One critical aim of product family design is to offer distinct variants that attract a variety of market segments while maximizing the number of common parts to reduce manufacturing cost. Several indices have been developed for measuring the degree of commonality in existing product lines to compare product families or assess improvement of a redesign. In the product family optimization literature, commonality metrics are used to define the multi-objective tradeoff between commonality and individual variant performance. These metrics for optimization differ from indices in the first group: While the optimization metrics provide desirable computational properties, they generally lack the desirable properties of indices intended to act as appropriate proxies for the benefits of commonality, such as reduced tooling and supply chain costs. In this paper, we propose a method for computing the commonality index introduced by Martin and Ishii using the available input data for any product family without predefined configuration. The proposed method for computing the commonality index, which was originally defined for binary formulations (common / not common), is relaxed to the continuous space in order to solve the discrete problem with a series of continuous relaxations, and the effect of relaxation on the metric behavior is investigated. Several relaxation formulations are examined, and a new function with desirable properties is introduced and compared with prior formulations. The new properties of the proposed metric enable development of an efficient and robust single-stage gradient-based optimization of the joint product family platform selection and design problem, which is examined in a companion paper.

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