Comparing Fundamentals of Additive and Multiplicative Aggregation in Ratio Scale Multi-Criteria Decision Making

Abstract

Additive and multiplicative aggregations of ratio scale preferences are frequently used in multi-criteria decision making models. In this paper, we compare the advantages and limitations of these two aggregation rules by exploring only their fundamental properties after ratio scaled local priorities and criteria weights have been successfully generated from the decision maker. The comparisons of these properties are therefore independent of ancillary procedures such as interac- tive elicitations from decision makers, pairwise comparisons and calculations of local priorities and criteria weights. We compare six fundamental properties of the two aggregation rules. The criteria weights used in the multiplicative aggrega- tion have complicated meanings which are not well understood and often mixed up in the ambiguous notion of "criteria importance". As the scaling factors of the local preference values do not appear explicitly in the computations of the rela- tive ratios of the overall preferences in the multiplicative aggregation model, the relative ratios remain unchanged when the scaling factors are changed or an alternative is added or deleted. Furthermore, the relative ratios in the multiplicative aggregation do not depend on similar local preference values which cancel each other out mathematically. It is quite evi- dent that the additive aggregation model is superior and easier for decision makers to use and understand. We recommend the additive aggregation rule over the multiplicative aggregation rule. Fundamental basic elements of the MCDM framework are first depicted without any specific interpretations im- posed on these elements. It is assumed that no relevant crite- rion is missed and each criterion is autonomous. In section 3, the measures of criteria weight, local and overall preferences are assumed to be in ratio scale. Some necessary conditions and the role of normalization are discussed. We then give a brief literature review, with particular attention to the differ- ent ancillary procedures and contradicting opinions in model interpretations. Additive and multiplicative aggregation rules are formally introduced in Section 5. In Section 6, we elabo- rate and compare the fundamental properties of these aggre- gation rules. Finally, we summarize and give some conclu- sions. 2. BASIC ELEMENTS OF MCDM MODEL The basic elements of a typical MCDM model include a set A={A1,A2,…,An} of n alternatives A1,A2,…,An and a set C={C1,C2,…,Cm} of m criteria C1,C2,…,Cm. The effect of the criteria C1,C2,…,Cm in C is represented by positive num- bers w1,w2,…,wm respectively. The vector w=(w1,w2,…,wm) is called the criteria weight vector of the criteria C1,C2,…,Cm in C. The criteria weight vector w is derived from question- ing the DM. The alternatives A1,A2,…,An can be evaluated under each individual criterion Cp, p=1,2,…,m. For each criterion Cp (p=1,2,…,m), the local preference of the alterna- tives A1,A2,…,An in A with respect to Cp is represented by positive numbers x1p,x2p,…,xnp, respectively. The vector xp=(x1p,x2p,…,xnp) is called the local preference vector of the alternatives A1,A2,…,An in A with respect to Cp. The local preference vectors x1,x2,…,xm are derived from questioning the DM.