# Papers

### Observations on multiplicative composition in the Analytic Hierarchy Process

**Author(s)****Thomas Saaty**

Joseph M. Katz Graduate School of Business

University of Pittsburgh

United States

**Publication date: **Jan, 1994

**Journal: **Proceedings of the 3rd International Symposium on the Analytic Hierarchy Process

**Abstract: **From the early years of development of the Analytic Hierarchy Process (AHP) I examined a number of alternatives for hierarchic composition that produced ratio scales. Ratio scales were always in my thinking because of my strong interest in physics. There one multiplies different ratio scales before and after raising them to powers. But even in mathematics, length, area and volume involve different powers of a metric. In my first book on the AHP I pointed out what is commonly known, that the product and quotient of different ratio scales is a ratio scale but not the sum or difference. The sum of elements from the same ratio scale is in fact a ratio scale. I also gave a theorem on the powers of a ratio scale which as products of ratio scales define a ratio scale. Later I worked out examples using multiplicative composition where to obtain the composite weight of an alternative, the weight of that alternative is raised to the power of its criterion and the product is taken over the criteria. In all the examples the outcome was close to that obtained by additive composition which is simply multiplying the weight of an alternative by that of its corresponding criterion and adding over the criteria. In this note I want to give several reasons for not using the multiplicative approach for hierarchic composition.

**Keywords: ***AHP**, Analytic Hierarchy Process*