# TyreOpt: Optimizing truck tyres

My name is Zuzana Nedělková and I am a PhD student in the optimization group at the Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg.  My research project TyreOpt – Fuel consumption reduction by tyre drag optimization financed by the Swedish Energy Agency is done in cooperation with Volvo Group Trucks Technology (GTT).  My main supervisors are Ann-Brith Strömberg, Michael Patriksson, and Bengt Jacobson, from Chalmers and Peter Lindroth from Volvo GTT.

Figure 1: The main goal of the TyreOpt project is to select tyres for each vehicle configuration and operating environment specification such that the vehicle’s fuel consumption is minimized.

Within the project TyreOpt we aim to find an optimal tyre configuration for each vehicle and operating environment combination that minimizes fuel consumption, while balancing the other tyre-dependent features.

The optimization model in TyreOpt is simulation-based, which means that the objective function and the constraints can only be estimated through computationally very expensive simulations. Optimization problems including such simulation-based functions are often solved by a response surface method (Jones, 2001), in which a surrogate model that mimics the behavior of the simulation-based function is iteratively shaped and optimized.

Multivariate functions can be effectively modeled using radial basis function (RBF) interpolation (Wendland, 2005), yielding good global representations of simulation-based functions and are therefore frequently used in algorithms for simulation-based optimization. Despite this, the surrogate model may contain inaccuracies and even physically absurd values, leading to a poor correspondence with the experts’ expectations. The latter phenomenon occurs in our particular tyre optimization application, whence we have developed an algorithm framework for obviating this problem (Nedělková et al., 2015).

The rolling resistance can be described as the effort required to keep a given tyre rolling, and is characterized through the rolling resistance coefficient (RRC), which forms the main ingredient of the optimization model developed to find the optimal truck tyres. A computationally heavy finite element analysis (FEA) model of a truck tyre, used to determine the RRC, was replaced by an RBF interpolation of sample points evaluated by the FEA model. The RBF interpolation of the RRC contains inaccuracies and is often even physically absurd, such as providing negative values. Therefore, we combined the RBF interpolation with the existing expert knowledge; leading to an RBF approximation; see Figure 2.

Simulation-based optimization models arise from many real optimization problems. They are often solved utilizing surrogate models. We have developed a new methodology for utilizing many kinds of expert knowledge in the construction of a surrogate function, which is here illustrated on the function describing the RRC of a truck tyre. The surrogate model of the rolling resistance function – constructed using the RBF approximation described above – is further used in the TyreOpt project to construct an optimization model, which needs to be solved in real time in Volvo GTT’s selling tool – in order to find the optimal set of tyres for each customer’s specifications of the vehicle and the operating environment.

Figure 2: A cut of the original surrogate model of the RRC illustrating that the RBF model does not conform to the expert knowledge. The updated surrogate model of the RRC is closer to the experts’ expectations.

##### References
• R. Jones, “A taxonomy of global optimization methods based on response surfaces,” Journal of Global Optimization 21(4), 2001, pp. 345–383.
• Nedělková, P. Lindroth, A.-B. Strömberg, and M. Patriksson, “Integration of expert knowledge into radial basis function surrogate models,” Optimization and Engineering, 2015. http://dx.doi.org/10.1007/s11081-015-9297-7
• Wendland, “Scattered Data Approximation,” Cambridge Monographs on Applied and Computational Mathematics, Cambridge, UK, 2005.