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At least 19 records

Short-Term Load Forecasting Considering EV Charging Loads with Prediction Interval Evaluation

Short-term load forecasting plays a critical role in power system planning and operation. Along with the electrification of various loads, electricity demands are becoming increasingly hard to predict. Notably, the recent rise in electric vehicles (EVs) has further contributed to this unpredictability. To address this issue, this paper proposes a probabilistic load forecasting strategy utilizing Gaussian process regression, structured in a day-ahead manner. While many works focus on deterministic prediction, probabilistic forecasting offers additional insights into variability and uncertainty, enabling more flexible and reliable operation for power systems. To enhance the accuracy of the load forecasting model, the inputs include features related to EV charging habits as well as commonly used weather information. The load forecasting results are evaluated using various metrics, including conventional ones that assess the accuracy of point forecasts, as well as additional metrics that test the reliability of prediction intervals. The proposed load forecasting method is finally tested on real residential power consumption data and EV charging data sampled from real-world sources. The results prove that the new features can greatly improve the performance of the load forecasting method.

electrical vehicle↗

Optimal parallel algorithms for problems modeled by a family of intervals

A family of intervals on the real line provides a natural model for a vast number of scheduling and VLSI problems. Recently, a number of parallel algorithms to solve a variety of practical problems on such a family of intervals have been proposed in the literature. Computational tools are developed, and it is shown how they can be used for the purpose of devising cost-optimal parallel algorithms for a number of interval-related problems including finding a largest subset of pairwise nonoverlapping intervals, a minimum dominating subset of intervals, along with algorithms to compute the shortest path between a pair of intervals and, based on the shortest path, a parallel algorithm to find the center of the family of intervals. More precisely, with an arbitrary family of n intervals as input, all algorithms run in O(log n) time using O(n) processors in the EREW-PRAM model of computation.

Olariu, Stephan↗

Optimal Approximation of Quadratic Interval Functions

Measurements are never absolutely accurate, as a result, after each measurement, we do not get the exact value of the measured quantity; at best, we get an interval of its possible values, For dynamically changing quantities x, the additional problem is that we cannot measure them continuously; we can only measure them at certain discrete moments of time t(sub 1), t(sub 2), ... If we know that the value x(t(sub j)) at a moment t(sub j) of the last measurement was in the interval [x-(t(sub j)), x + (t(sub j))], and if we know the upper bound D on the rate with which x changes, then, for any given moment of time t, we can conclude that x(t) belongs to the interval [x-(t(sub j)) - D (t - t(sub j)), x + (t(sub j)) + D (t - t(sub j))]. This interval changes linearly with time, an is, therefore, called a linear interval function. When we process these intervals, we get an expression that is quadratic and higher order w.r.t. time t, Such "quadratic" intervals are difficult to process and therefore, it is necessary to approximate them by linear ones. In this paper, we describe an algorithm that gives the optimal approximation of quadratic interval functions by linear ones.

Koshelev, Misha↗

Mechanisms underlying very-low-frequency RR-interval oscillations in humans

BACKGROUND: Survival of post-myocardial infarction patients is related inversely to their levels of very-low-frequency (0.003 to 0.03 Hz) RR-interval variability. The physiological basis for such oscillations is unclear. In our study, we used blocking drugs to evaluate potential contributions of sympathetic and vagal mechanisms and the renin-angiotensin-aldosterone system to very-low-frequency RR-interval variability in 10 young healthy subjects. METHODS AND RESULTS: We recorded RR intervals and arterial pressures during three separate sessions, with the patient in supine and 40 degree upright tilt positions, during 20-minute frequency (0.25 Hz) and tidal volume-controlled breathing after intravenous injections: saline (control), atenolol (0.2 mg/kg, beta-adrenergic blockade), atropine sulfate (0.04 mg/kg, parasympathetic blockade), atenolol and atropine (complete autonomic blockade), and enalaprilat (0.02 mg/kg, ACE blockade). We integrated fast Fourier transform RR-interval spectral power at very low (0.003 to 0.03 Hz), low (0.05 to 0. 15 Hz), and respiratory (0.2 to 0.3 Hz) frequencies. Beta-adrenergic blockade had no significant effect on very-low- or low-frequency RR-interval power but increased respiratory frequency power 2-fold. ACE blockade had no significant effect on low or respiratory frequency RR-interval power but modestly (approximately 21%) increased very-low-frequency power in the supine (but not upright tilt) position (P<0.05). The most profound effects were exerted by parasympathetic blockade: Atropine, given alone or with atenolol, abolished nearly all RR-interval variability and decreased very-low-frequency variability by 92%. CONCLUSIONS: Although very-low-frequency heart period rhythms are influenced by the renin-angiotensin-aldosterone system, as low and respiratory frequency RR-interval rhythms, they depend primarily on the presence of parasympathetic outflow. Therefore the prognostic value of very-low-frequency heart period oscillations may derive from the fundamental importance of parasympathetic mechanisms in cardiovascular health.

Non-NASA Center↗

Bootstrap Prediction Intervals in Non-Parametric Regression with Applications to Anomaly Detection

Prediction intervals provide a measure of the probable interval in which the outputs of a regression model can be expected to occur. Subsequently, these prediction intervals can be used to determine if the observed output is anomalous or not, conditioned on the input. In this paper, a procedure for determining prediction intervals for outputs of nonparametric regression models using bootstrap methods is proposed. Bootstrap methods allow for a non-parametric approach to computing prediction intervals with no specific assumptions about the sampling distribution of the noise or the data. The asymptotic fidelity of the proposed prediction intervals is theoretically proved. Subsequently, the validity of the bootstrap based prediction intervals is illustrated via simulations. Finally, the bootstrap prediction intervals are applied to the problem of anomaly detection on aviation data.

Kumar, Sricharan↗

Technical Specification Surveillance Interval Extension Using Self-Diagnostics

As part of the Light Water Reactor Sustainability program, an ongoing research effort is being conducted on technical specifications surveillance interval extension of digital equipment in nuclear power plants. The research team is led by Idaho National Laboratory and includes Pacific Northwest National Laboratory, Technology Resources, and Oak Ridge National Laboratory. This research focuses on developing methods for applying the U.S. Nuclear Regulatory Commission (NRC)–approved guidance to implement a licensee-controlled, risk-informed surveillance frequency change program in digital instrumentation and control (I&C) systems that include self-diagnostics and online monitoring (OLM) capabilities. Although approved methods exist for extending technical specifications (TS) surveillance test intervals (STIs) for general equipment, including analog I&C equipment, gaps remain in technology and guidance on crediting newer digital equipment’s internal self-diagnostics and OLM characteristics. Previous research described a general methodology for crediting internal self-diagnostics for extending surveillance test intervals. The methodology used self-diagnostics to detect—and credited recovery from—failure. Self-diagnostics were also applied for performance monitoring during the extended surveillance interval. This report discusses the status of recent activities to evaluate the previously developed methodology using a pilot study. Although both a utility partner for a pilot study and a specific digital asset were identified in FY2020, delays in obtaining proprietary information resulted in a limited ability to fully evaluate the methodology, and further interactions were complicated by the COVID pandemic. Therefore, at that time, the use of public-domain information—along with current processes for surveillance interval extension through a surveillance frequency control program—identified the need to fully assess diagnostic coverage as part of the pilot study. Furthermore, self-diagnostics were also identified as a potential option to replace the drift analyses conducted as part of current STI extension procedures. In FY2022, the project was reconstituted with the industry partner, and information and data were made available by the industry partner to the research team for review. The shared information included failure event descriptions and data for a digital I&C system since its implementation, as well as recent STI extension interval reports developed by the utility partner on that digital I&C system. This report presents an evaluation of this information and data and describes an application of the proposed methodology cited above. The methodology seeks to take advantage of the self-diagnostics and OLM capabilities to reduce risk or reduce the level of qualitative monitoring assessment needed to perform a risk-informed STI extension using existing NRC approved guidance or both. Addressing these issues of STI extension by crediting self-diagnostics is likely to result in benefits for current and future nuclear power plant (NPP) operations, including lowering the barriers to adoption of digital I&C systems and increasing cost savings by deferring or eliminating unneeded preventive maintenance (tasks or checks or activities). Specifically, self-diagnostic and OLM capabilities of newer digital equipment being installed in non-safety and safety applications are designed to detect failures, provide early warning of potential failures, and notify plant operators to take appropriate action to reduce out of service (OOS) time thus protecting safety margins. Moreover, the equipment is expected to provide information that time-related operational degradation is identified early to ensure timely and planned corrective actions instead of a reactive and unplanned approach ahead of an extended-surveillance interval.

42 ENGINEERING↗

A new technique to determine the correlation between the QT interval and heart-rate for control and SIDS babies

The ability of the autonomic nervous system to alter the QT interval in response to heart rate changes is essential to cardiovascular control. An accurate way to determine the relation between QT intervals and their corresponding RR intervals is described. A computer algorithm measures the RR intervals using digital filtering and cross-correlating the QRS sections of consecutive waveforms. The QT intervals is calculated by choosing a section of, the ECG that includes the T wave and cross-correlating it with all the consecutive T waves. At least 4000 pairs of QT-RR intervals are computed for each subject and a best fit correlation function determines the relations between the QT and RR intervals. This technique enables to establish a precise correlation between RR and QT in order to distinguish between control and SIDS babies.

Non-NASA Center↗

Fractal correlation properties of R-R interval dynamics and mortality in patients with depressed left ventricular function after an acute myocardial infarction

BACKGROUND: Preliminary data suggest that the analysis of R-R interval variability by fractal analysis methods may provide clinically useful information on patients with heart failure. The purpose of this study was to compare the prognostic power of new fractal and traditional measures of R-R interval variability as predictors of death after acute myocardial infarction. METHODS AND RESULTS: Time and frequency domain heart rate (HR) variability measures, along with short- and long-term correlation (fractal) properties of R-R intervals (exponents alpha(1) and alpha(2)) and power-law scaling of the power spectra (exponent beta), were assessed from 24-hour Holter recordings in 446 survivors of acute myocardial infarction with a depressed left ventricular function (ejection fraction </=35%). During a mean+/-SD follow-up period of 685+/-360 days, 114 patients died (25.6%), with 75 deaths classified as arrhythmic (17.0%) and 28 as nonarrhythmic (6.3%) cardiac deaths. Several traditional and fractal measures of R-R interval variability were significant univariate predictors of all-cause mortality. Reduced short-term scaling exponent alpha(1) was the most powerful R-R interval variability measure as a predictor of all-cause mortality (alpha(1) <0.75, relative risk 3.0, 95% confidence interval 2.5 to 4.2, P<0.001). It remained an independent predictor of death (P<0.001) after adjustment for other postinfarction risk markers, such as age, ejection fraction, NYHA class, and medication. Reduced alpha(1) predicted both arrhythmic death (P<0.001) and nonarrhythmic cardiac death (P<0.001). CONCLUSIONS: Analysis of the fractal characteristics of short-term R-R interval dynamics yields more powerful prognostic information than the traditional measures of HR variability among patients with depressed left ventricular function after an acute myocardial infarction.

Non-NASA Center↗

Cardiac interbeat interval dynamics from childhood to senescence : comparison of conventional and new measures based on fractals and chaos theory

BACKGROUND: New methods of R-R interval variability based on fractal scaling and nonlinear dynamics ("chaos theory") may give new insights into heart rate dynamics. The aims of this study were to (1) systematically characterize and quantify the effects of aging from early childhood to advanced age on 24-hour heart rate dynamics in healthy subjects; (2) compare age-related changes in conventional time- and frequency-domain measures with changes in newly derived measures based on fractal scaling and complexity (chaos) theory; and (3) further test the hypothesis that there is loss of complexity and altered fractal scaling of heart rate dynamics with advanced age. METHODS AND RESULTS: The relationship between age and cardiac interbeat (R-R) interval dynamics from childhood to senescence was studied in 114 healthy subjects (age range, 1 to 82 years) by measurement of the slope, beta, of the power-law regression line (log power-log frequency) of R-R interval variability (10(-4) to 10(-2) Hz), approximate entropy (ApEn), short-term (alpha(1)) and intermediate-term (alpha(2)) fractal scaling exponents obtained by detrended fluctuation analysis, and traditional time- and frequency-domain measures from 24-hour ECG recordings. Compared with young adults (<40 years old, n=29), children (<15 years old, n=27) showed similar complexity (ApEn) and fractal correlation properties (alpha(1), alpha(2), beta) of R-R interval dynamics despite lower spectral and time-domain measures. Progressive loss of complexity (decreased ApEn, r=-0.69, P<0.001) and alterations of long-term fractal-like heart rate behavior (increased alpha(2), r=0.63, decreased beta, r=-0.60, P<0.001 for both) were observed thereafter from middle age (40 to 60 years, n=29) to old age (>60 years, n=29). CONCLUSIONS: Cardiac interbeat interval dynamics change markedly from childhood to old age in healthy subjects. Children show complexity and fractal correlation properties of R-R interval time series comparable to those of young adults, despite lower overall heart rate variability. Healthy aging is associated with R-R interval dynamics showing higher regularity and altered fractal scaling consistent with a loss of complex variability.

NASA Discipline Cardiopulmonary↗

Abnormalities of the QT interval in primary disorders of autonomic failure

BACKGROUND: Experimental evidence shows that activation of the autonomic nervous system influences ventricular repolarization and, therefore, the QT interval on the ECG. To test the hypothesis that the QT interval is abnormal in autonomic dysfunction, we examined ECGs in patients with severe primary autonomic failure and in patients with congenital dopamine beta-hydroxylase (DbetaH) deficiency who are unable to synthesize norepinephrine and epinephrine. SUBJECTS AND METHODS: Maximal QT and rate-corrected QT (QTc) intervals and adjusted QTc dispersion [(maximal QTc - minimum QTc on 12 lead ECG)/square root of the number of leads measured] were determined in blinded fashion from ECGs of 67 patients with primary autonomic failure (36 patients with multiple system atrophy [MSA], and 31 patients with pure autonomic failure [PAF]) and 17 age- and sex-matched healthy controls. ECGs of 5 patients with congenital DbetaH deficiency and 6 age- and sex-matched controls were also analyzed. RESULTS: Patients with MSA and PAF had significantly prolonged maximum QTc intervals (492+/-58 ms(1/2) and 502+/-61 ms(1/2) [mean +/- SD]), respectively, compared with controls (450+/-18 ms(1/2), P < .05 and P < .01, respectively). A similar but not significant trend was observed for QT. QTc dispersion was also increased in MSA (40+/-20 ms(1/2), P < .05 vs controls) and PAF patients (32+/-19 ms(1/2), NS) compared with controls (21+/-5 ms(1/2)). In contrast, patients with congenital DbetaH deficiency did not have significantly different RR, QT, QTc intervals, or QTc dispersion when compared with controls. CONCLUSIONS: Patients with primary autonomic failure who have combined parasympathetic and sympathetic failure have abnormally prolonged QT interval and increased QT dispersion. However, QT interval in patients with congenital DbetaH deficiency was not significantly different from controls. It is possible, therefore, that QT abnormalities in patients with primary autonomic failure are not solely caused by lesions of the sympathetic nervous system, and that the parasympathetic nervous system is likely to have a modulatory role in ventricular repolarization.

Non-NASA Center↗

Fundamental relations between short-term RR interval and arterial pressure oscillations in humans

BACKGROUND: One of the principal explanations for respiratory sinus arrhythmia is that it reflects arterial baroreflex buffering of respiration-induced arterial pressure fluctuations. If this explanation is correct, then elimination of RR interval fluctuations should increase respiratory arterial pressure fluctuations. METHODS AND RESULTS: We measured RR interval and arterial pressure fluctuations during normal sinus rhythm and fixed-rate atrial pacing at 17.2+/-1.8 (SEM) beats per minute greater than the sinus rate in 16 healthy men and 4 healthy women, 20 to 34 years of age. Measurements were made during controlled-frequency breathing (15 breaths per minute or 0.25 Hz) with subjects in the supine and 40 degree head-up tilt positions. We characterized RR interval and arterial pressure variabilities in low-frequency (0.05 to 0.15 Hz) and respiratory-frequency (0.20 to 0.30 Hz) ranges with fast Fourier transform power spectra and used cross-spectral analysis to determine the phase relation between the two signals. As expected, cardiac pacing eliminated beat-to-beat RR interval variability. Against expectations, however, cardiac pacing in the supine position significantly reduced arterial pressure oscillations in the respiratory frequency (systolic, 6.8+/-1.8 to 2.9 +/-0.6 mm Hg2/Hz, P=.017). In contrast, cardiac pacing in the 40 degree tilt position increased arterial pressure variability (systolic, 8.0+/-1.8 to 10.8 +/-2.6, P=.027). Cross-spectral analysis showed that 40 degree tilt shifted the phase relation between systolic pressure and RR interval at the respiratory frequency from positive to negative (9 +/-7 degrees versus -17+/-11 degrees, P=.04); that is, in the supine position, RR interval changes appeared to lead arterial pressure changes, and in the upright position, RR interval changes appeared to follow arterial pressure changes. CONCLUSIONS: These results demonstrate that respiratory sinus arrhythmia can actually contribute to respiratory arterial pressure fluctuations. Therefore, respiratory sinus arrhythmia does not represent simple baroreflex buffering of arterial pressure.

NASA Discipline Cardiopulmonary↗

Important influence of respiration on human R-R interval power spectra is largely ignored

Frequency-domain analyses of R-R intervals are used widely to estimate levels of autonomic neural traffic to the human heart. Because respiration modulates autonomic activity, we determined for nine healthy subjects the influence of breathing frequency and tidal volume on R-R interval power spectra (fast-Fourier transform method). We also surveyed published literature to determine current practices in this burgeoning field of scientific inquiry. Supine subjects breathed at rates of 6, 7.5, 10, 15, 17.1, 20, and 24 breaths/min and with nominal tidal volumes of 1,000 and 1,500 ml. R-R interval power at respiratory and low (0.06-0.14 Hz) frequencies declined significantly as breathing frequency increased. R-R interval power at respiratory frequencies was significantly greater at a tidal volume of 1,500 than 1,000 ml. Neither breathing frequency nor tidal volume influenced average R-R intervals significantly. Our review of studies reporting human R-R interval power spectra showed that 51% of the studies controlled respiratory rate, 11% controlled tidal volume, and 11% controlled both respiratory rate and tidal volume. The major implications of our analyses are that breathing parameters strongly influence low-frequency as well as respiratory frequency R-R interval power spectra and that this influence is largely ignored in published research.

NASA Discipline Cardiopulmonary↗

A contracting-interval program for the Danilewski method

The concept of contracting-interval programs is applied to finding the eigenvalues of a matrix. The development is a three-step process in which (1) a program is developed for the reduction of a matrix to Hessenberg form, (2) a program is developed for the reduction of a Hessenberg matrix to colleague form, and (3) the characteristic polynomial with interval coefficients is readily obtained from the interval of colleague matrices. This interval polynomial is then factored into quadratic factors so that the eigenvalues may be obtained. To develop a contracting-interval program for factoring this polynomial with interval coefficients it is necessary to have an iteration method which converges even in the presence of controlled rounding errors. A theorem is stated giving sufficient conditions for the convergence of Newton's method when both the function and its Jacobian cannot be evaluated exactly but errors can be made proportional to the square of the norm of the difference between the previous two iterates. This theorem is applied to prove the convergence of the generalization of the Newton-Bairstow method that is used to obtain quadratic factors of the characteristic polynomial.

Harris, J. D.↗

IBM system/360 assembly language interval arithmetic software

Computer software designed to perform interval arithmetic is described. An interval is defined as the set of all real numbers between two given numbers including or excluding one or both endpoints. Interval arithmetic consists of the various elementary arithmetic operations defined on the set of all intervals, such as interval addition, subtraction, union, etc. One of the main applications of interval arithmetic is in the area of error analysis of computer calculations. For example, it has been used sucessfully to compute bounds on sounding errors in the solution of linear algebraic systems, error bounds in numerical solutions of ordinary differential equations, as well as integral equations and boundary value problems. The described software enables users to implement algorithms of the type described in references efficiently on the IBM 360 system.

Phillips, E. J.↗

Minimax confidence intervals in geomagnetism

The present paper uses theory of Donoho (1989) to find lower bounds on the lengths of optimally short fixed-length confidence intervals (minimax confidence intervals) for Gauss coefficients of the field of degree 1-12 using the heat flow constraint. The bounds on optimal minimax intervals are about 40 percent shorter than Backus' intervals: no procedure for producing fixed-length confidence intervals, linear or nonlinear, can give intervals shorter than about 60 percent the length of Backus' in this problem. While both methods rigorously account for the fact that core field models are infinite-dimensional, the application of the techniques to the geomagnetic problem involves approximations and counterfactual assumptions about the data errors, and so these results are likely to be extremely optimistic estimates of the actual uncertainty in Gauss coefficients.

Stark, Philip B.↗

Altered fractal dynamics of gait: reduced stride-interval correlations with aging and Huntington's disease

Fluctuations in the duration of the gait cycle (the stride interval) display fractal dynamics and long-range correlations in healthy young adults. We hypothesized that these stride-interval correlations would be altered by changes in neurological function associated with aging and certain disease states. To test this hypothesis, we compared the stride-interval time series of 1) healthy elderly subjects and young controls and of 2) subjects with Huntington's disease and healthy controls. Using detrended fluctuation analysis we computed alpha, a measure of the degree to which one stride interval is correlated with previous and subsequent intervals over different time scales. The scaling exponent alpha was significantly lower in elderly subjects compared with young subjects (elderly: 0.68 +/- 0.14; young: 0.87 +/- 0.15; P < 0.003). The scaling exponent alpha was also smaller in the subjects with Huntington's disease compared with disease-free controls (Huntington's disease: 0.60 +/- 0.24; controls: 0.88 +/-0.17; P < 0.005). Moreover, alpha was linearly related to degree of functional impairment in subjects with Huntington's disease (r = 0.78, P < 0.0005). These findings demonstrate that strike-interval fluctuations are more random (i.e., less correlated) in elderly subjects and in subjects with Huntington's disease. Abnormal alterations in the fractal properties of gait dynamics are apparently associated with changes in central nervous system control.

NASA Discipline Cardiopulmonary↗

Finding Every Root of a Broad Class of Real, Continuous Functions in a Given Interval

One of the most pervasive needs within the Deep Space Network (DSN) Metric Prediction Generator (MPG) view period event generation is that of finding solutions to given occurrence conditions. While the general form of an equation expresses equivalence between its left-hand and right-hand expressions, the traditional treatment of the subject subtracts the two sides, leaving an expression of the form Integral of(x) = 0. Values of the independent variable x satisfying this condition are roots, or solutions. Generally speaking, there may be no solutions, a unique solution, multiple solutions, or a continuum of solutions to a given equation. In particular, all view period events are modeled as zero crossings of various metrics; for example, the time at which the elevation of a spacecraft reaches its maximum value, as viewed from a Deep Space Station (DSS), is found by locating that point at which the derivative of the elevation function becomes zero. Moreover, each event type may have several occurrences within a given time interval of interest. For example, a spacecraft in a low Moon orbit will experience several possible occultations per day, each of which must be located in time. The MPG is charged with finding all specified event occurrences that take place within a given time interval (or pass ), without any special clues from operators as to when they may occur, for the entire spectrum of missions undertaken by the DSN. For each event type, the event metric function is a known form that can be computed for any instant within the interval. A method has been created for a mathematical root finder to be capable of finding all roots of an arbitrary continuous function, within a given interval, to be subject to very lenient, parameterized assumptions. One assumption is that adjacent roots are separated at least by a given amount, xGuard. Any point whose function value is less than ef in magnitude is considered to be a root, and the function values at distances xGuard away from a root are larger than ef, unless there is another root located in this vicinity. A root is considered found if, during iteration, two root candidates differ by less than a pre-specified ex, and the optimum cubic polynomial matching the function at the end and at two interval points (that is within a relative error fraction L at its midpoint) is reliable in indicating whether the function has extrema within the interval. The robustness of this method depends solely on choosing these four parameters that control the search. The roots of discontinuous functions were also found, but at degraded performance.

Tausworthe, Robert C.↗

Incremental Interval Assignment by Integer Linear Algebra with Improvements

Interval Assignment (IA) is the problem of selecting the number of mesh edges (intervals) for each curve for conforming quad and hex meshing. The intervals x is fundamentally integer-valued. Many other approaches perform numerical optimization then convert a floating-point solution into an integer solution, which is slow and error prone. We avoid such steps: we start integer, and stay integer. Incremental Interval Assignment (IIA) uses integer linear algebra (Hermite normal form) to find an initial solution to the meshing constraints, satisfying the integer matrix equation Solving for reduced row echelon form provides integer vectors spanning the nullspace of A. Here we add vectors from the nullspace to improve the initial solution, maintaining Ax = b Heuristics find good integer linear combinations of nullspace vectors that provide strict improvement towards variable bounds or goals. IIA always produces an integer solution if one exists. In practice we usually achieve solutions close to the user goals, but there is no guarantee that the solution is optimal, nor even satisfies variable bounds, e.g. has positive intervals. We describe several algorithmic changes since first publication that tend to improve the final solution. The software is freely available.

97 MATHEMATICS AND COMPUTING↗