Author's personal copy Estimating ratio scale values when units are unspecified q Eng U. Choo, William C. Wedley Faculty of Business Administration, Simon Fraser University, Burnaby, BC, Canada V5A 1S6 article info Article history: Received 31 May 2009 Received in revised form 24 January 2010 Accepted 1 April 2010 Available online 4 April 2010 Keywords: Methodology Column averaging Ratio scaled preferences Relative ratios Unit of measurement abstract Multiplication of a ratio scale by a positive constant (a similarity transformation) causes the values to change to a different unit of measure without upsetting relative ratios between objects. Because the rel- ative ratios of any two objects are uniquely represented, a column vector estimating the same objects can be expressed as a comparison matrix which has no effect of any similarity transformation. Using these principles for several vector estimates, this paper evaluates column averaging approaches for aggregating estimates with unknown units into overall values. The geometric mean is shown to be the only method that is truly independent of the arbitrary unit of measure. Measures of clarity are proposed for the derived scale from the multiple columns. These new modeling ideas are compared with the common techniques used in the Analytic Hierarchy Process. Ó 2010 Elsevier Ltd. All rights reserved. 1. Introduction A regular ratio scale is characterized by a well-defined object whose intensity from natural zero is the unit of measure. All other objects possessing the property are measured relative to the well- defined object that acts as the standard of measurement. Ratio scales, however, are fickle. A similarity transformation achieved by simple multiplication by a positive constant (b > 0, b – 1) alters scale values and repositions the unit with the relative ratios un- changed. The object that was formerly the unit no longer has unit value. Quite likely, the repositioned, new unit is an obscure and even abstract object. Non-standard ratio scales occur when the unit of measure is ob- scure or unspecified. Such scales are often encountered in Multiple Criteria Decision Making (MCDM) when we attempt to estimate a ratio scale for subjective factors such as importance, preference and likelihood. For such scales there is no well-defined unit known beforehand (Saaty & Sodenkamp, 2008). Since non-standard scales do not provide complete and clear information about the unit of measure, they are tricky and more challenging to use. They do, however, provide useful information. Because the ratio of objects is invariant under different similarity transformations, we always have fixed relative ratios. If necessary after the relative ratios have been derived, a specific object can be assigned as the unit of measure. This paper analyzes the fundamental problem of estimating a non-standard ratio scale, Vb, with unspecified b > 0, where V =(v 1 ,v 2 , ... , v n ) T is a column vector of the values. With any b > 0, only the relative ratios bv i /bv j = v i /v j (i, j = 1,2, ... , n) are dimensionless and uniquely represented. Thus, Vb is completely represented by the n  n matrix M =[v i /v j ]. For two ratio scaled col- umn vectors Vb and V 0 b 0 , where V 0 ¼ðv 0 1 ; v 0 2 ; ... ; v 0 n Þ T , we can mea- sure the difference between Vb and V 0 b 0 in terms of the difference between M and M 0 ¼½v 0 i =v 0 j . Suppose C 1 ,C 2 , ... , C q are column vec- tors that estimate the column vectors Vb 1 ,Vb 2 , ... , Vb q with some positive numbers b 1 ,b 2 , ... , b q respectively. Then C 1 ,C 2 , ... , C q pro- vide q estimates for the relative ratios v i /v j (i,j = 1,2, ... , n) with some embedded errors. These embedded errors can be ‘‘averaged” out by aggregating C 1 ,C 2 , ... , C q appropriately into a column vector which estimates Vb for some b > 0. We analyze the column vector Vb directly and differentiate its relative ratios from the relative ratios of another column vector V 0 b 0 of ratio scaled measures of the same property. Due to the obscurity of the unit of measurement caused by the unspecified b > 0 and b 0 > 0, this is not a trivial process. A column averaging approach is introduced and its advantages are discussed. Then we evaluate some specific column averaging approaches to estimate v i /v j by aggregating C 1 ,C 2 , ... , C q into a column vector which estimates Vb for some b > 0. The geometric mean method is shown to be the only method that is truly independent of the unspecified unit of measurement and arbitrary b > 0. As well, we introduce the intuitive concept for measuring the clarity of an underlying column vector C being estimated by the values in C 1 ,C 2 , ... , C q . This clarity concept is based totally on the values in C 1 ,C 2 , ... , C q and not a priori measured from any particu- lar method of aggregation. This is in contrast to the consistency in- dex in AHP which depends heavily on the Eigenvalue for a measure of the consistency of the pairwise comparison matrix. 0360-8352/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2010.04.001 q This manuscript was processed by Area Editor Imed Kacem. E-mail addresses: choo@sfu.ca (E.U. Choo), wedley@sfu.ca (W.C. Wedley) Computers & Industrial Engineering 59 (2010) 200–208 Contents lists available at ScienceDirect Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie