Mathematical Optimization Techniques Research Articles

Automated treatment planning for a dedicated multi‐source intracranial radiosurgery treatment unit using projected gradient and grassfire algorithms

The purpose of this work is to develop a framework to the inverse problem for radiosurgery treatment planning on the Gamma Knife(®) Perfexion™ (PFX) for intracranial targets. The approach taken in the present study consists of two parts. First, a hybrid grassfire and sphere-packing algorithm is used to obtain shot positions (isocenters) based on the geometry of the target to be treated. For the selected isocenters, a sector duration optimization (SDO) model is used to optimize the duration of radiation delivery from each collimator size from each individual source bank. The SDO model is solved using a projected gradient algorithm. This approach has been retrospectively tested on seven manually planned clinical cases (comprising 11 lesions) including acoustic neuromas and brain metastases. In terms of conformity and organ-at-risk (OAR) sparing, the quality of plans achieved with the inverse planning approach were, on average, improved compared to the manually generated plans. The mean difference in conformity index between inverse and forward plans was -0.12 (range: -0.27 to +0.03) and +0.08 (range: 0.00-0.17) for classic and Paddick definitions, respectively, favoring the inverse plans. The mean difference in volume receiving the prescribed dose (V(100)) between forward and inverse plans was 0.2% (range: -2.4% to +2.0%). After plan renormalization for equivalent coverage (i.e., V(100)), the mean difference in dose to 1 mm(3) of brainstem between forward and inverse plans was -0.24 Gy (range: -2.40 to +2.02 Gy) favoring the inverse plans. Beam-on time varied with the number of isocenters but for the most optimal plans was on average 33 min longer than manual plans (range: -17 to +91 min) when normalized to a calibration dose rate of 3.5 Gy/min. In terms of algorithm performance, the isocenter selection for all the presented plans was performed in less than 3 s, while the SDO was performed in an average of 215 min. PFX inverse planning can be performed using geometric isocenter selection and mathematical modeling and optimization techniques. The obtained treatment plans all meet or exceed clinical guidelines while displaying high conformity.

A generalised guideline for process changes for resource conservation networks

In the past one and a half decades, resource conservation network (RCN) synthesis has been well accepted by both academics and industrial practitioners in enhancing sustainability aspect for the process industry. Various insight-based pinch analysis and mathematical optimisation techniques have been proposed to synthesise optimum RCN. For the former, the developments to date have been focusing on various targeting and design tools, with very little emphasis on the systematic assessment of process changes. The latter involves the modification of process operating conditions (e.g., flowrate, concentration, temperature, etc.), which leads to further reduction of minimum fresh resource flowrates. In this study, the plus-minus principle in heat exchanger network synthesis is extended for use with graphical targeting tool in assessing opportunities for process changes in the RCN of fixed-flowrate problems, aiming to further reduce its minimum fresh resource flowrate. Literature examples are used for illustration.

Sensitivity analysis of separation circuits

Several methods for the design of separation circuits have been presented in the literature. These methodologies can be classified as those that use either heuristics or mathematical optimization techniques. However, none of these methods is currently used in industry. This is, the former is very simple to incorporate the complexities of the problem and the latter requires that the designer has more specialized training.In this work, we use sensitivity analysis to analyze and design separation circuits. We study the effect of each stage on the general circuit by identifying the relationships between recovery at each stage and global recovery of the circuit. Based on these results, we propose a novel methodology to analyze and design separation circuits. This new method is a hybrid because it uses mathematical analysis and the designer's experience. An example is provided for flotation separation in copper mining.

ACL Strain and Tensile Forces for Weight Bearing and Non—Weight-Bearing Exercises After ACL Reconstruction: A Guide to Exercise Selection

There is a growing body of evidence documenting loads applied to the anterior cruciate ligament (ACL) for weight-bearing and non-weight-bearing exercises. ACL loading has been quantified by inverse dynamics techniques that measure anterior shear force at the tibiofemoral joint (net force primarily restrained by the ACL), ACL strain (defined as change in ACL length with respect to original length and expressed as a percentage) measured directly in vivo, and ACL tensile force estimated through mathematical modeling and computer optimization techniques. A review of the biomechanical literature indicates the following: ACL loading is generally greater with non-weight-bearing compared to weight-bearing exercises; with both types of exercises, the ACL is loaded to a greater extent between 10° to 50° of knee flexion (generally peaking between 10° and 30°) compared to 50° to 100° of knee flexion; and loads on the ACL change according to exercise technique (such as trunk position). Squatting with excessive forward movement of the knees beyond the toes and with the heels off the ground tends to increase ACL loading. Squatting and lunging with a forward trunk tilt tend to decrease ACL loading, likely due to increased hamstrings activity. During seated knee extension, ACL force decreases when the resistance pad is positioned more proximal on the anterior aspect of the lower leg, away from the ankle. The evidence reviewed as part of this manuscript provides objective data by which to rank exercises based on loading applied to the ACL. The biggest challenge in exercise selection post-ACL reconstruction is the limited knowledge of the optimal amount of stress that should be applied to the ACL graft as it goes through its initial incorporation and eventual maturation process. Clinicians may utilize this review as a guide to exercise selection and rehabilitation progression for patients post-ACL reconstruction.

Sequential robust optimization of a V-bending process using numerical simulations

The coupling of finite element simulations to mathematical optimization techniques has contributed significantly to product improvements and cost reductions in the metal forming industries. The next challenge is to bridge the gap between deterministic optimization techniques and the industrial need for robustness. This paper introduces a generally applicable strategy for modeling and efficiently solving robust optimization problems based on time consuming simulations. Noise variables and their effect on the responses are taken into account explicitly. The robust optimization strategy consists of four main stages: modeling, sensitivity analysis, robust optimization and sequential robust optimization. Use is made of a metamodel-based optimization approach to couple the computationally expensive finite element simulations with the robust optimization procedure. The initial metamodel approximation will only serve to find a first estimate of the robust optimum. Sequential optimization steps are subsequently applied to efficiently increase the accuracy of the response prediction at regions of interest containing the optimal robust design. The applicability of the proposed robust optimization strategy is demonstrated by the sequential robust optimization of an analytical test function and an industrial V-bending process. For the industrial application, several production trial runs have been performed to investigate and validate the robustness of the production process. For both applications, it is shown that the robust optimization strategy accounts for the effect of different sources of uncertainty onto the process responses in a very efficient manner. Moreover, application of the methodology to the industrial V-bending process results in valuable process insights and an improved robust process design.

Tailoring the high-harmonic emission in two-level systems and semiconductors by pulse shaping

The generation of specific high harmonics for an optical two-level system is elucidated. The desired emitted radiation can be induced by a carefully designed excitation pulse, which is found by a multiparameter optimization procedure. The presented mechanism can also be applied to semiconductor structures for which the calculations result in much higher emission frequencies. The optimization procedure is either performed using a genetic algorithm or a rigorous mathematical optimization technique.

Finite Capacity Scheduling - Large System Design

This paper presents the design principles for a large finite capacity scheduling system which takes into account alternative tools and machines. The system will produce tightly packed and stable schedules which in many cases will be close to optimal. The system can then be used as a platform for research into the application of decision making techniques, mathematical optimization, and other modern techniques.

Recent trends indicate rapid growth of nature-inspired optimization in academia and industry

Although researchers often comment on the popularity of nature-inspired meta-heuristics (NIM), there has been very little empirical data to support the claim that NIM are growing in prominence compared to other optimization techniques. This paper presents strong evidence that the use of NIM is not only growing, but indeed appears to have surpassed mathematical optimization techniques and other meta-heuristics in several metrics related to academic research activity (publication frequency) and commercial activity (patenting frequency). Motivated by these findings, this article discusses some of the possible origins of this growing popularity including historical bias, conceptual appeal, simplicity of implementation, and algorithm utility.

Survival of the flexible: explaining the recent popularity of nature-inspired optimization within a rapidly evolving world

Researchers often comment on the popularity and potential of nature-inspired meta-heuristics (NIM), however there has been a paucity of data to directly support the claim that NIM are growing in prominence compared to other optimization techniques. In a companion article published in this special issue, I reported evidence that the use of NIM is not only growing, but indeed has surpassed mathematical optimization techniques (MOT) and other metaheuristics in several metrics related to academic research activity (publication frequency) and commercial activity (patenting frequency). Motivated by these findings, this article reviews several theories of algorithm utility and discusses why these arguments remain unsatisfying. I argue that any explanation of NIM popularity should directly account for the manner in which most NIM success has actually been achieved: through hybridization and customization to specific problems. By taking a problem lifecycle perspective, this paper provides simple yet important insights into how nature-inspired meta-heuristics might derive utility by being flexible. Given global trends in the evolution of business products and services where optimization algorithms are applied, I speculate that highly flexible algorithm frameworks will become increasingly popular within our rapidly changing world.

Integrated Management of Non-conventional Water Resources in Anhydrous Islands

Anhydrous islands are dependent either on non-conventional water resources, such as desalinated seawater or reclaimed water from wastewater, or on water importation from the mainland. The latter option is often expensive and non-sustainable. Desalinated water can be used for potable and non-potable water applications, while reclaimed water can be used for non-potable water applications. Thus all water needs can be satisfied by an optimal blend of desalinated and reclaimed water. It is important to calculate the optimal capacities and locations of seawater desalination plants, wastewater treatment plants and water reclamation plants, and to estimate the water/wastewater conveyance system, in order to minimise water production and distribution costs. Mathematical modelling and optimisation techniques can be employed for calculating the optimum scenario: the satisfaction of all water needs at minimum cost. In this article, we have estimated the water demands taking into account water quality for the anhydrous Greek island of Syros, in the Aegean Sea. Syros has been subdivided into 6 regions, taking into account geographical and population distribution criteria. All water needs are to be satisfied by desalinated seawater and reclaimed water. A mixed-integer linear programming algorithm is used here to calculate the optimal scenario (location and capacities of desalination plants and wastewater treatment and water reclamation plants, as well as the desalinated water, reclaimed water and wastewater conveyance infrastructure needed) by minimising the annualised total cost including capital and operating costs.

A mathematical optimization technique for managing selective catalytic reduction for coal-fired power plants

Selective catalytic reduction (SCR) is an emissions control technique that primarily reduces harmful emissions of oxides of nitrogen (NOx). To maintain SCR performance, catalyst layers maybe added, removed, or replaced to improve NOx reduction efficiency. To make these changes, power plants must be temporarily shut down, and SCR maintenance during scheduled power plant outages can be very expensive. Consequently, developing a fleet-wide SCR management plans that are both efficient at reducing NOx and limiting operating costs would be extremely desirable. We propose an SCR management framework that finds an optimal SCR management plan that minimizes NOx emissions using integer programming. The SCR management tool consists of two main modules—the SCR schedule generation module and the SCR optimization module. Furthermore, the SCR management framework addresses decision making from the fleet-wide perspective as well as a single plant as opposed to only a single plant, which is currently commercially available. We demonstrate the effectiveness of the tool and provide a tradeoff between NOx reduction and operating cost using Pareto optimal efficient frontiers.

Energy Allowances for Solid Fats and Added Sugars in Nutritionally Adequate U.S. Diets Estimated at 17–33% by a Linear Programming Model

The 2010 Dietary Guidelines Advisory Committee has recommended that no more than 5–15% of total dietary energy should be derived from solid fats and added sugars (SoFAS). The guideline was based on USDA food pattern modeling analyses that met the Dietary Reference Intake recommendations and Dietary Guidelines and followed typical American eating habits. This study recreated food intake patterns for 6 of the same gender-age groups by using USDA data sources and a mathematical optimization technique known as linear programming. The analytic process identified food consumption patterns based on 128 food categories that met the nutritional goals for 9 vitamins, 9 minerals, 8 macronutrients, and dietary fiber and minimized deviation from typical American eating habits. Linear programming Model 1 created gender- and age-specific food patterns that corresponded to energy needs for each group. Model 2 created food patterns that were iso-caloric with diets observed for that group in the 2001–2002 NHANES. The optimized food patterns were evaluated with respect to MyPyramid servings goals, energy density [kcal/g (1 kcal = 4.18 kJ)], and energy cost (US$/2000 kcal). The optimized food patterns had more servings of vegetables and fruit, lower energy density, and higher cost compared with the observed diets. All nutrient goals were met. In contrast to the much lower USDA estimates, the 2 models placed SoFAS allowances at between 17 and 33% of total energy, depending on energy needs.

On Synthesis and Optimization of Cooling Water Systems with Multiple Cooling Towers

Cooling water systems are generally designed with a set of heat exchangers arranged in parallel. This arrangement results in higher cooling water flow rate and low cooling water return temperature, thus reducing cooling tower efficiency. Previous research on cooling water systems has focused mainly on heat exchanger network thus excluding the interaction between heat exchanger network and the cooling towers. This paper presents a technique for grassroot design of cooling water system for wastewater minimization, which incorporates the performances of the cooling towers involved. The study focuses mainly on cooling systems consisting of multiple cooling towers that supply a common set of heat exchangers. The heat exchanger network is synthesized using the mathematical optimization technique. This technique is based on superstructure in which all opportunities for cooling water reuse are explored. The cooling tower model is used to predict the thermal performance of the cooling towers. Two case studies are presented to illustrate the proposed technique. The first case resulted in nonlinear programming (NLP) formulation and the second case yield mixed integer nonlinear programming (MINLP). The nonlinearity in both cases is because of the bilinear terms present in the energy balance constraints. In both cases, the cooling towers operating capacity were debottlenecked without compromising the heat duties.

Continuous optimization via simulation using Golden Region search

Simulation Optimization (SO) is a class of mathematical optimization techniques in which the objective function can only be numerically evaluated through simulation. In this paper, a new SO approach called Golden Region (GR) search is developed for continuous problems. GR divides the feasible region into a number of (sub) regions and selects one region in each iteration for further search based on the quality and distribution of simulated points in the feasible region and the result of scanning the response surface through a metamodel. Monte Carlo experiments show that the GR method is efficient compared to three well-established approaches in the literature. We also prove the asymptotic convergence in probability to a global optimum for a large class of random search methods in general and GR in particular.

Fuzzy Systems for Multicriteria Decision Making


 
 
 One of the techniques used to support decisions in uncertain environments is the Fuzzy TOPSIS method. However, from crisp data, this method considers only one fuzzy set in their analysis, besides being a strictly mathematical optimization technique. This article proposes extensions to the original Fuzzy TOPSIS, exploring two distinct versions: to increase the method with the necessary resources for the mathematics process to consider the membership values of the input data in more than one fuzzy set and to aggregate to method the empiric knowledge of an expert represented through fuzzy rules. In such case, the method, named by Fuzzy F-TOPSIS (Fuzzy Flexible TOPSIS), is proposed with the objective of improving the Fuzzy TOPSIS ability to deal with uncertainty through the combination of the mathematical process involved in the original Fuzzy TOPSIS with the expert empirical knowledge. A case study is presented to validate the proposal.
 
 

Rigorous algorithmic targeting methods for hydrogen networks—Part I: Systems with no hydrogen purification

Hydrogen utility consumption minimization is one of the key strategies for reducing refinery operating costs. Process integration techniques have been widely accepted as an effective tool for reducing hydrogen consumption. To date, various pinch based conceptual methods have been developed. Applying the pinch analysis is simpler than solving the mathematical model. However, the relationship between the pinch simplification and the mathematical model has seldom been addressed. In addition, existing pinch analysis paid little attention to the rare but realistic cases of threshold problem. In this article, the original mathematical model is associated with the pinch concept by proving simplified equivalent formulations. A mathematically rigorous systematic targeting approach is then obtained by mathematical deduction. The proposed method combines pinch insight with a rigorous mathematical optimization technique. The proposed method addresses both threshold and pinch problems, and initially the analysis is restricted to systems with no purification unit.

Parameter‐based mathematical optimization using MATLAB and CATT‐A software.

Mathematical optimization techniques have been applied to room acoustic modeling within CATT‐A. Stochastic optimization methods such as genetic algorithms, pattern search, and ant colony optimization are used via software integration between MATLAB and CATT‐A, allowing for acoustic parameter specific optimization of variable room geometry. The optimization routines rely on results from geometrical ray tracing, which informs the designer on aspects of room acoustics that empirical formulas cannot take into account. An example is presented showing an algorithm that optimizes placement of sound absorbing treatment to obtain a fixed speech transmission index using minimal sound absorbing material in a classroom setting.

Multistage Transmission Expansion Planning Using Local Branching Method

This paper proposes a method for solving a large-scale multistage transmission expansion planning problem. The formulated problem is of a large-scale mixed integer linear programming which is difficult to handle by general mathematical optimization techniques. A disjunctive model for a single stage transmission expansion planning is extended for solving the multistage model. In addition, a local branching method is employed with an initial solution obtained from the single stage transmission expansion planning problem. The proposed method has been tested with IEEE 24-bus and a northeastern Thailand power system to show it is efficient to implement for actual power system planning.

Closed bioregenerative life support systems: Applicability to hot deserts

AbstractWater scarcity in hot deserts, which cover about one-fifth of the Earth’s land area, along with rapid expansion of hot deserts into arable lands is one of the key global environmental problems. As hot deserts are extreme habitats characterized by the availability of solar energy with a nearly complete absence of organic life and water, space technology achievements in designing closed ecological systems may be applicable to the design of sustainable settlements in the deserts. This review discusses the key space technology findings for closed biogenerative life support systems (CBLSS), which can simultaneously produce food, water, nutrients, fertilizers, process wastes, and revitalize air, that can be applied to hot deserts. Among them are the closed cycle of water and the acceleration of the cycling times of carbon, biogenic compounds, and nutrients by adjusting the levels of light intensity, temperature, carbon dioxide, and air velocity over plant canopies. Enhanced growth of algae and duckweed at higher levels of carbon dioxide and light intensity can be important to provide complete water recycling and augment biomass production. The production of fertilizers and nutrients can be enhanced by applying the subsurface flow wetland technology and hyper-thermophilic aerobic bacteria for treating liquid and solid wastes. The mathematical models, optimization techniques, and non-invasive measuring techniques developed for CBLSS make it possible to monitor and optimize the performance of such closed ecological systems. The results of long-duration experiments performed in BIOS-3, Biosphere 2, Laboratory Biosphere, and other ground-based closed test facilities suggest that closed water cycle can be achieved in hot-desert bioregenerative systems using the pathways of evapotranspiration, condensation, and biological wastewater treatment technologies. We suggest that the state of the art in the CBLSS design along with the possibility of using direct sunlight for photosynthesis and recent advances in photovoltaic engineering can be used as a basis for building sustainable settlements producing food, water, and energy in hot deserts.

Application and Comparison of Regression and Markov Chain Methods in Bridge Condition Prediction and System Benefit Optimization

This study is designed to assist highway planners and engineers in identifying and choosing an appropriate method for bridge condition predictions. Mathematical optimization techniques, such as linear programming and dynamic programming, can be used for the evaluation of a bridge system. For each bridge, the input data of the bridge project selection model includes the predicted bridge condition in future years, the recommended bridge repair action, the estimated cost of recommended bridge repair action, and the expected improvement or benefit from the repair action. Through mathematical manipulation, bridge projects are selected to maximize the total expected benefit of the bridge system while a number of constraints are simultaneously satisfied. This optimization process is based on the predicted bridge conditions. Bridge condition predictions will affect bridge project selections and the corresponding system benefits. Using the analysis method presented in this paper and bridge condition data from Indiana, Markov chain methods were shown to yield more accurate condition predictions than the polynomial regression method.