Academic journal article Annals of Management Science

# Weight Stability Intervals in Multicriteria Decision Aid under Semiorder Preference Structures

Academic journal article Annals of Management Science

# Weight Stability Intervals in Multicriteria Decision Aid under Semiorder Preference Structures

## Article excerpt

(ProQuest: ... denotes formulae omitted.)

1. Introduction

In multiple criteria decision aid the assessment of the relative importance of the different criteria plays a crucial role. A number of methods which are mainly focused on the definition, determination and influence of the criteria weights have been proposed in literature. Criteria weight is a kind of quantity that tries to express the decision maker's subjective preference but the definition itself is not always very precise.

Due to the fact that decision makers are not very clear to assign a weight each criterion in the beginning, the weight of the criteria will vary continuously through the process of the ranking of alternatives and the ranking results may change with the alteration of the weight.

It is true that the use of criteria weight gives the opportunity to a decision maker to modelize or express his feeling/judgment about a decision problem but it is necessary to be very careful when determining criteria weights. To help decision makers understand how the change of weight influence the ranking results, the concept of weight stability intervals for the weights of different criteria was introduced by Mareschal (1988b) and the PROMETHEE Methods where selected to operationalize the new approach.

The extension of the PROMETHEE Methods that we have proposed introduces a preference structure that is even more complex than the traditional one's which notably enrich the modelization phase: a Semiorder Preference Structure. In this way, the Weight Stability Intervals requires a new definition in correspondence with the preference structure.

It will be very interesting to study the extent to which the consideration of Semiorder Preference Structures within PROMETHEE Methods improves the decision making process in its entirety.

Having exhaustively analyzed the referred preference structure, it can be said that it gives more flexibility, amplitude and certainty to the preference formulations, as they tend to abandon The Complete Transitive Comparability Axiom of the Preferences to replace it by the Partial Comparability Axiom of the Preferences. Going form an axiom to the other allows us to introduce, in the analysis, the Incomparability that are basically present when: (1) the decision-maker is not able to discriminate between two alternatives since the information that he has, is too subjective or too incomplete to produce a judgment of Indifference or Strict Preference; (2) the decision-maker is in a position that not allow him to determine the preferences since the last responsible for the decision may be inaccessible, being either a remote entity or a loose entity with ill-defined and/or contradictory preferences; and (3) the decision-maker does not want to discriminate and he prefers to remain removed from the decision process and wait until a later stage when he has more reliable and sure information about the preferences.

The New Weight Stability Intervals (NWSI) is presented in section 2. We develop the new intervals in PROMETHEE II which is considered as an additive method of first order but under a semiorder preference structure. Different types of sensibility are defined and studied.

In order to point out the contributions of PROMETHEE methods with NWSI under a Semiorder Preference Structure different numerical applications are presented in Section 3.

2. New Weight Stability Intervals (NWSI) and PROMETHEE Methods

The use of sensibility analysis has been introduced in PROMETHEE Methods in order to help facilitate the interpretation of the results. In this way it is possible to study the consequences of the modifications of initially specified weights on the results. These sensitivity analyses require the determination of weight stability intervals, polygons and areas (Mareschal, 1989). On one hand, they provide sufficient information on the stability of the ranking and, on the other hand, they do no give insight in the way the ranking changes if the stability limits are exceeded. …

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