Use Of Binary Variables Research Articles

Integer programming using a single atom

Integer programming (IP), as the name suggests is an integer-variable-based approach commonly used to formulate real-world optimization problems with constraints. Currently, quantum algorithms reformulate the IP into an unconstrained form through the use of binary variables, which is an indirect and resource-consuming way of solving it. We develop an algorithm that maps and solves an IP problem in its original form to any quantum system possessing a large number of accessible internal degrees of freedom that are controlled with sufficient accuracy. This work leverages the principle of superposition to solve the optimization problem. Using a single Rydberg atom as an example, we associate the integer values to electronic states belonging to different manifolds and implement a selective superposition of different states to solve the full IP problem. The optimal solution is found within a few microseconds for prototypical IP problems with up to eight variables and four constraints. This also includes non-linear IP problems, which are usually harder to solve with classical algorithms when compared to their linear counterparts. Our algorithm for solving IP is benchmarked by a well-known classical algorithm (branch and bound) in terms of the number of steps needed for convergence to the solution. This approach carries the potential to improve the solutions obtained for larger-size problems using hybrid quantum–classical algorithms.

Support vector machines within a bivariate mixed-integer linear programming framework

Support vector machines (SVMs) are a powerful machine learning paradigm, performing supervised learning for classification and regression analysis. A number of SVM models in the literature have made use of advances in mixed-integer linear programming (MILP) techniques in order to perform this task efficiently. In this work, we present three new models for SVMs that make use of piecewise linear (PWL) functions. This allows effective separation of data points where a simple linear SVM model may not be sufficient. The models we present make use of binary variables to assign data points to SVM segments, and hence fit within a recently presented framework for machine learning MILP models. Alongside presenting an inbuilt feature selection operator, we show that the models can benefit from robust inbuilt outlier detection. Experimental results show when each of the presented models is effective, and we present guidelines on which of the models are preferable in different scenarios.

Coordinating Multiple Cooperative Vehicle Trajectories on Shared Road Networks

This paper addresses the problem of planning trajectories for multiple cooperative agents along specified paths through a static road network. Vehicle trajectories are coupled through interactions at shared areas, including intersections and road segments. A motion model is presented that includes the combined trajectories of all vehicles. The trajectories are optimized for travel time by solving a Mixed Integer Linear Program (MILP) that incorporates constraints to ensure collision avoidance. To improve the computation time of solving the MILP, two model modifications are introduced related to interactions: 1) The use of binary variables is reduced by grouping them at targeted travel times. 2) The relative travel order of interacting vehicles is constrained, guided by a heuristic solver. To further reduce computation time, an iterative method solves a sequence of smaller MILPs by targeting essential vehicle interactions with an algorithm that predicts interactions. Experiments are run on a simulated road network based on a real surface mine and haul truck trips. The proposed MILP methods are shown to significantly reduce solution costs compared to a reactive approach based on common driving practices. The MILP modifications significantly reduce computation time, trading off marginal solution quality. The reduction is extended by the iterative MILP scheme without degradation of solution quality.

On the effect of the selection of suppliers on the design of formulated products

An extended pooling problem is developed for the design of detergents, that include process and product design together with suppliers’ selection. It is a multiperiod and multi-objective optimization problem since it considers the economic benefit and environmental impact as well as the contract length. This problem can be considered as a previous step to the integrated product, process and supply chain design. This type of problems is non-linear and non-convex. Two formulations are developed to tackle this problem. An MINLP and a reformulated NLP using a decision vector avoiding the use of binary variables. The NLP shows better computational performance in spite of the larger problem size. Furthermore, it is demonstrated that when considering different ingredients, formulations, pricing policies and suppliers, their selection can be adjusted until a reduction of almost 40% of CO2 emissions is achieved without the benefit decreasing more than 1.5%.

Rate-Latency Optimization for NB-IoT With Adaptive Resource Unit Configuration in Uplink Transmission

Narrowband Internet of Things (NB-IoT) is a cellular IoT communication technology standardized by 3rd Generation Partnership Project (3GPP) for supporting massive machine type communication and its deployment can be realized by a simple firmware upgrade on existing long term evolution (LTE) networks. The NB-IoT requirements in terms of energy efficiency, achievable rates, latency, extended coverage, make the resource allocation, in a limited bandwidth, even a more challenging problem w.r.t. to legacy LTE. The allocation, done with subcarrier (SC) granularity in NB-IoT, should maintain adequate performance for the devices while keeping the power consumption as low as possible. Nevertheless, the optimal solution of the resource allocation problem is typically unfeasible since nonconvex, NP-hard and combinatorial because of the use of binary variables. In this article, after the formulation of the optimization problem, we study the resource allocation approach for NB-IoT networks aiming to analyze the tradeoff between rate and latency. The proposed suboptimal algorithm allocates radio resource (i.e., SCs) and transmission power to the NB-IoT devices for the uplink transmission and the performance is compared in terms of latency, rate, and power. By comparing the proposed allocation to a conventional round robin (RR) and to a brute-force approach, we can observe the advantages of the formulated allocation problem and the limited loss of the suboptimal solution. The proposed algorithm outperforms the RR by a factor 2 in terms of spectral efficiency and, moreover, the study includes techniques that reduce the dropped packets from 29% to 1.6%.

A comparison of mixed integer programming formulations of the capacitated lot-sizing problem

We propose a novel mixed integer programming formulation for the capacitated lot-sizing problem with set-up times and set-up carryover. We compare our formulation to two earlier formulations, the Classical and Modified formulations, and a more recent formulation due to Suerie and Stadtler. Extensive computational experiments show that our formulation consistently outperforms the Classical and Modified formulations in terms of CPU time and solution quality. It is competitive with the Suerie–Stadtler (S&S) formulation, but outperforms all other formulations on the most challenging instances, those with low-capacity slack and a dense jobs matrix. We show that some of the differences in the performance of these various formulations arise from their different use of binary variables to represent production or set-up states. We also show that the LP relaxation of our Novel formulation provides a tighter lower bound than that of the Modified formulation. Our experiments demonstrate that, while the S&S formulation provides a much tighter LP bound, the Novel formulation is better able to exploit the intelligence of the CPLEX solution engine.

Teaching Use of Binary Variables in Integer Linear Programs: Formulating Logical Conditions

Binary variables are often needed in linear programming models to indicate whether particular alternatives should be implemented and to impose logical relations among decisions. However, it is usually not obvious to students how to use binary variables to transform conditional statements of logic into linear relations. We propose to address this difficulty with a simple two-step approach. It provides rules for decomposing a conditional requirement into a group of elementary implications and then translating each of these into linear constraints. Pre- and post-test results from a sample of undergraduate business students are presented to support the effectiveness of this pedagogical approach. Supplemental files available online at https://doi.org/10.1287/ited.2017.0177 .

Shape and size of parcels and transport costs as a mixed integer programming problem in optimization of land consolidation

This article presents a new approach to the problem of optimization of land arrangement. The new element is the use of binary variables in the calculation model using the principles of linear programming. For this reason the optimization model can be classified into the category of mixed integer programming problems (MIP). The proposed optimization model takes into account a number of very important factors in the process of land consolidation, including map of the land diversification or the actual shape of the transportation network. That also implies taking into account the real, not the straight-lined distances between different parts of the model. The solution of the model is an arrangement of plots minimizing the costs associated with the cultivation of these plots, depending on their shapes and distances from homesteads. The value of the land of individual farms before and after the consolidation is preserved. Optimization model has been saved and executed in the environment of the GLPK (GNU Linear Programming Kit) software package.The presented method has been used to carry out the optimization process on the test object of the area of 587 hectares. The outcome is a new layout of plots in a form which meets all the technical requirements required in the real project process.

R&D Project Selection at Northbanctec Inc

Suitable as an assignment or as an exam case, this short case is effective in illustrating the use of linear and integer programming in an assignment problem, wrapped around R&D project selection in a global IT firm. A twist in the case requires the use of binary variables to implement certain logical conditions on which projects can be undertaken by which project teams. The case is suitable for graduate or advanced undergraduate courses or modules on optimization, decision models, management science and the like, both for class discussion and as a part of assignment or exam. A spreadsheet is available to enhance student learning. Excerpt UVA-QA-0762 Rev. Jun. 14, 2011 R&D Project selection at NorthBancTec Inc. NorthBancTec Inc. was a high-end boutique IT firm. Headquartered in Washington, D.C., NorthBancTec had three teams: in Georgetown (United States), in Dubai (United Arab Emirates), and in Novosibirsk (Russia). Because of its small size, NorthBancTec had to be very selective in choosing R&D projects. Table 1 lists the possibilities it had identified for the upcoming year. Table 1. Possible projects, their type, and their projected net benefit. . . .

A Characterization of the SSD-Efficient Frontier of Portfolio Weights by Means of a Set of Mixed-Integer Linear Constraints

In this paper, the set of all second-order stochastic dominance (SSD)-efficient portfolios is characterized by using a series of mixed-integer linear constraints. Our derivation employs a combination of the first-order conditions of the utility maximization problem together with a judicious use of binary variables. This result opens the door to the formulation of optimizations whose objective function is free to select a particular portfolio out of the entire SSD-efficient set. This paper was accepted by Jerome Detemple, finance.

Constrained spacecraft reorientation using mixed integer convex programming

A constrained attitude guidance (CAG) system is developed using convex optimization to autonomously achieve spacecraft pointing objectives while meeting the constraints imposed by on-board hardware. These constraints include bounds on the control input and slew rate, as well as pointing constraints imposed by the sensors. The pointing constraints consist of inclusion and exclusion cones that dictate permissible orientations of the spacecraft in order to keep objects in or out of the field of view of the sensors. The optimization scheme drives a body vector towards a target inertial vector along a trajectory that consists solely of permissible orientations in order to achieve the desired attitude for a given mission mode. The non-convex rotational kinematics are handled by discretization, which also ensures that the quaternion stays unity norm. In order to guarantee an admissible path, the pointing constraints are relaxed. Depending on how strict the pointing constraints are, the degree of relaxation is tuneable. The use of binary variables permits the inclusion of logical expressions in the pointing constraints in the case that a set of sensors has redundancies. The resulting mixed integer convex programming (MICP) formulation generates a steering law that can be easily integrated into an attitude determination and control (ADC) system. A sample simulation of the system is performed for the Bevo-2 satellite, including disturbance torques and actuator dynamics which are not modeled by the controller. Simulation results demonstrate the robustness of the system to disturbances while meeting the mission requirements with desirable performance characteristics.

U.S. stock market interaction network as learned by the Boltzmann machine

We study historical dynamics of joint equilibrium distribution of stock returns in the U.S. stock market using the Boltzmann distribution model being parametrized by external fields and pairwise couplings. Within Boltzmann learning framework for statistical inference, we analyze historical behavior of the parameters inferred using exact and approximate learning algorithms. Since the model and inference methods require use of binary variables, effect of this mapping of continuous returns to the discrete domain is studied. The presented analysis shows that binarization preserves market correlation structure. Properties of distributions of external fields and couplings as well as industry sector clustering structure are studied for different historical dates and moving window sizes. We found that a heavy positive tail in the distribution of couplings is responsible for the sparse market clustering structure. We also show that discrepancies between the model parameters might be used as a precursor of financial instabilities.

On the continuum approximation of the on-and-off signal control on dynamic traffic networks

In the modeling of traffic networks, a signalized junction is typically treated using a binary variable to model the on-and-off nature of signal operation. While accurate, the use of binary variables can cause problems when studying large networks with many intersections. Instead, the signal control can be approximated through a continuum approach where the on-and-off control variable is replaced by a continuous priority parameter. Advantages of such approximation include elimination of the need for binary variables, lower time resolution requirements, and more flexibility and robustness in a decision environment. It also resolves the issue of discontinuous travel time functions arising from the context of dynamic traffic assignment.Despite these advantages in application, it is not clear from a theoretical point of view how accurate is such continuum approach; i.e., to what extent is this a valid approximation for the on-and-off case. The goal of this paper is to answer these basic research questions and provide further guidance for the application of such continuum signal model. In particular, by employing the Lighthill–Whitham–Richards model (Lighthill and Whitham, 1955; Richards, 1956) on a traffic network, we investigate the convergence of the on-and-off signal model to the continuum model in regimes of diminishing signal cycles. We also provide numerical analyses on the continuum approximation error when the signal cycles are not infinitesimal. As we explain, such convergence results and error estimates depend on the type of fundamental diagram assumed and whether or not vehicle spillback occurs to the signalized intersection in question. Finally, a traffic signal optimization problem is presented and solved which illustrates the unique advantages of applying the continuum signal model instead of the on-and-off model.

A Characterization of the SSD-Efficient Frontier of Portfolio Weights with an Application to SSD-Spanning

In this paper it is shown how the set of all portfolios which are second-order stochastic dominance efficient can be characterized by using a series of mixed-integer linear constrains. Our derivation employs a combination of the first-order conditions of the utility maximization problem together with a judicious use of binary variables. Our main result is illustrated by presenting a theoretical and empirical application, which tackles in a general way spanning and intersection in the context of SSD-efficiency.

Adjustable Profile Blocks With Spatial Relations in the Day-Ahead Electricity Market

The concept of adjustable combinatorial products within the scope of the day-ahead electricity market is introduced in this paper. These adjustable products comprise simple and linked profile block offers/bids, with a variable energy quantity profile during the block periods. In the proposed formulation, these blocks can be partially cleared, namely, the indivisibility of the classical profile blocks that are present in European markets is relaxed. This flexibility allows the elimination of paradoxically accepted and rejected blocks and, thus, the elimination of iterative heuristic procedures for their handling. Such modeling is not possible as a mixed-integer linear programming problem, due to the inherent indivisibility denoted by the use of binary variables. Additionally, advanced profile bids are formulated for demand entities, called “joint demand profile block bids” in this paper, which involve not only inter-temporal relations, but also spatial relations with each other. This two-dimensional relation between demand profile blocks is novel and is deemed feasible with the model presented here. These advanced profile orders are incorporated in a day-ahead market problem, formulating a mixed complementarity problem model, which is solved using commercial solvers. The model is validated using the IEEE RTS-96 system; nevertheless, its scalability to large real-world power systems is straightforward.

Climate change‐sensitive hydrologic design under uncertain future precipitation extremes

Precipitation as an important component of hydrologic cycle bears a significant influence on hydrologic design and water resources management. The uncertainties associated with future climate change coupled with limitations of climate change models and uncertainties in projections, our inability to quantify these introduce additional complexities in hydrologic design using future precipitation extremes. A new optimal compromise hydrologic design of a stormsewer system using a fuzzy mixed integer nonlinear mathematical programming (MINLP) model with discrete, binary variables and logical constraints is developed and evaluated in this study. Preferences of hydrologists toward expected uncertain future changes in precipitation extremes are modeled using linear, nonlinear, triangular and Gaussian fuzzy membership functions. Methods for deriving membership functions based on multimodel multiple scenario GCM‐based simulations are also suggested. Incorporation of triangular functions in optimization formulation required the use of binary variables. A hypothetical hydrologic design example with realistic parameter values is used to obtain compromise elements of stormsewer system. Results from the study suggest a compromise design is possible based on the preferences and a balance between over and under design of stormsewer infrastructure is achievable. The nature of membership functions reflecting the preferences attached to future extremes influence the optimal solutions obtained resulting in changes in cost‐based performance measures.

Localization of Sources of Brain Activity: A MILP Approach

The purpose of this work is to propose and test a mixed integer linear programming formulation for the problem of brain source localization. Such technique allows the localization of the minimum number of currents that are able to reconstruct evoked potentials recorded at the scalp. The algorithm makes use of binary variables in order to be less sensible to noise on input data. Some preliminary simulation results show that the algorithm is effective in source localization and robust with respect to errors on measurement data.

Time, Cost, and Quality in a Road Building Project

Time, cost and quality are the three factors that play a significant role in the planning and controlling of construction projects. The main barriers for their achievement are the changes in the project environment necessitating cost, time, and quality trade-offs. Literature has mainly focused on analyzing time and cost with little or no reported research focusing on models for optimizing construction time, cost, and quality jointly. Government agencies have recently started using new types of contracting methods which have placed an increasing pressure on decision makers in the construction industry to search for an optimal/near optimal resource utilization plan that minimizes construction cost and time while maximizing its quality. In this paper, a 0-1 Integer Programming model which enables meeting quality output standards and time and budget objectives respectively is developed. The clever use of binary variables allows us to solve many interesting and difficult problems and our ability to model complex problems increases tremendously when we use binary variables. A version of the model to minimize time meeting quality and cost objectives is applied to a road building project. Two alternatives versions of the model which enable to minimize cost meeting quality and time and to maximize quality meeting time and cost, are shown in an Appendix.

Optimizing Timetable Synchronization for Rail Mass Transit

In most urban public transit rail systems, passengers may need to make several interchanges between different lines to reach their destination. The design of coordinated timetables that enable smooth interchanges with minimal delay for all passengers is a very difficult task. This paper presents a mixed-integer-programming optimization model for this schedule synchronization problem for nonperiodic timetables that minimizes the interchange waiting times of all passengers. A novelty in our formulation is the use of binary variables that enable the correct representation of the waiting times to the “next available” train at the interchange stations. By adjusting trains' run times and station dwell times during their trips and their dispatch times, turnaround times at the terminals, and headways at the stations, our model can construct high-quality timetables that minimize transfer waiting times. We also discuss an optimization-based heuristic for the model. We have tested our algorithm for the Mass Transit Railway (MTR) system in Hong Kong, which runs six railway lines with many cross-platform interchange stations. Preliminary numerical results indicate that our approach improves the synchronization significantly compared with the current practice of using fixed headways and trip times. We also explore the trade-offs among different operational parameters and flexibility and their impact on overall passenger waiting times.

Multi-Period Near-Equilibrium in a Pool-Based Electricity Market Including On/Off Decisions

Based on the well-known concept of single-period equilibrium in an electricity market, this paper defines, analyzes and illustrates the concept of a multi-period equilibrium. Within this equilibrium framework and a multi-period horizon, market participants simultaneously optimize their respective individual and conflicting objectives. Constraints involving prices can be incorporated into the problems of the market participants. To avoid the limitations imposed by the necessary use of binary variables to model on/off decisions, the conditions to attain a multi-period equilibrium are formulated through Benders decomposition, which allows for efficiently solving the resulting equilibrium problem. The proposed procedure is illustrated using a realistic case study based on the IEEE Reliability Test System.