its wide range of applications and growing demand. AM offers the advantages of production flexibility and design freedom. In this study, we considered a practical variant of the batch-processing-machine (BPM) scheduling problem that arises in AM industries, where an AM machine can process multiple parts simultaneously, as long as the two-dimensional rectangular packing constraint is not violated. Based on the set-partitioning formulation of our mixed-integer programming (MIP) model, a branch-and-price (B&P) algorithm was developed by embedding a column-generation technique into a branch-and-bound framework. Additionally, a novel labelling algorithm was developed to accelerate the column-generation process. Ours is the first study to provide a B&P algorithm to solve the BPM scheduling problem in the AM industry. We tested the performance of our algorithm using a modern MIP solver (Gurobi) and real data from a 3D printing factory. The results demonstrate that for most instances tested, our algorithm produces results similar or identical to those of Gurobi with reasonable computation time and outperforms Gurobi in terms of solution quality and running time on some large instances.

Fairness over time in dynamic resource allocation with an application in healthcare

Decision making problems are typically concerned with maximizing efficiency. In contrast, we address problems where there are multiple stakeholders and a centralized decision maker who is obliged to decide in a fair manner. Different decisions give different utility to each stakeholder. In cases where these decisions are made repeatedly, we provide efficient mathematical programming formulations to identify both the maximum fairness possible and the decisions that improve fairness over time, for reasonable metrics of fairness. We apply this framework to the problem of ambulance allocation, where decisions in consecutive rounds are constrained. With this additional complexity, we prove structural results on identifying fair feasible allocation policies and provide a hybrid algorithm with column generation and constraint programming-based solution techniques for this class of problems. Computational experiments show that our method can solve these problems orders of magnitude faster than a naive approach.

Computational Performance Evaluation of Column Generation and Generate-and-Solve Techniques for the One-Dimensional Cutting Stock Problem

The Cutting Stock Problem (CSP) is an optimisation problem that roughly consists of cutting large objects in order to produce small items. The computational effort for solving this problem is largely affected by the number of cutting patterns. In this article, in order to cope with large instances of the One-Dimensional Cutting Stock Problem (1D-CSP), we resort to a pattern generating procedure and propose a strategy to restrict the number of patterns generated. Integer Linear Programming (ILP) models, an implementation of the Column Generation (CG) technique, and an application of the Generate-and-Solve (G&S) framework were used to obtain solutions for benchmark instances from the literature. The exact method was capable of solving small and medium sized instances of the problem. For large sized instances, the exact method was not applicable, while the effectiveness of the other methods depended on the characteristics of the instances. In general, the G&S method presented successful results, obtaining quasi-optimal solutions for the majority of the instances, by employing the strategy of artificially reducing the number of cutting patterns and by exploiting them in a heuristic framework.

Scheduling and Power Control for Connectivity Enhancement in Multi-Hop I2V/V2V Networks

Infrastructure-to-vehicle (I2V) and vehicle-to-vehicle (V2V) communications are often combined to extend the connectivity and coverage in the Intelligent Transportation System (ITS) and its applications, e.g., augmented reality, real-time parking management and online shopping. Through multi-hop I2V and V2V communications, requesting vehicles are always connected to road side units (RSUs) even when they do not reside within the RSUs’ coverage range. However, there may be not adequate network resource for several I2V and V2V links when multiple vehicles request services simultaneously. In this paper, we propose a joint frequency scheduling and power control scheme to enhance connectivity in multi-hop I2V/V2V networks. We associate I2V and V2V links with tuple-links, then formulate an NP-hard problem in which a frequency scheduler and a power controller are jointly designed for the tuple-links. The NP-hard problem is decomposed into two separate subproblems by employing the delayed column generation technique. Then, we employ a method for linear programming and a greedy algorithm to address these subproblems. Through numerical experiments with practical parameter settings, we demonstrate the proposed scheme outperforms several existing ones in terms of connectivity enhancement, measured by the service resumption number and average achieved throughput. Furthermore, the efficiency of our scheme is further enhanced when the number of available channels is high, and buffer size equipped to the requesting vehicles is large.

A practical algorithm for VMAT optimization using column generation techniques.

As a challenging but important optimization problem, the inverse planning for volumetric modulated arc therapy (VMAT) has attracted much research attention. The column generation (CG) type method is so far one of the most effective solution schemes. However, it often relies on simplifications leading to significant gaps between the output and the actual feasible plan. This paper presents a novel column generation (NCG) approach to push the planning results substantially closer topractice. The proposed NCG algorithm is equipped with multiple new quality-enhancing and computation-facilitating modules as below: (1) Flexible constraints are enabled on both dose rates and treatment time to adapt to machine capabilities as well as planner's preferences, respectively; (2) a cross-control-point intermediate aperture simulation is incorporated to better conform to the underlying physics; (3) new pricing and pruning subroutines are adopted to achieve better optimization outputs. To evaluate the effectiveness of this NCG, five VMAT plans, that is, three prostate cases and two head-and-neck cases, were computed using proposed NCG. The planning results were compared with those yielded by a historical benchmark planningscheme. The NCG generated plans of significantly better quality than the benchmark planning algorithm. For prostate cases, NCG plans satisfied all planning target volume (PTV) criteria whereas CG plans failed on D10% criteria of PTVs for over 9 Gy or more on all cases. For head-and-neck cases, again, NCG plans satisfied all PTVs criteria while CG plans failed on D10% criteria of PTVs for over 3 Gy or more on all cases as well as the max dose criteria of both cord and brain stem for over 13 Gy on one case. Moreover, the pruning scheme was found to be effective in enhancing the optimizationquality. The proposed NCG inherits the computational advantages of the traditional CG, while capturing a more realistic characterization of the machine capability and underlying physics. The output solutions of the NCG are substantially closer to practicalimplementation.

On Efficient Service Function Chaining in Hybrid Software Defined Networks

To minimize operating expenses (OPEX) and enhance flexibility for network service provisioning, network function virtualization (NFV) is used to chain an ordered sequence of virtualized network functions (VNFs), also known as service function chain (SFC), which can be placed on commodity servers. As a highly complementary to NFV, software-defined networking (SDN) can offer an agile way of VNF orchestration with the capability of fine-granularity network control over flows. However, one-step migration to SDN is impossible in ISP networks. Thus, in this paper we resort to hybrid SDN to implement SFC provisioning. By jointly optimizing SDN deployment that largely determines capital expenditures (CAPEX) and VNF placement that decides OPEX, we manage to make a tradeoff between CAPEX and OPEX, and to find out an appropriate SDN deployment rate for network operator. We first formulate the problem as an integer linear programming (ILP) model <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\boldsymbol {P}$ </tex-math></inline-formula> , then, reformulate it with column generation technique and decomposition theory to develop a distributed approximation algorithm CGPD which obtains a tight upper bound of optimal solution to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\boldsymbol {P}$ </tex-math></inline-formula> . In order to better decrease CAPEX, a dynamic programming-based heuristic EOES giving importance to effective resource sharing is further designed based on a hidden Markov model. The simulation results indicate that, the difference between <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\boldsymbol {P}$ </tex-math></inline-formula> and CGPD is marginal. CGPD outperforms EOES in terms of total cost, average delay and bandwidth utilization, and EOES on the other hand demands 20% SDN deployment rate which is half of that of CGPD.

A branch-and-price approach to a variant of the cognitive radio resource allocation problem

Radio-frequency portion of the electromagnetic spectrum is a scarce resource. Cognitive radio (CR) technology has emerged as a promising solution to overcome the spectrum scarcity bottleneck. Through this technology, secondary users (SUs) sense the spectrum opportunities free from primary users (PUs), and opportunistically take advantage of these (temporarily) idle portions, known as spectrum holes. In this correspondence, we consider a variant of the cognitive radio resource allocation problem posed by Martinovic et al. in 2017. The distinguishing feature of this version of the problem is that each SU, due to its hardware limitations, imposes the requirement that the to-be-aggregated spectrum holes cannot be arbitrarily far from each other. We call this restriction as the Maximal Aggregation Range (MAR) constraint, and refer to this variant of the problem as the MAR-constrained hole assignment problem. The problem can be formalized as an NP-hard combinatorial optimization problem. We propose a novel binary integer linear programming (ILP) formulation to the problem. The number of constraints in this formulation is the number of spectrum holes plus the number of SUs. On the other hand, the number of binary decision variables in the formulation can be prohibitively large, as for each legitimate spectrum allocation to each SU, one variable is needed. Due to this difficulty, the resulting 0-1 ILP instances may not even be explicitly describable, let alone be solvable by an ILP solver. We propose a branch-and-price (B&P) framework to tackle this challenge. This framework is in fact a branch-and-bound procedure in which at each node of the search tree, we utilize the so-called (delayed) column generation technique for solving the LP relaxation of the corresponding subproblem. As evidenced by the numerical results, the LP relaxation bounds are very tight. This allows for a very effective pruning of the search space. Compared to the previously suggested formulations, the proposed technique can require much less computational effort.

Be Scalable and Rescue My Slices During Reconfiguration

AbstractModern 5G networks promise more bandwidth, less delay and more flexibility for an ever increasing number of users and applications, with Software Defined Networking, Network Function Virtualization and Network Slicing as key enablers. Within that context, efficiently provisioning the network and cloud resources of a wide variety of applications with dynamic user demand is a real challenge. We study here the network slice reconfiguration problem. Reconfiguring network slices from time to time reduces network operational costs and increases the number of slices that can be managed within the network. However, this affect the Quality of Service of users during the reconfiguration step. To solve this issue, we study solutions implementing a make-before-break scheme. We propose new models and scalable algorithms (relying on column generation techniques) that solve large data instances in few seconds.

Design of robust programmable networks with bandwidth-optimal failure recovery scheme

More than ever, data networks have demonstrated their central role in the world economy, but also in the well-being of humanity that needs fast and reliable networks. In parallel, with the emergence of Network Function Virtualization (NFV) and Software Defined Networking (SDN), efficient network algorithms considered too hard to be put in practice in the past now have a second chance to be considered again. In this context, as new networks will be deployed and current ones get significant upgrades, it is thus time to rethink the network dimensioning problem with protection against failures. In this paper, we consider a path-based protection scheme with the global rerouting strategy in which, for each failure situation, there may be a new routing of all the demands. Our optimization task is to minimize the needed amount of bandwidth. After discussing the hardness of the problem, we develop two scalable mathematical models that we handle using both Column Generation and Benders Decomposition techniques. Through extensive simulations on real-world IP network topologies and on randomly generated instances, we show the effectiveness of our methods: they lead to savings of 40 to 48% of the bandwidth to be installed in a network to protect against failures compared to traditional schemes. Finally, our implementation in OpenDaylight demonstrates the feasibility of the approach. Its evaluation with Mininet shows that our solution provides sub-second recovery times, but the way it is implemented may greatly impact the amount of signaling traffic exchanged. In our evaluations, the recovery phase requires only a few tens of milliseconds for the fastest implementation, compared to a few hundreds of milliseconds for the slowest one.