Mixed-integer Formulation Research Articles

Energy-Efficient Computation Offloading and Resource Management in Ultradense Heterogeneous Networks

To meet the demand of green communications of ultradense mobile devices (MDs), an energy-efficient mechanism, jointly considering the computation offloading and resource management, is designed to minimize the network-wide weighted energy consumption under the delay constraints of MDs for ultradense heterogeneous networks. Such a mechanism tightly integrates with the adjustment of computation capability and transmission power of MDs. Considering that the finally formulated problem is in a nonlinear and mixed-integer form, we design an effective algorithm to solve it. Specifically, by utilizing the powerful global searching capability of genetic algorithm (GA) and the accurate local searching capability of particle swarm optimization (PSO), the adaptive GA with diversity-guided is firstly used for coarse-grained search, and then adaptive PSO is utilized for fine-grained search. After that, some detailed analyses on the convergence, computation complexity and parallel implement are provided for algorithms. Simulation results show that the designed algorithm can achieve a lower network energy consumption than other offloading algorithms in general. At the same time, the numerical simulation also reveals that the designed algorithm may be more suitable for ultradense networks than existing algorithms.

Agent Scheduling in Unrelated Parallel Machines with Sequence- and Agent–Machine–Dependent Setup Time Problem

Assignation-sequencing models have played a critical role in the competitiveness of manufacturing companies since the mid-1950s. The historic and constant evolution of these models, from simple assignations to complex constrained formulations, shows the need for, and increased interest in, more robust models. Thus, this paper presents a model to schedule agents in unrelated parallel machines that includes sequence and agent–machine-dependent setup times (ASUPM), considers an agent-to-machine relationship, and seeks to minimize the maximum makespan criteria. By depicting a more realistic scenario and to address this NP-hard problem, six mixed-integer linear formulations are proposed, and due to its ease of diversification and construct solutions, two multi-start heuristics, composed of seven algorithms, are divided into two categories: Construction of initial solution (designed algorithm) and improvement by intra (tabu search) and inter perturbation (insertions and interchanges). Three different solvers are used and compared, and heuristics algorithms are tested using randomly generated instances. It was found that models that linearizing the objective function by both job completion time and machine time is faster and related to the heuristics, and presents an outstanding level of performance in a small number of instances, since it can find the optimal value for almost every instance, has very good behavior in a medium level of instances, and decent performance in a large number of instances, where the relative deviations tend to increase concerning the small and medium instances. Additionally, two real-world applications of the problem are presented: scheduling in the automotive industry and healthcare.

Optimizing the multiple trip vehicle routing plan for a licensee green tea dealer in Sri Lanka

PurposeThis research is designed to optimize the business process of a green tea dealer, who is a key supply chain partner of the Sri Lankan tea industry. The most appropriate trips for each vehicle in multiple trip routing systems are identified to minimize the total cost by considering the traveling distance.Design/methodology/approachThe study has followed the concepts in vehicle routing problems and mixed-integer programming mathematical techniques. The model was coded with the Python programming language and was solved with the CPLEX Optimization solver version 12.10. In total, 20 data instances were used from the subjected green tea dealer for the validation of the model.FindingsThe result of the numerical experiment showed the ability to access supply over the full capacity of the available fleet. The model achieved optimal traveling distance for all the instances, with the capability of saving 17% of daily transpiration cost as an average.Research limitations/implicationsThis study contributes to the three index mixed-integer programing model formulation through in-depth analysis and combination of several extensions of vehicle routing problem.Practical implicationsThis study contributes to the three index mixed-integer programming model formulation through in-depth analysis and combination of several extensions of the vehicle routing problem.Social implicationsThe proposed model provides a cost-effective optimal routing plan to the green tea dealer, which satisfies all the practical situations by following the multiple trip vehicle routing problems. Licensee green tea dealer is able to have an optimal fleet size, which is always less than the original fleet size. Elimination of a vehicle from the fleet has the capability of reducing the workforce. Hence, this provides managerial implication for the optimal fleet sizing and route designing.Originality/valueDeveloping an optimization model for a tea dealer in Sri Lankan context is important, as this a complex real world case which has a significant importance in export economy of the country and which has not been analyzed or optimized through any previous research effort.

Bilinear matrix equation characterizes Laplacian and distance matrices of weighted trees

It is known from algebraic graph theory that if L is the Laplacian matrix of some tree G with a vertex degree sequence d=(δ1,…,δn)⊤ and D is its distance matrix, then LD+2I=(2⋅1−d)1⊤, where 1 is an all-ones column vector. We prove the converse proposition: if this identity holds for the Laplacian matrix of some graph G with a degree sequence d and for some matrix D, then G is essentially a tree, and D is its distance matrix. This result immediately generalizes to weighted graphs. Therefore, the above bilinear matrix equation in L, D, and d characterizes trees in terms of their Laplacian and distance matrices, so it can be used as a constraint in mixed-integer formulations of distance-related tree topology design problems (e.g., optimum communication spanning tree or hop-constrained minimum spanning tree problems). If the matrix D is symmetric, the lower triangular part of this matrix identity is redundant and can be omitted, which halves the number of constraints in an optimization problem. Applications to the extremal graph theory (especially, to topological index optimization and to optimal tree problems) and to road topology design are discussed.

A framework for railway transit network design with first-mile shared autonomous vehicles

Providing a railway transit system (RTS) in less populated areas is a challenging task for transportation agencies due to its high construction and operating costs. With the advent of automation, shared autonomous vehicles (SAVs) as an integral part of public transit services has the potential to enhance the design of transit systems. In this paper, we present a joint optimization framework of railway transit network design and SAV first-mile service that minimizes the total cost of the combined RTS-SAV services and commuters’ waiting time, while serving a dynamic travel demand in the network. The proposed model optimizes the SAV fleet size and the RTS alignment while enabling vehicle relocations to tackle the vehicle imbalance issue in the SAV service. Due to the non-linear and mixed-integer formulation, we develop a fixed-point algorithm for this joint RTS-SAV problem where we transform the original problem into a mixed-integer linear programming (MILP) formulation. Our results indicate that the joint RTS-SAV services can be constructed and operated at a lower cost than either of the RTS or SAV services alone. Furthermore, the resulting joint RTS-SAV services are underpinned by a shorter railway alignment and larger fleet size rather than a multi-link extension. Additionally, the joint RTS-SAV services is robust to the variation in total demand, with respect to the railway alignment, SAV utilization and commuters’ waiting time.

The minimum cost network upgrade problem with maximum robustness to multiple node failures

The design of networks which are robust to multiple failures is gaining increasing attention in areas such as telecommunications. In this paper, we consider the problem of upgrading an existent network in order to enhance its robustness to events involving multiple node failures. This problem is modeled as a bi-objective mixed linear integer formulation considering both the minimization of the cost of the added edges and the maximization of the robustness of the resulting upgraded network. As the robustness metric of the network, we consider the value of the Critical Node Detection (CND) problem variant which provides the minimum pairwise connectivity between all node pairs when a set of c critical nodes are removed from the network. We present a general iterative framework to obtain the complete Pareto frontier that alternates between the minimum cost edge selection problem and the CND problem. Two different approaches based on a cover model are introduced for the edge selection problem. Computational results conducted on different network topologies show that the proposed methodology based on the cover model is effective in computing Pareto solutions for graphs with up to 100 nodes, which includes four commonly used telecommunication networks.

Mixed-Integer Convex Optimization for DC Microgrid Droop Control

Droop control is a viable method for the operation of island DC microgrids in a decentralized architecture. This paper presents a mixed-integer conic optimization formulation for the design of generator droop control, comprising the parameters of a piecewise linear droop curve. The mixed-integer formulation originates from a stochastic optimization framework that considers several operating scenarios for finding the optimal design. The convexity of the mixed-integer problem continuous relaxation gives global optimality guarantees for the design problem. The paper presents computational results using a tight polyhedral approximation of the conic program, leading to a mixed-integer linear programming (MILP) problem that is solved using a state-of-the-art commercial solver. The results from the proposed approach are contrasted with both a classic linear droop control design and a recent piecewise linear formulation. The Monte-Carlo simulation results quantify the extent to which the MILP solution is superior in reducing voltage violations and power loss, and the degree to which the loss is close to that from a conic optimal power flow solution.

Coalitional Game-Based User Association Integrated With Open Loop Power Control for Green Communications in Uplink HCNs

In heterogeneous cellular networks (HCNs), the energy reduction has a great impact on the operational cost and global carbon footprint, which has attracted more and more attention. To enhance the network energy efficiency (EE), we design two types of association mechanisms integrated with open loop power control (OLPC) for uplink code-division-multiple-access (CDMA) enabled HCNs. Specifically, they are formulated into two optimization problems that maximize the sum of users’ EEs and the one of users’ EE utilities. It is noteworthy that the one of these two mechanisms guarantees the user fairness by employing a logarithmic function. Considering that these formulated problems are in a fractional mixed-integer form and hard to tackle, we utilize a coalitional game to design two types of association algorithms with guaranteed convergence. To show the effectiveness of designed algorithms, another one is introduced for comparison in the simulation. As for them, we mainly investigate the impacts of user and BS densities on the association performance.

A robust SVM-based approach with feature selection and outliers detection for classification problems

This paper proposes a robust classification model, based on support vector machine (SVM), which simultaneously deals with outliers detection and feature selection. The classifier is built considering the ramp loss margin error and it includes a budget constraint to limit the number of selected features. The search of this classifier is modeled using a mixed-integer formulation with big M parameters. Two different approaches (exact and heuristic) are proposed to solve the model. The heuristic approach is validated by comparing the quality of the solutions provided by this approach with the exact approach. In addition, the classifiers obtained with the heuristic method are tested and compared with existing SVM-based models to demonstrate their efficiency.

Disjunctive linear separation conditions and mixed-integer formulations for aircraft conflict resolution

We address the aircraft conflict resolution problem in air traffic control. We introduce new mixed-integer programming formulations for aircraft conflict resolution with speed, heading and altitude control which are based on disjunctive linear separation conditions. We first examine the two-dimensional aircraft conflict resolution problem with speed and heading control represented as continuous decision variables. We show that the proposed disjunctive linear separation conditions are equivalent to the classical nonlinear conditions for aircraft separation. Further, we characterise conflict-free trajectories based on aircraft velocity bounds and propose a simple pre-processing algorithm to identify aircraft pairs which are either always conflict-free, or which cannot be separated using speed and heading control only. We then incorporate altitude control and propose a lexicographic optimisation formulation that aims to minimise the number of flight level changes before resolving outstanding conflicts via two-dimensional velocity control. The proposed mixed-integer programming formulations are nonconvex, and we propose convex relaxations, decomposition methods and constraint generation algorithms to solve the two-dimensional and lexicographic optimisation formulations to guaranteed optimality. Numerical experiments on four types of conflict resolution benchmarking instances are conducted to test the performance of the proposed mixed-integer formulations. Further, the proposed method is compared against two benchmarks based on state-of-the-art approaches for the aircraft conflict resolution problem. Our numerical results show that the proposed method largely outperforms both benchmarks in terms of runtime and is able to solve significantly more instances to global optimality.

Leveraged least trimmed absolute deviations

The design of regression models that are not affected by outliers is an important task which has been subject of numerous papers within the statistics community for the last decades. Prominent examples of robust regression models are least trimmed squares (LTS), where the k largest squared deviations are ignored, and least trimmed absolute deviations (LTA) which ignores the k largest absolute deviations. The numerical complexity of both models is driven by the number of binary variables and by the value k of ignored deviations. We introduce leveraged least trimmed absolute deviations (LLTA) which exploits that LTA is already immune against y-outliers. Therefore, LLTA has only to be guarded against outlying values in x, so-called leverage points, which can be computed beforehand, in contrast to y-outliers. Thus, while the mixed-integer formulations of LTS and LTA have as many binary variables as data points, LLTA only needs one binary variable per leverage point, resulting in a significant reduction of binary variables. Based on 11 data sets from the literature, we demonstrate that (1) LLTA’s prediction quality improves much faster than LTS and as fast as LTA for increasing values of k and (2) that LLTA solves the benchmark problems about 80 times faster than LTS and about five times faster than LTA, in median.

On the Implementation of OPF-Based Setpoints for Active Distribution Networks

In the context of active distribution networks, AC Optimal Power Flow (OPF) has shown great potential to calculate setpoints for controllable devices. Although considerable literature exists, temporal aspects that may affect the actual execution of these setpoints are rarely investigated. Due to the diverse operating characteristics of controllable devices (i.e., delays, ramp rates and deadbands), when these setpoints are executed by multiple devices without adequate considerations, the resulting outcome can differ drastically from what is expected; leading to violations of network constraints and excessive control actions. Therefore, this work proposes a series of necessary adaptations within the controllers of existing devices as well as in the OPF formulation to cater for the diversity in operating characteristics, ensuring that calculated setpoints are adequately implemented by controllable devices. This involves the direct control of conventional devices and enforcing a new ramping behavior for inverter-interfaced devices. Furthermore, a linear, mixed-integer formulation is proposed to handle discrete devices and improve scalability in large networks. Co-simulation results (using a U.K. test network with the objective of maximizing renewable energy production and considering 1s time-step) demonstrate that, by catering for the operating characteristics of controllable devices, the expected outcome from OPF-based setpoints can be achieved.

Baggage Carousel Assignment at Airports: Model and Case Study

Baggage operations at an airport are split into several different sub problems. The problem of assigning baggage carousels to each arriving flight is part of the important passenger arrival process. This article presents a mixed integer formulation for the underlying decision problem which aims at balancing customer satisfaction with operational needs. We introduce a static model that fulfills the different real-world requirements and balances the different objective criteria. It is shown how the model can be used in the dynamic environment of an airport. Because of the high degree of uncertainty at an airport, the issue of stability is discussed in detail. In a case study several scenarios with different time horizons and times of decision were studied. The model was verified with data from Frankfurt Airport and is successfully running as part of the decision support system at Frankfurt Airport.

Mixed integer formulations using natural variables for single machine scheduling around a common due date

While almost all existing works which optimally solve just-in-time scheduling problems propose dedicated algorithmic approaches, we propose in this work mixed integer formulations. We consider a single machine scheduling problem that aims at minimizing the weighted sum of earliness–tardiness penalties around a common due date. Using natural variables, we provide one compact formulation for the unrestrictive case and, for the general case, a non-compact formulation based on non-overlapping inequalities. We show that the separation problem related to the latter formulation is solved polynomially. In this formulation, solutions are only encoded by extreme points. We establish a theoretical framework to show the validity of such a formulation using non-overlapping inequalities, which could be used for other scheduling problems. A Branch-and-Cut algorithm together with an experimental analysis is proposed to assess the practical relevance of this mixed integer programming based methods.

A Large Neighborhood Search for the Pickup and Delivery Problem with Time Windows, Split Loads and Transshipments

In the pickup and delivery problem (PDP) a fleet of capacitated vehicles must be routed in order to satisfy a set of customer requests. In this article, we address a variant of the one-to-one PDP where each customer location can be visited several times and load may be transshipped from one vehicle to another at specific locations. The goal is to minimize the sum of travel costs and transshipment costs. The corresponding problem is called the PDP with time windows, split loads and transshipments. We present an arc-based mixed-integer formulation for the problem and propose a large neighbourhood search metaheuristic for solving it. Extensive computational experiments using both benchmark instances from the literature and instances based on real-world data illustrate the performance of the solution method and give insights regarding the benefits of combining split loads and transshipments.

An optimization model to maximize energy generation in short-term hydropower unit commitment using efficiency points

This paper presents a linear mixed-integer formulation to solve the short-term hydropower unit commitment problem. It uses the pair of maximum efficiency points of water discharge and the power produced at the maximum storage for all possible combinations of turbines. The goal is to maximize total energy production for all periods. The objective function is calculated using the correction between the power produced at the current volume and the maximum storage and penalizes unit start-ups. Constraints on the maximum number of turbine changes are imposed to find a viable solution. Computational results are reported for 65 instances with two powerhouses of five turbines each located in the Saguenay-Lac-St-Jean region of the province of Quebec in Canada.

Remarks on mixed-integer formulations for hyper-reduction schemes in nonlinear dynamics

The hyper-reduction problem for reduced-order internal forces evaluation in transient, nonlinear, explicit dynamics is reformulated, employing Mixed-Integer Programming (MIP), taking into account consistency constraints. Constraint reduction is introduced. Resulting quadratures, as well as reduced runs, are compared against the standard Energy Conserving Sampling and Weighting (ECSW) scheme, on a reference example. Rather than searching for optimal performance, the goal is to provide a benchmark solution, for evaluation of heuristic hyper-reduction formulations along with a non-greedy approach.

Integration of Electric Vehicles in Low-Voltage Distribution Networks Considering Voltage Management

The expected growth of the number of electric vehicles can be challenging for planning and operating power systems. In this sense, distribution networks are considered the Achilles’ heel of the process of adapting current power systems for a high presence of electric vehicles. This paper aims at deciding the maximum number of three-phase high-power charging points that can be installed in a low-voltage residential distribution grid. In order to increase the number of installed charging points, a mixed-integer formulation is proposed to model the provision of decentralized voltage support by electric vehicle chargers. This formulation is afterwards integrated into a modified AC optimal power flow formulation to characterize the steady-state operation of the distribution network during a given planning horizon. The performance of the proposed formulations have been tested in a case study based on the distribution network of La Graciosa island in Spain.

Control Optimization of Pump Cycles in Onshore Oilfields With Network and Electric Power Constraints

Abstract In onshore oilfields, sucker rod pumps are deployed to lift fluids from wells to the surface, which are then transferred by a pipeline network to a central facility. Each pump operates according to a cyclic control policy that alternates between a period of production and a period of recess, allowing the reservoir to replenish the fluids. To minimize the peak in the electric power drawn by the sucker rods, previous works have considered the scheduling of the control policies over time. However, when the production system sprawls over a large area, the energy required to force the flow through the pipelines and the pressure drops are significant. To this end, this article develops mixed-integer nonlinear formulations and mixed-integer linear approximations for baseline control-policy scheduling and reconfiguration.

A two-stage stochastic mixed-integer program for reliable supply chain network design under uncertain disruptions and demand

Having a reliable supply chain network is one of the most effective ways to plan and deal with facility failures and demand uncertainty due to very limited options available in a facility location problem. This research addresses issues bordering on capacitated supply chain problems, specifically on how reliable supply chain networks can be designed in the face of random facility disruptions and uncertain demand with a combination of hardening selected facilities and product reassignment. The proposed multi-period capacitated facility location and allocation problem is modeled as a two-stage stochastic mixed-integer formulation that minimizes the total establishing and transportation cost. The L-shaped method of stochastic linear programming is applied by integrating with two types of optimality and feasibility cuts for solving the stochastic model. This research aims to improve the proposed algorithm in two different ways: replacing the single-cut approach with a multi-cut and showing relatively complete recourse in the stochastic model by reformulating the original model. Finally, to illustrate the applicability of the model and computational effectiveness of the improved algorithm, a case study with different scenarios is presented and the results are discussed. Our computational results show that the proposed solution algorithm is not only capable of solving large-scale problems, but also avoids long run times. It is also demonstrated that substantial improvements in reliability of the system can often be possible with minimal increases in facility cost.