Plotting a penalty function as a line

Plot a penalty function to identify optimal materials along a trade-off curve.

To use penalty functions with a trade-off curve chart:

  1. Create a trade-off curve on a chart with linear scales.
  2. Calculate or look up the value of the exchange constant.
  3. Create a display line with slope -1/α.
    1. Click Index and display lines.
    2. Type in the slope value in decimal form.
    3. Select Minimize the index.
    4. Click OK, and then click the chart to position the line through that point.
  4. Move the line so it is tangential to the trade-off curve, and then drag the line upwards until the first set of materials are below the line.

    The optimal materials are the ones below the penalty function line, close to the trade-off curve.

    The chart above is for selecting materials to build a truck, using an exchange constant α=$5/kg (which means that the user/manufacturer is prepared to pay an extra $5 for each kilogram of weight saving). The best materials would be magnesium alloys and aluminum alloys, as these are closest to the display line of low Z (the line tangential to the trade-off curve). Polymers and composites would be least appropriate, as these materials have higher values of Z.

    The choice of optimal material for an application will vary depending on the value of the exchange constant. Although magnesium alloys and aluminum alloys are good materials for making a truck, they are not so good where saving weight is much more important than minimizing cost. If the exchange constant for a different vehicle is α=$500/kg, optimal materials would be composites like CFRP.

    This type of chart allows you to plot lines for different values of exchange constant on the same chart, and to directly compare material choices for different applications.