Pareto multi-criteria decision making☆
Introduction
Design is generally governed by multiple conflicting criteria, which requires designers to look for good compromise designs by performing tradeoff studies involving the criteria. The competing criteria are often non-commensurable, and their relative importance is generally not definable. This suggests the use of non-dominated optimization to identify a set of feasible designs that are equal-rank optimal, in the sense that no design in the set is dominated by any other feasible design for all criteria. This approach, referred to as Pareto1 optimization, has been extensively applied in the literature concerned with multi-criteria design (e.g., Grierson [2], [3], Osyczka [4], Mackenzie and Gero [5], Koski [6], Khajehpour [7], Grierson and Khajehpour [8], and Yoo and Hajela [9]).
A Pareto design optimization problem, involving n conflicting objective criteria expressed as functions of the design variables, can be concisely stated as,where fi (i = 1, n) are the objective functions, expressed in terms of the design variable vector z in the feasible domain Ω for the n-dimensional criteria space. A design z∗ ∈ Ω is a Pareto-optimal solution to the problem posed by Eq. (1), if there does not exist any other design z ∈ Ω such that,The number m of Pareto-optimal design solutions to Eq. (1) can be quite large, and it is yet necessary to select the best compromise design(s) from among them.
For example, consider the simply-supported plate with uniformly distributed loading shown in Fig. 1. The design of the plate is governed by the two conflicting criteria, to minimize structural weight f1(z) = W and midpoint deflection f2(z) = Δ, for variables z taken as the thicknesses z1, z2, …, z6 of six pre-specified zones of the plate (see Koski [6] for details). The function f1(z) is expressed explicitly in terms of the properties of the plate, while the function f2(z) implies displacement analysis of the plate. For any plate design z∗, its weight W∗ is found by evaluating the explicit function f1(z∗), while its midspan deflection Δ∗ is found by evaluating the implicit function f2(z∗).
Koski [6] solved Eq. (1) for n = 2, to find the m = 10 alternative Pareto-optimal design scenarios having weight W∗ and deflection Δ∗ listed in columns 2 and 3 of Table 1. The 10 Pareto designs define the Pareto (front) curve in Fig. 2; in fact, any one of the theoretically infinite number of points along this curve corresponds to a Pareto design. Therefore, it essentially remains to select a best-compromise plate design from among a theoretically infinite set of Pareto designs.
The various methods proposed in the literature for searching among Pareto optima to select best-compromise designs are somewhat informal, in that the selection process is primarily driven by heuristic methods and/or designer preferences (e.g., Koski [6], Yoo and Hajela [9], Rahimi-Vahed et al. [10]). Recent studies by the author (Grierson [2], [3]), employed a formal mathematical tradeoff-analysis technique adapted from the theory of social welfare economics (e.g., Boadway and Bruce [11]), to identify competitive equilibrium states corresponding to Pareto-compromise designs; i.e., designs that represent a precise Pareto tradeoff between the competing criteria. The present paper further develops the tradeoff analysis into a general multi-criteria decision making (MCDM) strategy, capable of identifying Pareto-compromise solutions to design problems involving any number n of conflicting objective criteria. The MCDM strategy is initially presented in detail for design governed by n = 2 objective criteria, and illustrated for the two-criteria flexural plate design discussed in the foregoing. The MCDM strategy is then extended to design governed by two or more objective criteria, and an underlying theorem is formally stated and proved. The concepts are further illustrated for a bridge maintenance-intervention protocol design governed by n = 3 criteria concerning bridge maintenance cost, condition and safety, and for a media centre envelop design governed by n = 11 criteria concerning lighting, thermal and visual performance, and overall cost. To begin, some relevant characteristics of Pareto data are first discussed in the following section.
The MCDM strategy involves manipulation of Pareto data, the characteristics of which are both quantitative and qualitative. The solution of the optimization problem Eq. (1) is an n-dimensional Pareto data set of m-dimensional objective criteria vectors . Quantitatively, it is assumed the data set exists in finite-dimensional Euclidian space. Thus, the dimensions m and n have positive finite value. The individual entries of each criteria vector also have finite value, which may be positive or negative or exactly zero.
Qualitatively, it is the ordinal positions-not the cardinal values-of the vector entries that define the Pareto-optimal character of the data. For n = 2 criteria vectors and , for example, if the m entries of any one vector are sequentially ordered from their minimum to maximum values, the Pareto character of the data set is maintained by sequentially arranging the m entries for the other vector in reverse order, i.e.,or vice versa. That is, the Pareto character of the data set represented by the two vectors in Eq. (3) is maintained if,or vice versa. In other words, so long as the vector entries are ordinally maintained, the Pareto character of the data set is not dependent on their cardinal values.
The criteria vectors are often non-commensurable among themselves, with possibly large numerical differences in their entries. These incompatibilities are overcome by normalizing the entries of each vector over the positive unit range [0, 1], without changing their ordinal positions. In general, for a vector f∗ with entries , this is accomplished through the normalization calculation,where fmin and fmax are the minimum and maximum entry values for the original vector f∗. Note that the minimum and maximum entry values for the normalized vector X = [X1, …, Xm]T are Xmin = 0 and Xmax = 1.
The normalization defined by Eq. (5) applies regardless of whether the individual vector entries are positive, negative or zero valued. If all the entries for the original vector f∗ have non-negative values , the normalized vector may be alternatively found by the simpler calculation,where, the minimum and maximum entry values for the normalized vector x = [x1, …, xm]T are xmin ⩾ 0 and xmax = 1. Unless fmin = 0, the two normalized vectors x and X are not the same.
In Section 2 following, the simple normalization defined by Eq. (6) is initially adopted for the development of the MCDM strategy for design governed by n = 2 objective criteria, because the corresponding normalized vectors xi (i = 1, 2) for the illustrative flexural plate design facilitate a compelling presentation of the underlying tradeoff-analysis technique. It is subsequently shown in Section 2.4 that, when the general normalization defined by Eq. (5) is adopted for the development, there is no change in the formulation of the MCDM strategy.
Section snippets
MCDM in 2-D criteria space
Consider a scenario in which two designers A and B are bargaining with each other to achieve a compromise tradeoff between n = 2 competing objective criteria, represented by two m-dimensional vectors and , whose entries are found through Eq. (1) to correspond to m alternative Pareto-optimal designs of an artifact or entity (e.g., see columns 2 and 3 of Table 1 for the flexural plate design). It is initially assumed the two vectors have only non-negative
MCDM in n-D criteria space
Consider now the design of an artifact or entity governed by n ⩾ 2 objective criteria, represented by a n-dimensional data set of m-dimensional vectors , found through solution of Eq. (1) to correspond to m alternative Pareto-optimal design scenarios. The individual entries of each criteria vector may be positive, negative or zero valued. Similar to Eq. (19a), (19b), the vectors are each normalized as,where the entries of each normalized
Application of the PEG-MCDM procedure
It is important to recognize that, the proposed PEG-MCDM computational procedure to find a unique Pareto-compromise design involves two approximate calculations. Firstly, for n ⩾ 2, an approximate linear interpolation is employed through Eq. (14) to identify the point at which the Pareto curve intersects the diagonal OA–OB of the PEG-square (see Fig. 5, Fig. 6). Secondly, for n > 2, each aggregate criterion vector yi formed through Eqs. (27), (28) is but an approximate simulation of n
Concluding commentary
The PEG-theorem and corresponding PEG-MCDM computational procedure resolve an important issue related to multi-criteria decision making, that of rigorously selecting a compromise design from among a potentially large number of alternative feasible designs. However, even though it is understood to represent a mathematically-derived Pareto tradeoff, that is equally beneficial for all objective criteria, it is still possible that the Pareto-compromise design may not be acceptable to, or among, all
Acknowledgements
This study is supported by the Natural Science and Engineering Research Council of Canada. For implementation of the multi-criteria decision making software used for the example applications of the study, credit and thanks are due to Kevin Xu, Department of Electrical and Computer Engineering, University of Waterloo, Canada. For providing insight concerning the concepts of social welfare economics that underlie the design tradeoff principles of the study, the author is grateful to Kathleen
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Presented in part at the 14th EG-ICE Workshop, Maribor, Slovenia, June 26–29, 2007.