#### Document Type

Thesis

#### Degree Name

Master of Science (MSc)

#### Department

Mathematics

#### Faculty/School

Faculty of Science

#### First Advisor

Marc Kilgour

#### Advisor Role

Thesis Supervisor

#### Abstract

In Multiple-Criteria Decision Analysis (MCDA), a good way to find the best alternative is to construct a value function that represents a Decision Maker’s (DM) preferences. For multidimensional alternatives, an additive value function is easiest to work with because it assesses the alternatives in a simple and transparent manner. A DM’s preferences over consequences on a subset of the set of criteria may or may not depend on consequences on the rest of the criteria. Preferences that are free from all such interdependence are said to be separable. The existence of an additive value function implies separability and, when consequences form a continuum in each dimension and preference is continuous, the converse is also true. But we concentrate on orderings of binary alternatives (only two possible consequences on each criterion), for which the converse is known to be false unless there are four or fewer criteria.

On binary alternatives, the probability of a separable order arising at random decreases rapidly as the number of criteria increases. However, there are different degrees of non-separability; many combinations of separable and non-separable subsets of criteria are possible. Here, we introduce notions of partial separability and partial additivity, which could be appropriate if criteria can be grouped into two or more natural classes. We establish that partial additivity implies partial separability, but that the converse is true only when the number of criteria is less than or equal to three. We also show that, when the number of criteria is more than three, partial separability with respect to a singleton set of criteria implies partial additivity with respect to this same subset.

#### Recommended Citation

Bashar, Md. Abul, "Partial Separability and Partial Additivity for Orderings of Binary Alternatives" (2007). *Theses and Dissertations (Comprehensive)*. 860.

https://scholars.wlu.ca/etd/860

#### Convocation Year

2007