Vehicle crashworthiness
==========================

The crash safety design problem has three objectives, which need to be minimized.
Objectives are :math:`(f_1)` the mass of the vehicle, 
:math:`(f_2)` acceleration-induced biomechanical damage of occupants and
:math:`(f_3)` the toe board intrusion in the 'offset-frontal crash'.
More details about the problem can be found in [1]_.

**Definition**

.. math::

  \min f_1(x) = & \; 1640.2823 + 2.3573285x_1 + 2.3220035x_2 \\[2mm]
  & + 4.5688768x_3 + 7.7213633x_4 + 4.4559504x_5 \\[2mm]
  \min f_2(x) = & \; 6.5856 + 1.15x_1 - 1.0427x_2 + 0.9738x_3 \\[2mm]
  & + 0.8364x_4 - 0.3695x_1x_4 + 0.0861x_1x_5 + 0.3628x_2x_4 \\[2mm]
  & - 0.1106x_1^2 - 0.3437x_3^2 + 0.1764x_4^2 \\[2mm]
  \min f_3(x) = & -0.0551 + 0.0181x_1 + 0.1024x_2 \\[2mm]
  & + 0.0421x_3 - 0.0073x_1x_2 + 0.024x_2x_3 - 0.0118x_2x_4 \\[2mm]
  & - 0.0204x_3x_4 - 0.008x_3x_5 - 0.0241x_2^2 + 0.0109x_4^2 \\[2mm]
  \\
  & s.t. \; 1mm \leq x \leq 3mm

The design variables represent the thickness of the five reinforced members around 
the frontal structure, which could significantly affect crash safety.

.. [1] Liao, X., Li, Q., Yang, X., Zhang, W. & Li, W. 
  (2007). Multiobjective optimization for crash safety design of vehicles 
  using stepwise regression model. Structural and multidisciplinary 
  optimization, 35(6), 561-569. https://doi.org/10.1007/s00158-007-0163-x