Pareto Efficiency - Use in Engineering

Use in Engineering

The notion of Pareto efficiency is also useful in engineering. Given a set of choices and a way of valuing them, the Pareto frontier or Pareto set or Pareto front is the set of choices that are Pareto efficient. By restricting attention to the set of choices that are Pareto-efficient, a designer can make tradeoffs within this set, rather than considering the full range of every parameter. The Pareto frontier is defined formally as follows. Consider a design space with n real parameters (corresponding to the allocation of goods in the economics interpretation), and for each design space point there are m different criteria by which to judge that point (corresponding to the utility of the different agents in the economics interpretation). Let be the function which assigns, to each design space point x, a criteria space point f(x). This represents the way of valuing the designs. Now, it may be that some designs are infeasible; so let X be a set of feasible designs in, which must be a compact set. Then the set which represents the feasible criterion points is f(X), the image of the set X under the action of f. Call this image Y. Now construct the Pareto frontier as a subset of Y of the feasible criterion points. It is often assumed in engineering that the preferable values of each criterion parameter are the lesser ones (e.g. lower emissions or lower cost), thus minimizing each dimension of the criterion vector is desired. Then compare criterion vectors as follows: One criterion vector y strictly dominates (or "is preferred to") a vector y* if each parameter of y is not strictly greater than the corresponding parameter of y* and at least one parameter is strictly less: that is, for each i and for some i. This is written as to mean that y strictly dominates y*. Then the Pareto frontier is the set of points from Y that are not strictly dominated by any other point in Y. Formally, this defines a partial order on Y, namely the product order on (more precisely, the induced order on Y as a subset of ), and the Pareto frontier is the set of maximal elements with respect to this order. Algorithms for computing the Pareto frontier of a finite set of alternatives have been studied in computer science, sometimes referred to as the maximum vector problem or the skyline query.Kung, H.T.; Luccio, F.; Preparata, F.P. (1975). "On finding the maxima of a set of vectors.". Journal of the ACM 22 (4): 469–476. doi:10.1145/321906.321910Godfrey, Parke; Shipley, Ryan; Gryz, Jarek (2006). "Algorithms and Analyses for Maximal Vector Computation". VLDB Journal 16: 5–28. doi:10.1007/s00778-006-0029-7