Modelling the Pareto-optimal set using B-spline basis functions for continuous multi-objective optimization problems

In the past few years, multi-objective optimization algorithms have been extensively applied in several fields including engineering design problems. A major reason is the advancement of evolutionary multi-objective optimization (EMO) algorithms that are able to find a set of non-dominated points spread on the respective Pareto-optimal front in a single simulation. Besides just finding a set of Pareto-optimal solutions, one is often interested in capturing knowledge about the variation of variable values over the Pareto-optimal front. Recent innovization approaches for knowledge discovery from Pareto-optimal solutions remain as a major activity in this direction. In this article, a different data-fitting approach for continuous parameterization of the Pareto-optimal front is presented. Cubic B-spline basis functions are used for fitting the data returned by an EMO procedure in a continuous variable space. No prior knowledge about the order in the data is assumed. An automatic procedure for detecting gaps in the Pareto-optimal front is also implemented. The algorithm takes points returned by the EMO as input and returns the control points of the B-spline manifold representing the Pareto-optimal set. Results for several standard and engineering, bi-objective and tri-objective optimization problems demonstrate the usefulness of the proposed procedure.     

A model equation for steady surface waves over a bump

The objective of this paper is to study the solutions of a model equation for steady surface waves on an ideal fluid over a semicircular or semielliptical bump. For upstream Froude number F>1, we show that the numerical solution of the equation has two branches and there is a cut-off value of F below which no solution exists. For F<1, the problem is reformulated to overcome the so-called infinite-mass dilemma. A branch of solutions and a cut-off value of F, above which no solution exists, are found. Furthermore, we also obtain a branch of hydraulic-fall solutions which decrease monotonically from upstream to downstream.     

Preference-based Pareto optimization in certain and noisy environments

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.     

Analytic hierarchy process-based approach for selecting a Pareto-optimal solution of a multi-objective,multi-site supply-chain planning problem

The current manufacturing environment has changed from traditional single-plant to multi-site supply chain where multiple plants are serving customer demands. In this article, a tactical multi-objective, multi-period, multi-product, multi-site supply-chain planning problem is proposed. A corresponding optimization model aiming to simultaneously minimize the total cost, maximize product quality and maximize the customer satisfaction demand level is developed. The proposed solution approach yields to a front of Pareto-optimal solutions that represents the trade-offs among the different objectives. Subsequently, the analytic hierarchy process method is applied to select the best Pareto-optimal solution according to the preferences of the decision maker. The robustness of the solutions and the proposed approach are discussed based on a sensitivity analysis and an application to a real case from the textile and apparel industry.     

Class-modelling techniques that optimize the probabilities of false noncompliance and false compliance

The work presents two approaches for the construction of empirical class-models for a given category C. The attention is centred on the information provided by the sensitivity and specificity, the two usual parameters employed to qualify a class-model. In fact, not only a class-model is built for C but a set of class-models which differ in their sensitivity and specificity. Therefore the range of possible jointly available values is described, allowing the user to select the model that best adapt to specific situations or particular needs.One of the approaches, PLS-CM (Partial Least Squares Class-Modelling), is based on the modelling of the distribution of the values obtained by a PLS model fitted with binary response (belong/do not belong to C). In that way, the corresponding hypothesis test permits the computation of the probabilities α and β of type I and type II errors when deciding whether a sample belongs to C. These probabilities, expressed as percentages, are 100 minus sensitivity and 100 minus specificity, respectively. The representation of β versus α is the risk curve that describes the PLS-CM capability of modelling category C.The other approach comes from setting the problem as a multi-objective optimization problem, the one that corresponds to simultaneously maximize sensitivity and specificity, which usually behave oppositely. The trading-off solutions (again, different class-models) are computed to be Pareto-optimal solutions, that is, the set of the optimal solutions in at least one of the conflicting objectives, what is known as the Pareto-optimal front, POF.Additionally, a procedure to cross-validate the risk curve and the Pareto-optimal front is proposed for the first time in order to evaluate the prediction ability of both methods.Two case-studies are used to drive the discussion: 1) the characterization of wines that official wine-tasters regarded as compliant ones according to the quality characteristics stated by a Denomination of Origin and 2) The characterization of breast tumours defined as benign (compliant class) from 9 cytological variables.Finally, the performance of the methods is tested using several data sets from the literature.     

Migration NSGA: method to improve a non-elitist searching of Pareto front,with application in magnetics

The paper describes a migration strategy to improve classical non-dominated sorting genetic algorithm (NSGA) to find optimal solution of a multi-objective problem. Migration NSGA has been tested to assess its performance using analytical functions for which the Pareto front is known in analytical form, as well as two case studies in electromagnetics, for which the Pareto front is not known a priori. This strategy improves the approximation of the Pareto-optimal solutions of a multi-objective problem by introducing new individuals in the population miming the effect of migrations.     

Optimal power system generation scheduling by multi-objective genetic algorithms with preferences

Power system generation scheduling is an important issue both from the economical and environmental safety viewpoints. The scheduling involves decisions with regards to the units start-up and shut-down times and to the assignment of the load demands to the committed generating units for minimizing the system operation costs and the emission of atmospheric pollutants.As many other real-world engineering problems, power system generation scheduling involves multiple, conflicting optimization criteria for which there exists no single best solution with respect to all criteria considered. Multi-objective optimization algorithms, based on the principle of Pareto optimality, can then be designed to search for the set of nondominated scheduling solutions from which the decision-maker (DM) must a posteriori choose the preferred alternative. On the other hand, often, information is available a priori regarding the preference values of the DM with respect to the objectives. When possible, it is important to exploit this information during the search so as to focus it on the region of preference of the Pareto-optimal set.In this paper, ways are explored to use this preference information for driving a multi-objective genetic algorithm towards the preferential region of the Pareto-optimal front. Two methods are considered: the first one extends the concept of Pareto dominance by biasing the chromosome replacement step of the algorithm by means of numerical weights that express the DM’ s preferences; the second one drives the search algorithm by changing the shape of the dominance region according to linear trade-off functions specified by the DM.The effectiveness of the proposed approaches is first compared on a case study of literature. Then, a nonlinear, constrained, two-objective power generation scheduling problem is effectively tackled.     

Rescheduling with controllable processing times for number of disrupted jobs and manufacturing cost objectives

We consider a machine rescheduling problem that arises when a disruption such as machine breakdown occurs to a given schedule. Machine unavailability due to a breakdown requires repairing the schedule as the original schedule becomes infeasible. When repairing a disrupted schedule a desirable goal is to complete each disrupted job on time, i.e. not later than the planned completion time in the original schedule. We consider the case where processing times of jobs are controllable and compressing the processing time of a job requires extra processing cost. Usually, there exists a nonlinear relation between the processing time and manufacturing cost. We solve a bicriteria rescheduling problem that trades off the number of on-time jobs and manufacturing cost objectives. We give a mixed-integer second-order cone programming formulation for the problem. We develop a heuristic search algorithm to generate efficient solutions for the problem. Heuristic algorithm searches solution space by moving and swapping jobs among machines. We develop cost change estimates for job moves and swaps so that the heuristic implements only promising moves and hence generates a set of efficient solutions in reasonably short CPU times.     

A k, k-ε optimality selection based multi objective genetic algorithm with applications to vehicle engineering

In the paper, a multi objective genetic algorithm based on the concept of k-optimality and k-ε-optimality (KEMOGA) is introduced and applied. Pareto optimality alone is not always adequate for selecting a final solution because the Pareto optimal set can be very large. The k-optimality approach and the more general k-ε-optimality method, can be used to rank the Pareto-optimal solutions. The two methods have been included into a genetic algorithm selection procedure. The k-optimality method searches for points which remain Pareto-optimal when all of the subsets of n-k objectives (n is the number of objective functions) are optimised. The k-ε approach considers not only if an objective is worse than the others but also the entity of this variation through the introduction of a vector of indifference thresholds. The KEMOGA has been applied for the solution of two engineering problems. The selection of the stiffness and damping of a passively suspended vehicle in order to get the best compromise between discomfort, road holding and working space and a complex problem related to the optimisation of the tyre/suspension system of a sport car. The final design solution, found by means of the KEMOGA seems consistent with the solution selected by skilled suspensions specialists. The proposed approach has been tested and validated on a complex optimization problem. The solved problem deals with the optimization of the tyre/suspension system of a sport car. The proposed approach (KEMOGA) has shown to be very effective in terms of computational efficiency and accuracy.     

Forced-convection heat transfer over a circular cylinder with Newtonian heating

A mathematical model for the forced convection boundary-layer flow over a circular cylinder is considered when there is Newtonian heating on the surface of the cylinder through which the heat transfer is proportional to the local surface temperature. The dimensionless version of the boundary-layer equations involve two parameters, the Prandtl number σ and γ measuring the strength of the surface heating. The solution near the stagnation point is considered first and this reveals that, to get a physically acceptable solution, γ must be less than some critical value γ _c , dependent on σ. Numerical solutions to the full boundary-layer problem are obtained which show that the surface temperature increases as the flow develops from the stagnation point.     

Variations of stress intensity factor of a semi-elliptical surface crack subjected to mixed mode loading

Maximum stress intensity factors of a surface crack usually appear at the deepest point of the crack, or a certain point along crack front near the free surface depending on the aspect ratio of the crack. However, generally it has been difficult to obtain smooth distributions of stress intensity factors along the crack front accurately due to the effect of corner point singularity. It is known that the stress singularity at a corner point where the front of 3 D cracks intersect free surface is depend on Poisson's ratio and different from the one of ordinary crack. In this paper, a singular integral equation method is applied to calculate the stress intensity factor along crack front of a 3-D semi-elliptical surface crack in a semi-infinite body under mixed mode loading. The body force method is used to formulate the problem as a system of singular integral equations with singularities of the form r ⁻³ using the stress field induced by a force doublet in a semi-infinite body as fundamental solution. In the numerical calculation, unknown body force densities are approximated by using fundamental density functions and polynomials. The results show that the present method yields smooth variations of mixed modes stress intensity factors along the crack front accurately. Distributions of stress intensity factors are indicated in tables and figures with varying the elliptical shape and Poisson's ratio.     

Coordination of set-ups between two stages of a supply chain using multi-objective genetic algorithms

A multi-objective genetic algorithm (MOGA) solution approach for a sequencing problem to coordinate set-ups between two successive stages of a supply chain is presented in this paper. The production batches are processed according to the same sequence in both stages. Each production batch has two distinct attributes and a set-up occurs in the upstream stage every time the first attribute of the new batch is different from the previous one. In the downstream stage, there is a set-up when the second attribute of the new batch is different from that of the previous one. Two objectives need to be considered in sequencing the production batches including minimizing total set-ups and minimizing the maximum number of set-ups between the two stages. Both problems are NP-hard so attainment of an optimal solution for large problems is prohibited. The solution approach starts with an initialization stage followed by evolution of the initial solution set over generations. The MOGA makes use of non-dominated sorting and a niche mechanism to rank individuals in the population. Selected individuals taken from a given population form the succeeding generation using four genetic operators as: reproduction, crossover, mutation and inversion. Experiments in a number of test problems show that the MOGA is capable of finding Pareto-optimal solutions for small problems and near Pareto-optimal solutions for large instances in a short CPU time.     

Multiobjective optimization with an adaptive weight determination scheme using the concept of hyperplane

Finding an optimum design that satisfies all performances in a design problem is very challenging. To overcome this problem, multiobjective optimization methods have been researched to obtain Pareto optimum solutions. Among the different methods, the weighted sum method is widely used for its convenience. However, since the different weights do not always guarantee evenly distributed solutions on the Pareto front, the weights need to be determined systematically. Therefore, this paper presents a multiobjective optimization using a new adaptive weight determination scheme. Solutions on the Pareto front are gradually found with different weights, and the values of these weights are adaptively determined by using information from the previously obtained solutions' positions. For an n-objective problem, a hyperplane is constructed in n -dimensional space, and new weights are calculated to find the next solutions. To confirm the effectiveness of the proposed method, benchmarking problems that have different types of Pareto front are tested, and a topology optimization problem is performed as an engineering problem. A hypervolume indicator is used to quantitatively evaluate the proposed method, and it is confirmed that optimized solutions that are evenly distributed on the Pareto front can be obtained by using the proposed method.     

The propagation and basal solidification of two-dimensional and axisymmetric viscous gravity currents

The effect of basal solidification on viscous gravity currents is analysed using continuum models. A Stefan condition for basal solidification is incorporated into the Navier-Stokes equations. A simplified version of this model is determined in the lubrication and large-Bond-number limit. Asymptotic solutions are obtained in three parameter régimes. (i) A similarity solution is possible in the following cases: the two-dimensional problem when volume per unit length (V) is proportional to time (t) raised to the power 7/4(V = qt ^7/4) and the Julian number (v ³ g ² /q ⁴ ) is large, where v is kinematic viscosity, q is a constant of proportionality and g is the acceleration due to gravity; the axisymmetric problem when volume is proportional to time raised to the power 3 (V = Qt ³) and the dimensionless group vg/Q is large, where Q is a constant of proportionality. In both cases, the front is found to depend on time raised to the power 5/4, as it does in the absence of solidification, but the constant of proportionality satisfies a modified system of equations. (ii) In the case of large Stefan number and small modified Peclet number (Pe ² 1, where Pe is the Peclet number and is the aspect ratio), asymptotic and numerical methods are combined to produce the most revealing results. The temperature of the fluid approaches the melting point over a short time-scale. Over the long time-scale, the solid/liquid interface is determined from the conduction of latent heat into the solid. Strong coupling is observed with the basal solidification modifying the flow at leading order. The solidification may retard and eventually arrest the front motion long before complete phase change has taken place. (iii) In the case of constant volume and large modified Peclet number (Pe ² 1), similarity solutions are found for the solidification at the base of the gravity current on the short time-scale. The coupling is weak on this time-scale with the solidification being dependent on the front position but not influencing the fluid motion at leading order. Over the long time-scale, the drop completely solidifies. Analytical solutions are not obtained on this time-scale, but scalings are deduced.     

Relaxed Simultaneous Congruences

 In certain digital signal processing applications, residues of an input integer signal x are measured with respect to integer moduli . The signal x is then recovered by solving the system of simultaneous linear congruences . Since the residues r _i are measured quantities, they are subject to noise contamination. To provide noise protection, the moduli m _i may be chosen to possess common factors. Accurate approximations for x may then be determined by solving approximate or ``relaxed" simultaneous congruences. This paper presents a coherent mathematical theory for the approximate solution of simultaneous congruences with inaccurate residues when no exact solution exists. After precisely formulating the notion of relaxed congruences, it is found that, under nonrestrictive technical assumptions, unique solutions to these congruences always exist. A variety of examples illustrating characteristics of solutions of relaxed congruences are provided, and a fast, efficient algorithm for solving them numerically is presented. The problem of finding an optimal approximate solution is then discussed. Several optimality criteria are proposed and procedures for finding optimal approximate solutions are outlined. Error bounds are derived which specify the maximum amount that an approximate solution based upon inaccurate residues may differ from the corresponding true solution based upon the exact residues. Received: January 12, 2001; revised version: December 23, 2002 Keywords: Residue number systems, Error correction, Error detection, Measurement noise, Linear simultaneous congruences.     

An entropy-based three-stage approach for multi-objective system reliability optimization considering uncertainty

This article presents a three-stage approach for solving multi-objective system reliability optimization problems considering uncertainty. The reliability of each component is considered in the formulation as a component reliability estimate in the form of an interval value and discrete values. Component reliability may vary owing to variations in the usage scenarios. Uncertainty is described by defining a set of usage scenarios. To address this problem, an entropy-based approach to the redundancy allocation problem is proposed in this study to identify the deterministic reliability of each component. In the second stage, a multi-objective evolutionary algorithm (MOEA) is applied to produce a Pareto-optimal solution set. A hybrid algorithm based on k-means and silhouettes is performed to select representative solutions in the third stage. Finally, a numerical example is presented to illustrate the performance of the proposed approach.     

On exact solutions of flow problems of a second grade fluid through two parallel porous walls

Exact analytical solutions of two problems of laminar flow of a second grade fluid through two parallel porous walls are obtained. For each problem the rate of injection at one wall is assumed equal to the rate of suction at the other wall. Two geometries are considered: rectangular - when the flow takes place between two parallel flat walls, and cylindrical - when the flow takes place through an annulus. For the exact solution no assumption is made on the size of K, the viscoelastic fluid parameter, or the cross-flow Reynolds number. However, it is assumed that a Taylor series expansion of the solution exists near K=0. Also assuming K small, perturbation solutions are developed for both the geometries. The exact solutions reveal that the viscoelasticity of the fluid tends to destroy the formation of the boundary layer at the wall where the suction takes place for large values of the cross-flow Reynolds number. It is further shown that the commonly used perturbation technique does not give satisfactory results even for moderate values of the cross-flow Reynolds number.     

Structural and dielectric properties of a novel Bi2O3-(1 −x)WO3-xCuO solid solution

A novel solid solution in the system Bi-W-Cu-O has been synthesized and its structural and dielectric properties studied. The solid solution Bi₂O₃-(1 −x)WO₃-xCuO exists up tox=0·7; the solid solutions up tox=0·65 are orthorhombic but stabilize in tetragonal structure in a narrow range aroundx=0·7. The solid solutions are non-centrosymmetric and exhibit ferroelectric behaviour similar to their parent phase Bi₂WO₆. The Curie point of the solid solution is found to decrease with increase inx.     

Generalized railway tank car safety design optimization for hazardous materials transport: addressing the trade-off between transportation efficiency and safety

North America railways offer safe and generally the most economical means of long distance transport of hazardous materials. Nevertheless, in the event of a train accident releases of these materials can pose substantial risk to human health, property or the environment. The majority of railway shipments of hazardous materials are in tank cars. Improving the safety design of these cars to make them more robust in accidents generally increases their weight thereby reducing their capacity and consequent transportation efficiency. This paper presents a generalized tank car safety design optimization model that addresses this tradeoff. The optimization model enables evaluation of each element of tank car safety design, independently and in combination with one another. We present the optimization model by identifying a set of Pareto-optimal solutions for a baseline tank car design in a bicriteria decision problem. This model provides a quantitative framework for a rational decision-making process involving tank car safety design enhancements to reduce the risk of transporting hazardous materials.     

Numerical verification of solutions for Signorini problems using Newton‐like method

We proposed a numerical method to verify the existence of solutions for a simplified Signorini problem (Comput. Math. Appl. 2000; 40 :1003–1013). Using sequential iteration method, we numerically constructed a set containing solutions that satisfies the hypothesis of Schauder's fixed point theorem in a certain Sobolev space. It is difficult to apply this method to the problem of which associated operator is not retractive in a neighborhood of the solution. In this paper, in order to overcome such a difficulty, we describe an alternative approach to this problem. Numerical examples are presented. Copyright © 2007 John Wiley & Sons, Ltd.