Material selection: penalty functions

One challenge when using trade-off curves is how to objectively identify which section of the curve contains materials optimal for your application.

All materials along the trade-off curve are candidate materials, but which material offers the best balance between minimizing mass and minimizing cost?

The optimum material can be determined by using a penalty function. This is a method that uses exchange constants to determine the relative importance of the conflicting objectives.

The penalty function, Z, is defined as Z = α₁M₁ + α₂M₂ where M₁ and M₂ are the performance indices for the two objectives, and α₁ and α₂ are the exchange constants. Optimum materials are those with the smallest value of Z.

The penalty function is measured in the same units as one of the objectives (often cost). The exchange constants convert the objectives into the same units as Z.

Using the bicycle example, where the objectives are to minimize both mass (M₂) and cost (M₁), Z is set to be measured in units of currency. The exchange constant for the cost is α₁=1, and the units for α₂ are currency/mass. The value of α₂ reflects how much the user is willing to pay for each unit of weight saving, and determines the relative importance of reducing cost over mass.

This allows you to simplify the penalty function to:

Z = α₁ M₁ + α₂ M₂

Z = M₁ + α M₂

Rearranging in terms of the y-axis index (M₂):

M₂ = - M₁/α + Z/α.

This is an equation for a straight line in the form of y=mx+c. This means you can identify optimal materials by adding a display line of slope -1/α to the trade-off plot with linear scales to find materials that minimize Z. This procedure is detailed below.