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June 13. 2002

Weighing the Choices

Dimensional weights can help decision makers allocate resources appropriately

By Erik Thomsen

Continued from Page 1

Instead of using past sales values to define the future allocation ratios, you could use a combination of the relative frequency of sales to the United States vs. sales to all countries, the relative frequency of sales of luxury goods vs. sales of all goods, and the relative frequency of sales to urban zones vs. sales to all zones. Let's work through an example to see how this value gets calculated.

The dimensional ratios, as shown in Table 2, are easy to calculate.

So how do you combine them to define cell-specific weights (keeping in mind that the sum of the weighting factors always needs to add up to one)? If you just multiply them together, you'll consistently over or under weight the entire data set (see the sidebar "The Ideal Weight"). Just adding them doesn't work either.

The trick is to divide each weight by the number of cells containing the weight and by the number of dimensions used for weighting (assuming that each dimensional weighting factor is to be equally weighted) and then sum that set of modified dimensional weights per cell as shown in Table 3. (Note the weight components in the cells in Table 3 refer to the product weight in the first line, the zone weight in the second, and the geography weight in the third.)

Table 4 compares the dimensional weights with the cell-based weights (in fifteenths).

Returning to our manager who felt that the cell-based weighting for luxury goods in urban zones in the United States was too low, notice how the dimensional weighting is twice as high. That's a huge difference! Where did that number come from? In this example, it came mostly from the fact that the sales were in the United States because the geography dimension had the greatest difference in weighting.

Making the Case

Managers can now argue in terms of trends in the sale of luxury goods vs. essentials, trends in sales to France vs. the United States, or trends in sales to urban zones vs. rural ones, instead of simply arguing for their particular cell (of responsibility). The manager for luxury goods in urban zones in the United States can now argue that low sales last year were the result of some special factors but that the data clearly shows that the United States is a growth market and her marketing dollars should reflect that fact.

(Although beyond the scope of this column, it's important to recognize that purely dimension-specific weightings don't capture any interdependencies that might exist between the various dimensions and that these interdependencies also need to be taken into account.)

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Of course, if you were the manager for luxury goods in urban zones in France and were liable to have your marketing budget slashed through the use of this dimensional weighting scheme, you might argue that the high sales you had last year were no fluke but rather that (in the presence of real detail data), it's incorrect to assign a single weight to luxury goods. Statistical analysis of luxury goods sales would show that French urban zones constitute a special market where the weightings need to be independently calculated and, furthermore, that the weighting applied to product type (which weightings were treated as equal, and thus were all 1/3 in the dimensional weighting scheme I showed) should be higher than the weighting applied to either geography or zone.

By using and arguing in terms of dimensional weights, key decision makers have a better way to come to an agreement over appropriate allocation weightings regardless of what they are. Thus, you should consider using dimensional weightings for your allocation functions both from an algorithmic and a human interaction perspective.

Erik Thomsen [ethomsen@dsslab.com] is cofounder of Power Thinking Tools, which developed the first OLAP engine with integrated statistics, visualization, text processing, and object management. He is a researcher and consultant for DSS Lab Inc. and focuses on integrated multitechnology analytic solutions. He is the author of OLAP Solutions, Second Edition (John Wiley & Sons, 2002) and coauthor of Microsoft OLAP Solutions (John Wiley & Sons, 1999).

THE IDEAL WEIGHT

To visualize why simply multiplying weighted cells will cause you to over or under weight the entire data set, imagine that the weights for each of the members in each of the three dimensions was 1/2 and that there were thousands of cells instead of the eight used in this column's example. With equal weighting everywhere, the weight for any cell would be (1/2)3 or 1/8. With anything other than 8 cells, the sum of the weights would either be too low or too high. Clearly the combination of the separate dimensional weights needs to reflect the number of actual data cells.