This example is similar, but now we show the result using three methods. Method 1 uses explicit mixed references. We could have done this one using implicit intersection, as we did the example "ConvolutionGraphically," but this illustrates another way. Method 2 also uses explicit references, but this one computes the correct cell of the productivity curve using the horizontal and vertical helper ranges. The advantage of this approach is that you can fill one cell across the entire upper triangle, without any additional editing. Method 3 uses the Convolve macro.

In Rows 40 and 41, we illustrate the use of the Convolve macro on a range that has multiple rows. Here, we're computing the load on the trainers, as well as the total productivity. The Convolve macro can easily handle this. Using Method 1 or Method 2 would require another triangular range calculation. You can see that the macro approach is much more compact.

Strictly speaking, though, this example doesn't meet the linearity criterion required for convolution to be valid. Specifically, when a "negative hire" occurs — a layoff — the contribution to productivity has a shape that differs somewhat from the shape of the productivity curve for a positive hire. For a layoff, the curve just drops to zero immediately and directly. Thus, when we compute the effects of layoffs, we have to use a different shape, and this is the essence of this kind of violation of linerarity. But as long as there are no negative hires, we're OK. Most business applications of convolution do have this same problem.