M. Pavan, R. Todeschini

Publisher Summary

This chapter describes the theory of the mostly known total-order ranking techniques. Total-order ranking methods are multicriteria decision making (MCDM) techniques used for the ranking of various alternatives on the basis of more than one criterion. The chapter reviews a number of total-order ranking methods that have been widely used to facilitate the structuring and understanding of the perceived decision problem. The chapter presents the simplest approaches, like the Pareto optimality and the simple additive ranking (SAR) approach followed by the approaches belonging to the so-called multiattribute value theory—that is, utility, desirability, and dominance functions. In these models, numerical scores are constructed to represent the degree to which an alternative may be preferred to another. These scores are developed initially for each criterion and aggregated into a higher level of preference models.

1 Introduction

Total-order ranking methods belong to multicriteria decision making (MCDM), a discipline in its own right, which deals with decisions involving the choice of a best alternative from several potential candidates in a decision, subject to several criteria or attribute that may be concrete or vague.

Typically, a decision problem is a situation where an individual has alternative courses of action available and has to select one of them, without an a priori knowledge of which is the best one. A decision process can be organized in three phases. The first phase is problem identification and structuring, which consists in the identification of the purpose of the decision, in the recognition of the problem to be solved, in the diagnosis of the cause–effect relationships for the decision situation and in the identification of the judgment criteria. The second phase is the so-called model development and use, which consist in the development of formal models, decision maker preferences, values, trade-offs, goals to compare the alternatives or actions under consideration to each other in a systematic and transparent way. The third phase is the development of action plans since the analysis does not solve the decision. The decision process, which results in the selection of the best solution, i.e. the solution where the positive outcomes outweigh possible losses, is efficient if the procedure to reach the solution is optimal. The aims of a decision process are to effectively generate information on the decision problem from available data, to effectively generate solutions and to provide a good understanding of the structure of a decision problem.

Multicriteria decision-making techniques are used for helping people in making their decision according to their preferences, in cases where there is more than one conflicting criterion, finding the optimal choice among the alternatives. Making a decision is not just a question of selecting a best alternative. Often the need is to rank all the alternatives for resource allocation or to combine the strengths of preferences of individuals to form a collective preference.

Mathematics applied to decision making provides methods to quantify or prioritize personal or group judgments that are typically intangible and subjective. Decision making requires comparing different kinds of alternatives by decomposing the preferences into many properties that the alternatives have, determining their importance, comparing and obtaining the relative preference of alternatives with respect to each property and synthesizing the results to get the overall preference. Therefore, the strategy consists in breaking down a complex problem into its smaller components and establishing importance or priority to rank the alternatives in a comprehensive and general way to look at the problem mathematically.

The key starting point of MCDM lies in attempting to represent often intangible goals in terms of the number of individual criteria. A challenging feature of MCDM methods is the identification of the set of criteria by which alternatives are to be compared. The selection criteria is part of the modeling and problem formulation, a significant phase often under-emphasized. A useful general definition of a criterion is the one provided by Bouyssou (1990) as a tool allowing comparison of alternatives according to a particular axis or point of view. It is generally assumed that each criterion can be represented by a surrogate measure of performance, represented by some measurable attribute of the consequences arising from the achievement of any particular decision alternative.

Some thoughts are to be considered in identifying the criteria: their value relevance, i.e. their link with the decision maker concept of their goals; their understandability and their measurability, i.e. the performance of the alternative against the criteria should be measurable; their non-redundancy in order to avoid that the concept they represent is in attributed greater importance; their judgmental independence, i.e. the preferences with respect to a single criterion should be independent from the level of another; their balancing between completeness and conciseness.

Subjectivity is intrinsic in all decision making and in particular in the choice of the criteria on which the decision is based and in their relative weight. Multicriteria decision making does not dissolve subjectivity, but it makes the need for subjective judgments explicit and the whole process by which they are considered is made transparent.

Over the years, several MCDM methods have been proposed (Hobbs and Horn, 1997) in different areas, with different theoretical background and facing different kinds of questions and providing different kinds of results (Hobbs and Meier, 1994).

Some of these methods have been developed to fulfill the need of specific problems; other methods are more general and have been used in different areas. The different MCDM methods are distinguished from each other in the nature of the model, in the information needed and in how the model is used. They have in common the aim to create a more formalized and better-informed decision-making process, the need to define alternatives to be considered, the criteria to guide the evaluation and the relative importance of the different criteria.

In this chapter, the theory of the mostly known total-order ranking techniques is described.

2 Total-Order Ranking Methods

Once the decision problem identification phase has generated a set of alternatives, which can be a discrete list of alternatives as well as be defined implicitly by a set of constraints on a vector of decision variables, and once the set of criteria against which the alternatives have to be analyzed and compared are defined, then a decision model has to be built to support decision makers in searching the optimal or the set of satisfactory solutions to the multicriteria decision problem. The decision model is made of two main components as described by Belton and Stewart (2003):

The purpose of the decision model is to create a view of decision maker preferences based on a defined set of assumptions and to guide the decision maker in the optimum solution search.

Before reviewing some total-order ranking methods, some further terms and basic principles are introduced.

A p-dimensional system is generally considered, with an associated (n×p) data matrix X. To each of the n alternatives, a set of p criteria relevant to the decision-making procedure is associated. Each criterion can then be weighted to take account of the different importance of the criteria in the decision rule. The strategies to reach the optimal choice require the development of a ranking of the different options. Within a set of alternatives A (a, b, c, d), a ranking (order) on A is a relation with the following...