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.     

Understanding knee points in bicriteria problems and their implications as preferred solution principles

A knee point is almost always a preferred trade-off solution, if it exists in a bicriteria optimization problem. In this article, an attempt is made to improve understanding of a knee point and investigate the properties of a bicriteria problem that may exhibit a knee on its Pareto-optimal front. Past studies are reviewed and a couple of new definitions are suggested. Additionally, a knee region is defined for problems in which, instead of one, a set of knee-like solutions exists. Edge-knee solutions, which behave like knee solutions but lie near one of the extremes on the Pareto-optimal front, are also introduced. It is interesting that in many problem-solving tasks, despite the existence of a number of solution methodologies, only one or a few of them are commonly used. Here, it is argued that often such common solution principles are knee solutions to a bicriteria problem formed with two conflicting goals of the underlying problem-solving task. The argument is illustrated on a number of tasks, such as regression, sorting, clustering and a number of engineering designs.     

Tunneling radial cracks in layered structures from contact loading

Glass/epoxy laminates glued onto a compliant substrate are indented with a hard ball. The damage is characterized by a set of transverse cracks which pop out from the subsurface of the glass layers due to flexure and propagate stably in the radial direction with load in a bell-shape front under a diminishing stress field. Compliant interlayers, even extremely thin ones, are effective in inhibiting crossover fracture. This leads to crack tunneling and crack multiplication in the hard layers, which enhances energy dissipation and reduces the spread of damage relative to the basic bilayer configuration. The experiments show that the fracture in a given layer is well approximated by a power-law relation of the form c^3/2K_C/P = δ, where P, c, and K_C are the indentation load, crack length and fracture toughness, in that order, and δ an implicit function of the layer position and material and geometric variables, derived with the aid of available tunnel crack solutions.The model specimen studied provides a useful insight into the fracture behavior of natural, biological and synthetic layered structures from concentrated loading. The analysis shows that the crack arrest capability of a thin interlayer increases in proportion to the modulus misfit ratio between the layer and interlayer, and that the spread of radial cracks in a laminate of given thickness reduces in proportion to n^1/3, where n is the number layers in the laminate.     

Oblique impact testing of bicycle helmets

The performance of bicycle helmets was investigated in oblique impacts with a simulated road surface. The linear and rotational accelerations of a headform, fitted with a compliant scalp and a wig, were measured. The peak rotational accelerations, the order of 5 krad s⁻² when the tangential velocity component was 4 m s⁻¹, were only slightly greater than in comparable direct impact tests. Oblique impact tests were possible on the front lower edge of the helmet, a site commonly struck in crashes, without the headform striking the ‘road’. Data characterizing the frictional response at the road/shell and helmet/head interfaces, were generated for interpretation via FEA modelling.     

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.     

Availability of a general k-out-of-n:G system with non-identical components considering shut-off rules using quasi-birth-death process

A general repairable k-out-of-n:G system with non-identical components, which is a common form of redundancy, is considered in this paper. The number of repairmen is assumed to be r (1≤r≤n−k+1) while components can have similar or different repair priorities. The objective of this work is to address the problem of efficient evaluation of the system's availability in a way that steady state solutions can be obtained systematically in a reasonable computational time. This problem is modeled as a finite state-dependent non-homogeneous quasi-birth-death (QBD) process. An algorithm is introduced to systematically generate the system state vectors and transition rate matrix and then an iterative method based on the Block Gauss-Seidel method is employed to determine the steady state probabilities. These are the novel contributions made in this paper. An analog Monte Carlo simulation is presented to demonstrate the correctness and the efficiency of the proposed method.     

Validation of stress intensity factors of diametrically opposed corner cracks in a hole

Population of the world’s largest database of stress intensity factor (K) solutions began in 2002 with the calculation of 5.6 million K solutions for diametrically opposed unsymmetric corner cracks at a straight shank hole in a finite width sheet subject to remote tension, remote bending, and bearing loading. Previous work to validate these K solutions was in the form of fatigue life predictions and crack shape development. The current work attempts to build on the previous validation efforts with the addition of comparing the calculated K solutions with K solutions obtained from carefully controlled laboratory experiments. The latter are obtained via fatigue striation measurements at high magnification, up to 40,000×, using a scanning electron microscope and crack growth rate data, in terms of da/dN vs. ΔK at the same test condition. The results show the numerical K solutions are within 20% of the experimentally derived K’s at discrete locations along the crack front. The relatively large error is due to the discontinuous crack extension process of the crack front. Moreover, the entire crack front does not instantaneously extend uniformly in a self similar fashion. The crack extends stepwise over discrete portions of the crack front. Possibly averaging the striation spacing over a specified arc length of the crack front would ameliorate the discontinuous nature of crack propagation resulting in better correlation between the numerical and experimental results. As a result of the current work, we have shown time consuming striation spacing measurements at high magnification are not required to validate K solutions. The best method for such validation efforts is using the fatigue life, crack history, and crack shape which can be obtained at 1/10th the cost of obtaining striation spacing measurements.     

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.     

The potential of textured front ZnO and flat TCO/metal back contact to improve optical absorption in thin Cu(In,Ga)Se2 solar cells

The role of additionally textured front transparent conductive oxide − TCO (ZnO:Al) and flat TCO/metal contact on optical improvements in thin Cu(In,Ga)Se₂ (CIGS) solar cells are investigated by means of numerical simulations. A de-coupled analysis of two effects related to additional texturing of front surface of ZnO:Al TCO − (i) enhancement of light scattering and (ii) decreased total reflectance (antireflective effect) − reveals that the improvements in quantum efficiency, QE, and short-circuit current, J_SC, of the solar cell originate from an antireflective effect only. In order to improve optical properties of the back contact the introduction of a TCO layer (undoped ZnO) between CIGS and back metal contact is investigated from the optical point of view. In addition to ZnO/Mo, a highly reflective ZnO/Ag contact (ZnO is also assumed to work as a protection layer for Ag) is also included in simulations. Results show significant increase in reflectance related to introduced ZnO in front of Mo. Drastically increased reflectance is obtained if ZnO/Mo is substituted with ZnO/Ag. The improvements in QE and J_SC of a thin CIGS solar cell, related to ZnO/metal contacts are presented.     

A weight-based multiobjective immune algorithm: WBMOIA

A novel immune algorithm is suggested for finding Pareto-optimal solutions to multiobjective optimization problems based on opt-aiNET, the artificial immune system algorithm for multi-modal optimization. In the proposed algorithm, a randomly weighted sum of multiple objectives is used as a fitness function, and a local search algorithm is incorporated to facilitate the exploitation of the search space. Specifically, a new truncation algorithm with similar individuals (TASI) is proposed to preserve the diversity of the population. Also, a new selection operator is presented to create the new population based on TASI. Simulation results on seven standard problems (ZDT2, ZDT6, DEB, VNT, BNH, OSY and KIT) show that the proposed algorithm is able to find a much better spread of solutions and better convergence near the true Pareto-optimal front compared to the vector immune algorithm and the elitist non-dominated sorting genetic system.