Showing papers in "Engineering Optimization in 2009"
TL;DR: This study proposes a modified harmony search algorithm incorporating particle swarm concept that was applied to the design of four bench-mark networks, with good results.
Abstract: The optimal design of water distribution networks is a non-linear, multi-modal, and constrained problem classified as an NP-hard combinatorial problem. Because of the drawbacks of calculus-based algorithms, the problem has been tackled by assorted stochastic algorithms, such as the genetic algorithm, simulated annealing, tabu search, shuffled frog-leaping algorithm, ant colony optimization algorithm, harmony search, cross entropy, and scatter search. This study proposes a modified harmony search algorithm incorporating particle swarm concept. This algorithm was applied to the design of four bench-mark networks (two-loop, Hanoi, Balerma, and New York City networks), with good results.
242 citations
TL;DR: Through comparative studies on sampling property and metamodel accuracy, the new approach is shown to outperform other sequential sampling methods for global meetamodelling and is comparable to the one-stage sampling method while providing more flexibility in a sequential metammodelling procedure.
Abstract: Space-filling and projective properties are desired features in the design of computer experiments to create global metamodels to replace expensive computer simulations in engineering design. The goal in this article is to develop an efficient and effective sequential Quasi-LHD (Latin Hypercube design) sampling method to maintain and balance the two aforementioned properties. The sequential sampling is formulated as an optimization problem, with the objective being the Maximin Distance, a space-filling criterion, and the constraints based on a set of pre-specified minimum one-dimensional distances to achieve the approximate one-dimensional projective property. Through comparative studies on sampling property and metamodel accuracy, the new approach is shown to outperform other sequential sampling methods for global metamodelling and is comparable to the one-stage sampling method while providing more flexibility in a sequential metamodelling procedure.
62 citations
TL;DR: The Non-dominated Archiving Ant Colony Optimization (NA-ACO), which benefits from the concept of a multi-colony ant algorithm and incorporates a new information-exchange policy, which results in a set of non-dominated solutions.
Abstract: Multi-objective optimization using heuristic methods has been established as a subdiscipline that combines the fields of heuristic computation and classical multiple criteria decision making. This article presents the Non-dominated Archiving Ant Colony Optimization (NA-ACO), which benefits from the concept of a multi-colony ant algorithm and incorporates a new information-exchange policy. In the proposed information-exchange policy, after a given number of iterations, different colonies exchange information on the assigned objective, resulting in a set of non-dominated solutions. The non-dominated solutions are moved into an offline archive for further pheromone updating. Performance of the NA-ACO is tested employing two well-known mathematical multi-objective benchmark problems. The results are promising and compare well with those of well-known NSGA-II algorithms used in real-world multi-objective-optimization problems. In addition, the optimization of reservoir operating policy with multiple objectives...
60 citations
TL;DR: In this article, the authors point out the real-world prevalence of the multiple-product, multiple-constraint newsboy problem with two objectives: total and incremental discounts on purchasing prices.
Abstract: This article points out the real-world prevalence of the multiple-product, multiple-constraint newsboy problem with two objectives—the ‘newsstand problem’—in which there are total and incremental discounts on purchasing prices. The constraints are the warehouse capacity and the batch forms of the order placements. The first objective of this problem is to find the order quantities that maximize expected profit, and the second objective is maximizing the service rate. It is assumed that the holding and shortage costs, modelled by a quadratic function, occur at the end of the period. Moreover, the decision variables are integer. A formulation of the problem is presented and shown to be an integer nonlinear programming model. Finally, an efficient hybrid algorithm is provided to solve the model and the results are illustrated with a numerical example.
57 citations
TL;DR: Recently developed pruning techniques for incremental search space reduction in combination with subdivision techniques for the approximation of the entire solution set, the so-called Pareto set, are used.
Abstract: A multi-objective problem is addressed consisting of finding optimal low-thrust gravity-assist trajectories for interplanetary and orbital transfers. For this, recently developed pruning techniques for incremental search space reduction – which will be extended for the current situation – in combination with subdivision techniques for the approximation of the entire solution set, the so-called Pareto set, are used. Subdivision techniques are particularly promising for the numerical treatment of these multi-objective design problems since they are characterized (amongst others) by highly disconnected feasible domains, which can easily be handled by these set oriented methods. The complexity of the novel pruning techniques is analysed, and finally the usefulness of the novel approach is demonstrated by showing some numerical results for two realistic cases.
56 citations
TL;DR: In this paper, an automated method for generating itineraries for the harvest vehicles facilitates the planning for autonomous agricultural vehicles, and the solution time of the infield logistics problem is considerably reduced by reformulating it as a modified minimum-cost network flow problem.
Abstract: Crop-harvesting operations are typically carried out with combine harvesters. The harvested product is transferred to one or more tractors every time the combine harvester's storage capacity is reached. The efficiency of the process can be significantly improved by computing optimal routes and interactions for the harvest vehicles in the field. Furthermore, an automated method for generating itineraries for the harvest vehicles facilitates the planning for autonomous agricultural vehicles. The infield logistics problem is formulated as an integer linear programming vehicle routing problem with additional turn penalty constraints, but, because of the high number of decision variables, it is not possible to solve cases of realistic field size. The solution time of the infield logistics problem is considerably reduced by reformulating it as a modified minimum-cost network flow problem. This specific structure allows the exact solution of intermediate-size planning problems in a much shorter time period. The ...
56 citations
TL;DR: In this paper, an interval-valued fuzzy robust programming (I-VFRP) model has been developed and applied to municipal solid-waste management under uncertainty, which can explicitly address system uncertainties with multiple presentations, and can directly communicate the waste manager's confidence gradients into the optimization process.
Abstract: In this study, an interval-valued fuzzy robust programming (I-VFRP) model has been developed and applied to municipal solid-waste management under uncertainty. The I-VFRP model can explicitly address system uncertainties with multiple presentations, and can directly communicate the waste manager's confidence gradients into the optimization process, facilitating the reflection of weak or strong confidence when subjectively estimating parameter values. Parameters in the I-VFRP model can be represented as either intervals or interval-valued fuzzy sets. Thus, variations of the waste manager's confidence gradients over defining parameters can be effectively handled through interval-valued membership functions, leading to enhanced robustness of the optimization efforts. The results of a theoretical case study indicate that useful solutions for planning municipal solid-waste-management practices can be generated. The waste manager's confidence gradients over various subjective judgments can be directly incorpora...
54 citations
TL;DR: In this article, the design of robust tapers for coupling power between uniform and slow-light periodic waveguides is discussed. But the authors do not consider the effect of parameter variations.
Abstract: This article concerns the design of tapers for coupling power between uniform and slow-light periodic waveguides. New optimization methods are described for designing robust tapers, which not only perform well under nominal conditions, but also over a given set of parameter variations. When the set of parameter variations models the inevitable variations typical in the manufacture or operation of the coupler, a robust design is one that will have a high yield, despite these parameter variations. The ideas of successive refinement, and robust optimization based on multi-scenario optimization with iterative sampling of uncertain parameters, using a fast method for approximately evaluating the reflection coefficient, are introduced. Robust design results are compared to a linear taper, and to optimized tapers that do not take parameter variation into account. Finally, robust performance of the resulting designs is verified using an accurate, but much more expensive, method for evaluating the reflection coeff...
54 citations
TL;DR: In this article, an efficient methodology is proposed to find the optimum shape of arch dams considering fluid-structure interaction subject to earthquake loading, where the earthquake load is considered by time variant ground acceleration applied in the upstream-downstream direction of the arch dam.
Abstract: An efficient methodology is proposed to find the optimum shape of arch dams considering fluid-structure interaction subject to earthquake loading. The earthquake load is considered by time variant ground acceleration applied in the upstream–downstream direction of the arch dam. The optimization is carried out by particle swarm optimization, employing real values of design variables. To reduce the computational cost of the optimization process, two strategies are adopted. In the first strategy, the most influential design variables on arch-dam response from original variables are selected using an adaptive neuro-fuzzy inference system. In the second, arch-dam response is predicted by a properly trained wavelet radial basis function neural network employing the influential design variables as the inputs. In order to assess the effectiveness of the suggested methodology, a real arch dam is considered as a test example. The numerical results demonstrate the computational advantages of the proposed methodology...
53 citations
TL;DR: Identified results showed that the hybrid HS–Solver algorithm requires fewer iterations and gives more effective results than other deterministic and stochastic solution algorithms.
Abstract: In this article, a hybrid global–local optimization algorithm is proposed to solve continuous engineering optimization problems. In the proposed algorithm, the harmony search (HS) algorithm is used as a global-search method and hybridized with a spreadsheet ‘Solver’ to improve the results of the HS algorithm. With this purpose, the hybrid HS–Solver algorithm has been proposed. In order to test the performance of the proposed hybrid HS–Solver algorithm, several unconstrained, constrained, and structural-engineering optimization problems have been solved and their results are compared with other deterministic and stochastic solution methods. Also, an empirical study has been carried out to test the performance of the proposed hybrid HS–Solver algorithm for different sets of HS solution parameters. Identified results showed that the hybrid HS–Solver algorithm requires fewer iterations and gives more effective results than other deterministic and stochastic solution algorithms.
50 citations
TL;DR: A study that advances posture prediction with a multi-objective optimization (MOO) approach that is used to both develop and combine the following human performance measures: joint displacement; musculoskeletal discomfort; and a variation on potential energy.
Abstract: With sufficient fidelity, the use of virtual humans can save time, money, and lives through improved product design, process design, and understanding of behaviour. Optimization-based posture prediction is a unique tool, and this article presents a study that advances posture prediction with a multi-objective optimization (MOO) approach. MOO is used to both develop and combine the following human performance measures: joint displacement; musculoskeletal discomfort; and a variation on potential energy. The following MOO methods are studied in the context of human modelling: objective sum; min–max; and global criterion. Using MOO yields realistic results. Of the independent performance measures, discomfort generally provides the most accurate postures. Potential energy, however, is not a significant factor in governing human posture and should be combined with other performance measures. The three MOO methods for combining performance measures yield similar results, but the objective sum provides slightly m...
TL;DR: This study proposes particle swarm optimization based algorithms to solve multi-objective engineering optimization problems involving continuous, discrete and/or mixed design variables and uses a closest discrete approach (CDA) to solve optimization problems with discrete design variables.
Abstract: This study proposes particle swarm optimization (PSO) based algorithms to solve multi-objective engineering optimization problems involving continuous, discrete and/or mixed design variables. The original PSO algorithm is modified to include dynamic maximum velocity function and bounce method to enhance the computational efficiency and solution accuracy. The algorithm uses a closest discrete approach (CDA) to solve optimization problems with discrete design variables. A modified game theory (MGT) approach, coupled with the modified PSO, is used to solve multi-objective optimization problems. A dynamic penalty function is used to handle constraints in the optimization problem. The methodologies proposed are illustrated by several engineering applications and the results obtained are compared with those reported in the literature.
TL;DR: In this article a method for including a priori preferences of decision makers into multicriteria optimization problems is presented and a set of Pareto-optimal solutions is determined via desirability functions of the objectives which reveal experts’ preferences regarding different objective regions.
Abstract: In this article a method for including a priori preferences of decision makers into multicriteria optimization problems is presented. A set of Pareto-optimal solutions is determined via desirability functions of the objectives which reveal experts’ preferences regarding different objective regions. An application to noisy objective functions is not straightforward but very relevant for practical applications. Two approaches are introduced in order to handle the respective uncertainties by means of the proposed preference-based Pareto optimization. By applying the methods to the original and uncertain Binh problem and a noisy single cut turning cost optimization problem, these approaches prove to be very effective in focusing on different parts of the Pareto front of the ori-ginal problem in both certain and noisy environments.
TL;DR: The computational experience gained from the first two experiments indicates that the algorithm proposed in this article is extremely competitive when compared with other MOEAs, being able to accurately, reliably and robustly approximate the true Pareto front in almost every tested case.
Abstract: In this article, a new proposal of using particle swarm optimization algorithms to solve multi-objective optimization problems is presented. The algorithm is constructed based on the concept of Pareto dominance, as well as a state-of-the-art ‘parallel’ computing technique that intends to improve algorithmic effectiveness and efficiency simultaneously. The proposed parallel particle swarm multi-objective evolutionary algorithm (PPS-MOEA) is tested through a variety of standard test functions taken from the literature; its performance is compared with six noted multi-objective algorithms. The computational experience gained from the first two experiments indicates that the algorithm proposed in this article is extremely competitive when compared with other MOEAs, being able to accurately, reliably and robustly approximate the true Pareto front in almost every tested case. To justify the motivation behind the research of the parallel swarm structure, the computational results of the third experiment confirm ...
TL;DR: In this paper, a behavioural decentralized approach that allows an unmanned aerial vehicle (UAV) formation flight to carry out a waypoint-passing mission effectively is proposed. But the objective of the proposed controller is to make each UAV in the formation fly through predefined waypoints while maintaining its distance from other UAVs.
Abstract: This article proposes a behavioural decentralized approach that allows an unmanned aerial vehicle (UAV) formation flight to carry out a waypoint-passing mission effectively. The objective of the proposed controller is to make each UAV in the formation fly through predefined waypoints while maintaining its distance from other UAVs. To perform these two tasks, which can conflict with each other, coupled dynamics of UAVs is considered; this combines the dynamics of all the members in the formation. To apply the behavioural decentralized controller on the basis of the coupled dynamics, a feedback linearization technique with a diffeomorphic transfer map is derived for a three-dimensional UAV kinematics model. A behavioural approach in which the control input is decided by the relative weight of each UAV's desired behaviour is considered, so that the UAVs can react promptly in various situations. Optimization techniques of the gain matrices are also performed to improve the performance of the formation flight ...
TL;DR: A series of techniques is presented for overcoming some of the numerical instabilities associated with SIMP materials and a robust topology optimization algorithm designed to be able to accommodate a large suite of problems that more closely resemble those found in industry applications is created.
Abstract: A series of techniques is presented for overcoming some of the numerical instabilities associated with SIMP materials. These techniques are combined to create a robust topology optimization algorithm designed to be able to accommodate a large suite of problems that more closely resemble those found in industry applications. A variant of the Kreisselmeier–Steinhauser (KS) function in which the aggregation parameter is dynamically increased over the course of the optimization is used to handle multi-load problems. Results from this method are compared with those obtained using the bound formulation. It is shown that the KS aggregation method produces results superior to those of the bound formulation, which can be highly susceptible to local minima. Adaptive mesh-refinement is presented as a means of addressing the mesh-dependency problem. It is shown that successive mesh-refinement cycles can generate smooth, well-defined structures, and when used in combination with nine-node elements, virtually eliminate...
TL;DR: A preliminary investigation is presented of the performance of a custom multi-objective genetic algorithm (MOGA) for the optimization of a fast cycle PSA operation – the separation of air for N2 production.
Abstract: Pressure Swing Adsorption (PSA) is a cyclic separation process, with advantages over other separation options for middle-scale processes. Automated tools for the design of PSA processes would be beneficial for the development of the technology, but their development is a difficult task due to the complexity of the simulation of PSA cycles and the computational effort needed to detect the performance in the cyclic steady state. A preliminary investigation is presented of the performance of a custom multi-objective genetic algorithm (MOGA) for the optimization of a fast cycle PSA operation – the separation of air for N2 production. The simulation requires a detailed diffusion model, which involves coupled nonlinear partial differential and algebraic equations (PDAEs). The efficiency of MOGA to handle this complex problem has been assessed by comparison with direct search methods. An analysis of the effect of MOGA parameters on the performance is also presented.
TL;DR: The aim of this article is to show how cellular automata can be used for the solution of reservoir-operation problems, and to demonstrate that the method is extraordinarily more efficient and effective than heuristic-search methods.
Abstract: A novel cellular-automata approach is developed in this article for the optimal solution of large-scale reservoir-operation problems. The aim of this article is to show how cellular automata can be used for the solution of reservoir-operation problems, and, more importantly, to demonstrate that the method is extraordinarily more efficient and effective than heuristic-search methods. Both penalized and non-penalized versions of the method are proposed and formulated for the solution of water-supply and hydropower reservoir-operation problems. The cells are defined as the discrete points chosen on the operation horizon of the problem and storage volumes are taken as the cell states. The optimization objective functions of the problems are used to derive the updating rule of the problems. In the non-penalized method, the problems constraints are satisfied explicitly by limiting the change in the cell states between one iteration and the next. In the penalized version, however, a penalty method is used to mod...
TL;DR: In this paper, three different problem analysis techniques (why-what-stopping analysis, fundamental objective hierarchy, and means objective network) are used to analyse the problem from various perspectives.
Abstract: The implementation of closed-loop supply chains to handle the problem of increasing levels of electronic and electrical equipment waste can be beneficial both economically and ecologically. Three different problem analysis techniques—why–what's stopping analysis, fundamental objective hierarchy, and means objective network—are used to analyse the problem from various perspectives. A non-preemptive goal-programming model and solution approach have also been developed, with goals being assigned different weights according to the respective goal priorities. The model considers multiple products, as well as operations associated with the product, subassembly, part, and material levels. A major contribution of the research involves the fact that the objectives and related constraints for each member of the supply chain are explicitly modeled. The results of the analysis show both the effect of varying the priority/weight associated with a goal, and how the values of the deviational variables can aid a decision...
TL;DR: In this article, a robust and efficient risk-based genetic-algorithm model for the optimal design of large, temporary flood-diversion systems, in which the routing effect may not be disregarded, is presented.
Abstract: This article presents a robust and efficient risk-based genetic-algorithm model for the optimal design of large, temporary flood-diversion systems, in which the routing effect may not be disregarded. This article integrates the flood-routing process and uncertainties in flood-magnitude estimator, as well as the hydraulic uncertainties, into an optimization model. A modification to parameter uncertainty modelling is proposed that is verified using Monte Carlo simulation technique. System design capacity and dimensions are explicitly treated as decision variables. The performance of the model is demonstrated using a hypothetical case example. Furthermore, a series of sensitivity analyses are conducted to assess the effect of uncertainties in damage cost and construction time on the final results. The results indicate that these factors, as well as consideration of flood routing, could have a significant effect on the optimum design capacity of the system.
TL;DR: In this article, a bi-level programming model is given in which the upper level model is to make the total system cost and flow entropy minimum, while the lower level is a stochastic user equilibrium assignment with an advanced traveller information system.
Abstract: In the case of two-way traffic, there are two opposite-direction flows on every road and serious unsymmetrical flows exist in rush-hour periods. One of the primary methods used for handling this kind of traffic flow is the use of reversible lanes. Which and how many lanes should be adjusted are optimization problems. They can be treated as a network design problem, i.e. the optimal decision on the resource distribution of a street and highway system in response to a growing travel demand. This article studies a new form of transportation network design problem by performing the strategy of reversible lanes. In order to describe this problem, a bi-level programming model is given in which the upper level model is to make the total system cost and flow entropy minimum, while the lower level is a stochastic user equilibrium assignment with an advanced traveller information system. Finally, the numerical example shows that, by adopting the reversible lane and advanced traveller information system, the total s...
TL;DR: The similarities in concepts, distinctions in philosophy and methodology and effectiveness as direct search methods for both mode-pursing sampling and the genetic algorithm will inspire the development of new direct optimization methods.
Abstract: Since the publication of the authors’ recently developed mode-pursing sampling method, questions have been asked about its performance as compared with traditional global optimization methods such as the genetic algorithm and when to use mode-pursing sampling as opposed to the genetic algorithm. This work aims to provide an answer to these questions. Similarities and distinctions between mode-pursing sampling and the genetic algorithm are presented. Then mode-pursing sampling and the genetic algorithm are compared via testing with benchmark functions and practical engineering design problems. These problems can be categorized from different perspectives such as dimensionality, continuous/discrete variables or the amount of computational time for evaluating the objective function. It is found that both mode-pursing sampling and the genetic algorithm demonstrate great effectiveness in identifying the global optimum. In general, mode-pursing sampling needs much fewer function evaluations and iterations than ...
TL;DR: In this article, a multi-objective finance-based scheduling for construction projects under uncertainty is proposed, which takes into account the line of credit to provide cash for implementation of a construction project.
Abstract: This article employs a new approach to investigate multi-objective finance-based scheduling for construction projects under uncertainty. It takes into consideration the line of credit to provide cash for implementation of a construction project. Using a finance-based scheduling concept and NSGA-II, the article presents a multi-objective model to search the non-dominated solutions considering total duration, required credit, and financing cost as three objectives. Fuzzy-sets theory is used to account for uncertainties in direct cost of each activity for determining the required credit and financing cost. The model fully embeds fuzzy presentation of the uncertainties in direct cost into the model structure. The α -cut approach is used to account for the accepted risk level of the project manager, for which a separate Pareto front with set of non-dominated solutions has been developed. Fuzzy numbers ranking is performed by the Hamming distance method. An example project is presented to validate the model and...
TL;DR: In this article, an alternative topology optimization method for the design of compliant actuators using mesh-free methods, in which the thermo-mechanical multi-physics modelling and geometrically non-linear analysis are included.
Abstract: This article presents an alternative topology optimization method for the design of compliant actuators using mesh-free methods, in which the thermo-mechanical multi-physics modelling and geometrically non-linear analysis are included. The relatively new mesh-free method rather than the standard finite element method (FEM) is used to discretize the design domain and interpolate the bulk density field, because the mesh-free method is in some cases more capable of modelling the large-displacement compliant mechanisms involving the geometrical non-linearity. An interpolation scheme is used to indicate the dependence of material properties on element pseudo densities which are distributed to the corresponding integration points, and the method for imposing essential boundary conditions in mesh-free methods is also discussed. Furthermore, the adjoint approach is incorporated into the mesh-free method to perform the design sensitivity analysis. The optimization problem is established mathematically as a non-lin...
TL;DR: In this article, a fuzzy linear programming (FLP) method is developed for dealing with uncertainties expressed as fuzzy sets that exist in the constraints' left-hand and right-hand sides and the objective function.
Abstract: In this study, a fuzzy linear programming (FLP) method is developed for dealing with uncertainties expressed as fuzzy sets that exist in the constraints’ left-hand and right-hand sides and the objective function. A direct transforming algorithm is advanced for solving the FLP model that improves upon the existing method through provision of a quantitative expression for uncertain relationships among a large number of fuzzy sets. The proposed solution method can greatly reduce computational requirements, which is particularly meaningful for the application of FLP to large-scale practical problems with many fuzzy sets. The developed FLP method is applied to a case of long-term waste-management planning. The results indicate that reasonable solutions have been obtained. They can be used for generating decision alternatives and to help managers identify desired policies for waste management under uncertainty. Compared with the conventional interval-parameter linear programming approach, FLP can provide more i...
TL;DR: The applicability and validity of the CMAES algorithm is demonstrated on three DED test systems with a sequential decomposition approach, and it is found that the obtained results satisfy the Karush–Kuhn–Tucker conditions and confirm optimality.
Abstract: This article presents a covariance matrix adapted evolution strategy (CMAES) algorithm to solve dynamic economic dispatch (DED) problems. The DED is an extension of the conventional economic dispatch problem, in which optimal settings of generator units are determined with a predicted load demand over a period of time. In this article, the applicability and validity of the CMAES algorithm is demonstrated on three DED test systems with a sequential decomposition approach. The results obtained using the CMAES algorithm are compared with results obtained using the real-coded genetic algorithm, the Nelder–Mead simplex method, and other methods from the literature. To compare the performance of the various algorithms, statistical measures like best, mean, worst, standard deviation, and mean computation time over 20 independent runs are taken. The effect of population size on the performance of the CMAES algorithm is also investigated. The simulation experiments reveal that the CMAES algorithm performs better i...
TL;DR: The evolution strategy, which is one of the evolutionary algorithms, is modified to solve mixed–discrete optimization problems and yields better solutions than other methods for most of the test problems.
Abstract: In this study, the evolution strategy, which is one of the evolutionary algorithms, is modified to solve mixed–discrete optimization problems. Three approaches are proposed for handling discrete variables. The first approach is to treat discrete variables as continuous variables and replace the latter with discrete variables that are closest to the continuous variables. The second approach is to compress the difference between discrete variables so that discrete variables far away from the current value will have a higher probability of being selected. The third approach is to represent the discrete variables as integers. As a result, the difference between neighbouring discrete variables becomes equal. This also increases the probability of selection of discrete variables far away from the current value through the mutation operation. Five examples are tested representing single objective, multi-objective, unconstrained, constrained, pure discrete and mixed–discrete variable problems. From the results ob...
TL;DR: Three new algorithms—the bisection algorithm, the three-step algorithm, and the improved three- step algorithm with gap—are developed to automatically generate fewer circles approximating the components to solve the 2D packing optimization problem.
Abstract: In this article, the finite-circle method is introduced for 2D packing optimization. Each component is approximated with a group of circles and the non-overlapping constraints between components are converted into simple constraints between circles. Three new algorithms—the bisection algorithm, the three-step algorithm, and the improved three-step algorithm with gap—are developed to automatically generate fewer circles approximating the components. The approximation accuracy, the circle number, and the computing time are analyzed in detail. Considering the fact that packing optimization is an NP-hard problem, both genetic and gradient-based algorithms are integrated in the finite-circle method to solve the problem. A mixed approach is proposed when the number of components is relatively large. Various tests are carried out to validate the proposed algorithms and design approach. Satisfactory results are obtained.
TL;DR: This article critically reviews grey model formulation and solution algorithms and identifies a fundamental shortcoming of grey stochastic programming with recourse and suggests new solution algorithms that give more risk-adverse solutions.
Abstract: A grey number is an uncertain number with fixed lower and upper bounds but unknown distribution. Grey numbers find use in optimization to systematically and proactively incorporate uncertainties expressed as intervals plus communicate resulting stable, feasible ranges for the objective function and decision variables. This article critically reviews their use in linear and stochastic programs with recourse. It summarizes grey model formulation and solution algorithms. It advances multiple counter-examples that yield risk-prone grey solutions that perform worse than a worst-case analysis and do not span the stable feasible range of the decision space. The article suggests reasons for the poor performance and identifies conditions for which it typically occurs. It also identifies a fundamental shortcoming of grey stochastic programming with recourse and suggests new solution algorithms that give more risk-adverse solutions. The review also helps clarify the important advantages, disadvantages, and distincti...
TL;DR: In this article, a game-theory approach has been used for the multi-objective optimum design of stationary flat-plate solar collectors, where three objectives are considered in the optimization-problem formulation: maximization of the annual average incident solar energy, maximisation of the lowest month incident solar power, and minimization of costs.
Abstract: A game-theory approach has been used for the multi-objective optimum design of stationary flat-plate solar collectors. The clear-day solar-beam radiation and diffuse radiation at the location of the solar collector (Miami) are estimated. Three objectives are considered in the optimization-problem formulation: maximization of the annual average incident solar energy; maximization of the lowest month incident solar energy; and minimization of costs. The game-theory methodology is used for the solution of the three objective-constrained optimization problems to find a balanced solution. This solution represents the best compromise in terms of the super-criterion selected. Two types of sensitivity analyses are conducted on the optimum solution in this work. The sensitivity analysis with respect to the design variables indicates which design valuables are more important to different objective functions. The sensitivity analysis with respect to the solar constant shows that small fluctuations of solar constant ...