We examine optimal-profit product design platforming problems for products sold across multiple markets. For many contexts, this problem can be represented as Nash equilibrium between multiple competing firms, each with a nonconvex mixed-integer nonlinear programing (MINLP) problem for maximizing their individual profits. Algorithms for solving this structure of platforming problem have not previously been investigated in engineering design or economics literature. We derive the Karush-Kuhn-Tucker (KKT) conditions for the problem and compare two algorithmic approaches: (1) an iterative MINLP approach that uses the BARON algorithm to solve each firm’s design and platforming problem and iterates until convergence to an equilibrium, and (2) an approach that solves the KKT conditions directly holding platforming decisions fixed, and compares profits for these platforming decisions to find an equilibrium. Results are presented for a case study of plug-in hybrid electric vehicles (PHEVs) where firms choose whether or not to platform the battery pack across the U.S. and China, and the optimal battery capacity.

We also investigate the effects of varying incentives to platform and other problem parameters on the market equilibria of the case study. To this end, we perform a parameter sweep of learning rates and consumers’ WTP for all electric range in China within the case study, holding WTP for all electric range of consumers in the U.S. fixed. We find that market equilibrium solutions vary depending on these parameters. When learning rates are relatively high, and consumer WTP for all electric range in China is low, firms choose not to platform. Otherwise, equilibrium solutions are for firms to platform the vehicles so that battery capacity is the same across the markets, and the optimal battery capacity depends on whether consumers in both markets have WTPs sufficient to balance costs for increasing all electric range within the constraint boundaries for battery capacity.

We compare solutions found using an algorithm that iteratively solves each firm’s MINLP profit optimization problem until convergence to an equilibrium problem with an algorithm that solves the KKT conditions directly. We calculate total firm costs for solutions found using both algorithms, and find that the cost discrepancy between the two algorithms is within 10% of profits for 98% of points on the parameter sweep. However, for some parameter values, the KKT algorithm converged to false solutions that did not meet all KKT conditions. Future work will investigate improvements to the algorithm to avoid convergence to false solutions, and additional comparisons of the two approaches to determine comparative performance under different conditions of the platforming problem.


Global Product Design Platforming: A Comparison of Two Methods to Find Equilibrium Solutions

Division / Sponsors: 
Design Engineering Division and Computers & Information In Engineering Division


Paper Proceedings: https://asmedigitalcollection.asme.org/IDETC-CIE/IDETC-CIE2020/volume/83907

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