Christopher A. Mattson

Normal Constraint Method with Guarantee of Even Representation of Complete Pareto Frontier

Multiobjective optimization is rapidly becoming an invaluable tool in engineering design. A particular class of solutions to the multiobjective optimization problem is said to belong to the Pareto frontier. A Pareto solution, the set of which comprises the Pareto frontier, is optimal in the sense that any improvement in one design objective can only occur with the worsening of at least one other. Accordingly, the Pareto frontier plays an important role in engineering design—it characterizes the tradeoffs between conflicting design objectives. Some optimization methods can be used to automatically generate a set of Pareto solutions from which a final design is subjectively chosen by the designer. For this approach to be successful, the generated Pareto set must be truly representative of the complete optimal design space (Pareto frontier). In other words, the set must not overrepresent one region of the design space, or neglect others. Some commonly used methods comply with this requirement, whereas others do not. This paper offers a new phase in the development of the normal constraint method, which is a simple approach for generating Pareto solutions that are evenly distributed in the design space of an arbitrary number of objectives. The even distribution of the generated Pareto solutions can facilitate the process of developing an analytical expression for the Pareto frontier in n dimension. An even distribution of Pareto solutions also facilitates the task of choosing the most desirable (final) design from among the set of Pareto solutions. The normal constraint method bears some similarities to the normal boundary intersection and � -constraint methods. Importantly, the developments presented in this paper define its critical distinction, namely, the ability to generate a set of evenly distributed Pareto solutions over the complete Pareto frontier. Examples are provided that show the normal constraint method to perform favorably under the new developments when compared with the normal boundary intersection method, as well as with the original normal constraint method.

Pareto Frontier Based Concept Selection Under Uncertainty, with Visualization

In a recent publication, we presented a new multiobjective decision-making tool for use in conceptual engineering design. In the present paper, we provide important developments that support the next phase in the evolution of the tool. These developments, together with those of our previous work, provide a concept selection approach that capitalizes on the benefits of computational optimization. Specifically, the new approach uses the efficiency and effectiveness of optimization to rapidly compare numerous designs, and characterize the tradeoff properties within the multiobjective design space. As such, the new approach differs significantly from traditional (non-optimization based) concept selection approaches where, comparatively speaking, significant time is often spent evaluating only a few points in the design space. Under the new approach, design concepts are evaluated using a so-calleds-Pareto frontier; this frontier originates from the Pareto frontiers of various concepts, and is the Pareto frontier for thesetof design concepts. An important characteristic of the s-Pareto frontier is that it provides a foundation for analyzing tradeoffs between design objectives and the tradeoffs between design concepts. The new developments presented in this paper include; (i) the notion ofminimally representingthe s-Pareto frontier, (ii) the quantification of concept goodness using s-Pareto frontiers, (iii) the development of an interactive design space exploration approach that can be used to visualizen-dimensional s-Pareto frontiers, and (iv) s-Pareto frontier-based approaches for considering uncertainty in concept selection. Simple structural examples are presented that illustrate representative applications of the proposed method.

Generating Well-Distributed Sets of Pareto Points for Engineering Design Using Physical Programming

Engineering design generally involves two, possibly integrated, phases: (i) generating design options, and (ii) choosing the most satisfactory option on the basis of some determined criteria. The depth, or lack, of integration between these two phases defines different design approaches, and differing philosophical views from the part of researchers in the field of computational design. Optimization-Based Design (OBD) covers the spectrum of this depth of integration. While most OBD approaches strongly integrate these two phases, some employ computational optimization only in the first or second phase. Regardless of where a method or researcher lies in this philosophical spectrum, some requisite characteristics are fundamental to the effectiveness of OBD methods. In particular, (i) the Aggregate Objective Function (AOF) used in the optimization must have the ability to generate all Pareto solutions, (ii) the generation of any existing Pareto solutions must be possible with reasonable ease, and (iii) even changes in the AOF parameters should yield a well distributed set of Pareto solutions. This paper examines the effectiveness of physical programming (PP) with respect to the latter, yielding favorable conclusions. Previous papers have led to similarly positive conclusions with respect to the former two. This paper also presents a comparative study featuring PP and other popular methods, where PP is shown to perform favorably. A PP-based method for generating the Pareto frontier is presented.

Smart Pareto filter: obtaining a minimal representation of multiobjective design space

Multiobjective optimization is a powerful tool for resolving conflicting objectives in engineering design and numerous other fields. One general approach to solving multiobjective optimization problems involves generating a set of Pareto optimal solutions, followed by selecting the most attractive solution from this set as the final design. The success of this approach critically depends on the designers ability to obtain, manage, and interpret the Pareto set—importantly, the size and distribution of the Pareto set. The potentially significant difficulties associated with comparing a significantly large number of Pareto designs can be circumvented when the Pareto set: (i) is adequately small, (ii) represents the complete Pareto frontier, (iii) emphasizes the regions of the Pareto frontier that entail significant tradeoff, and (iv) de-emphasizes the regions corresponding to little tradeoff. We call a Pareto set that possesses these four important and desirable properties a smart Pareto set. Specifically, a smart Pareto set is one that is small and effectively represents the tradeoff properties of the complete Pareto frontier. This article presents a general method to obtain smart Pareto sets for problems of n objectives, given previously generated sets of Pareto solutions. Under the proposed method, the designer uses a smart Pareto filter to control the size of the Pareto set and the degree of tradeoff representation among objectives. Importantly, the smart Pareto filter yields a Pareto set comprising a minimal number of solutions needed to adequately characterize the problems tradeoff properties. In this article, the smart Pareto filter is analytically developed, and mathematical and physical examples are presented to illustrate the filters effectiveness.

Concept Selection Using s-Pareto Frontiers

We introduce the notion of s-Pareto optimality and show how it can be used to improve concept selection in engineering design. Specie c design alternatives are classie ed as s-Pareto optimal when there are no other alternatives from the same or any other general design concept that exhibit improvement in all design objectives. Further, we say that the set of s-Pareto design alternatives comprises the s-Pareto frontier. Under the proposed approach the s-Pareto frontier plays a paramount role in the concept selection process, as it is used to dee ne and classify concept dominance. Thes-Pareto frontier-based concept selection method can becharacterized as onethat capitalizes on the benee ts of computational optimization during the conceptual phase of design, before a general design concept has been chosen. An introduction to s-Pareto optimality and a method for generating s-Pareto frontiersaredeveloped.Anapproachforusing s-Paretofrontierstoperformconceptselectionisalsopresented.The methods proposed can effectively aid in the elimination of dominated design concepts, keep competitive concepts, and ultimately choose a specie cdesign alternative from theselected design concept. A truss design problem is used to illustrate the usefulness of the method. Nomenclature g = vector of inequality constraints h = vector of equality constraints J = aggregate objective function mi = number of points along Ni Ni = ith vector dee ning the utopia plane n = number of design objectives nx = number of design variables T k = relaxation/slack variable for concept k X P = generic point on the utopia plane x = vector of design variables ± = increment by which feasible space is reduced π = vector of design objectives (or design metrics) π i¤ = ith anchor point π ¤k = optimum design objective value for concept k π si¤ = s-anchor point for the ith objective

Configuration Selection, Modeling, and Preliminary Testing in Support of Constant Force Electrical Connectors

The recent introduction and advancements in design of simple, constant-force mechanisms have created the potential for small-scale, low-cost. constant-force electronic connectors (CFECs). CFECs differ from traditional connectors by the separation or disassociation of contact normal force and contact deflection. By removing the traditional constraints imposed by forces and deflections that are dependent on each other, new types of electronic connectors can be explored. These new designs may lead to smaller and more reliable electronic connectors. In this paper, constant-force mechanisms are adapted to satisfy current industry practices for the design of electronic connectors. Different CFEC configurations are explored and one is selected, prototyped, and used as a proof-of-concept connector for a personal digital assistant (PDA) docking station. The modeling. optimization, and verification of the prototype CFEC is presented. Adaptation of constant-force technology to electronic connectors creates melt. possibilities in electronic connector designs, including allowing an optimal contact force to be utilized to decrease the effects of fretting and wear lowering required manufacturing tolerances, reducing the system s sensitivity to variations introduced by the user, and increasing the systems robustness in applications where movement or vibrations exist.

Development of Commercially Viable Compliant Mechanisms Using the Pseudo-Rigid-Body Model: Case Studies of Parallel Mechanisms

Analysis and synthesis of compliant mechanisms has recently been the subject of significant study in the research community. This focus has led to a number of design approaches for developing compliant mechanisms. This paper describes the value of using the Pseudo-Rigid-Body Model (PRBM) to design compliant mechanisms for commercial products. Application of the PRBM is illustrated through the development of two parallel mechanisms: a bicycle derailleur and parallel-motion bicycle brakes. The PRBM allows compliant mechanisms to be modeled and analyzed as rigid-body mechanisms and significantly reduces the complexity of analysis. Mechanisms with straightforward properties are used to demonstrate the use of the PRBM to design commercially viable compliant mechanisms for required motion and force-deflection characteristics.

New Decision Matrix Based Approach for Concept Selection Using Linear Physical Programming

Decision matrix based methods are perhaps the most popular concept selection approaches used in industry. Although potentially effective and simple to use, they are not without drawbacks. The weighted sum approach that these methods are based on presents serious risks, because it can falsely portray some concepts as being undesirable when in reality they may even be the most desirable. Additionally, the typical decision matrix construction requires that the decision maker specify physically meaningless weights and ratings in order to calculate the total scores of the concepts. This paper examines the drawbacks of the typical decision matrix construction, as well as the limitations of possible alternatives. In this paper, Linear Physical Programming (LPP) is proposed as an alternative to the typical construction of the decision matrix. The use of LPP overcomes the main drawbacks of this typical construction. A synopsis of LPP is presented, and a procedure to use it in concept selection is developed. Concept selection examples are provided that demonstrate the effectiveness of the LPP based approach for concept selection.

A Non-Deterministic Approach to Concept Selection Using S-Pareto Frontiers

The most significant design decisions are typically made during the conceptual phase of the engineering design process, when critical design features are proposed, evaluated and selected. In this paper, we explore the critical task of concept selection and propose a non-deterministic, optimization-based approach for selecting the most promising concept. The method presented in this paper builds upon the recently-proposed s-Pareto based concept selection approach. Within the framework of the s-Pareto approach, so-called s-Pareto frontiers are obtained by using the definition of Pareto optimality to identify Pareto optimal solutions that pertain to a set of distinct concepts. These s-Pareto frontiers are used to assess the tradeoffs between various proposed concepts during conceptual design. The s-Pareto approach is a marked departure from traditional concept selection methods and from the traditional use of Pareto frontiers. In this work the s-Pareto approach is extended to include uncertainties caused by stochastic design parameters as well as low model fidelity. More specifically, the reliability of design decisions is accounted for in the decision-making process. Two approaches are presented for performing non-deterministic concept selection. Two examples are given that support the approach.Copyright

Metrics for Evaluating the Barrier and Time to Reverse Engineer a Product

Reverse engineering, defined as extracting information about a product from the product itself, is a common industry practice for gaining insight into innovative products. Both the original designer and those reverse engineering the original design can benefit from estimating the time and barrier to reverse engineer a product. This paper presents a set of metrics and parameters that can be used to calculate the barrier to reverse engineer any product, as well as the time required to do so. To the original designer, these numerical representations of the barrier and time can be used to strategically identify and improve product characteristics so as to increase the difficulty and time to reverse engineer them. As the metrics and parameters developed in this paper are quantitative in nature, they can also be used in conjunction with numerical optimization techniques, thereby enabling products to be developed with a maximum reverse engineering barrier and time—at a minimum development cost. On the other hand, these quantitative measures enable competitors who reverse engineer original designs to focus their efforts on products that will result in the greatest return on investment.