Determination of criteria weights using ranking

Abstract

This paper presents possible ways for determining criteria weights based on the criteria ranks.The linear weights with a variable slope, reciprocal weights, rank order centroid weights, rank sum weights, geometric weights and rank order distribution weights are shown in this paper.

When the ranking process involves several experts, there is a possibility for aggregating criteria ranks (weights) using selected mathematical methods, social choice theory methods and the Kemeni median. The application of the selected methods is illustrated with one numerical example and the obtained results are analysed.

The main aim of this paper is a systematic review of possible ways for determining the criteria weights based on criteria ranks given by one or more experts.

Introduction

One of the most commonly used subjective approaches to determining criteria weights is the approach based on the criteria ranking. The main advantage of this method of determining the criteria weight is the fact that a decision maker finds it often much easier to make criteria ranking than to assign numerical values ​​to criteria weights.

In calculating criteria weights based on criteria ranking, it is necessary to determine the type of the function rank – weight.

When the process of determining the criteria weights involves many experts, then the aggregation individual criteria ranks (or weights) must be done as well as the formation of a unique group criteria weight using various aggregation methods.

The selected methods for determining the criteria weights using ranking

The linear weights with a variable slope, reciprocal weights, rank order centroid weights, rank sum weights, geometric weights and rank order distribution weights are shown in this paper.

This paper presents a brief review of the selected methods for determining the criteria weights on the basis of their rank: linear weights with a variable slope, reciprocal weights, rank order centroid weights, rank sum weights, geometric weights and rank order distribution, in order to form a complete notion of the possibilities of determining the criteria weights based on criteria ranks.

Roberts and Goodwin (2002) suggest the existence of clear theoretical evidence that weights obtained using the rank order centroid method represent the best approximation of the weights that can be obtained using the direct allocation of weights. Weights determined by the rank order distribution method best fit weights that can be obtained using the proportional method of determining the criteria weights.

To solve the problem of determining the weight of a large number of criteria, the application of the rank sum weights method is recommended.

Aggregation of individual criteria orders

Group values ​​of criteria weights can be obtained in two ways: by transforming the individual ranks into weights, and then aggregating the individual weights or aggregating individual ranks and transforming group criteria ranks into group criteria ​​weights. The formation of group values criteria ​​weights can be performed by applying mathematical aggregatin methods, by the implementation of the social choice theory and using the Kemeni median.

Selected mathematical aggregation methods

Based on the literature (Alfares, 2007.), the following mathematical methods for aggregating individual weights (ranks) are shown: a method of arithmetic averaging of criteria weights, method of geometric averaging of criteria weights and the median ranks method.

The use of the methods of social choice for the aggregation of individual criteria orders

The social choice theory identifies the group decision making problem (in this case it is an expert group) with the problem of finding a method for obtaining a unique group rank list from a set of differently ranked individual preferences. The requested method should enable the successful aggregation of individual orders of objects (variants, criteria, etc.).

Without going into a further theoretical consideration of the social choice theory, there are listed group decision making methods which as a result give a complete rank list. The best known methods are the Condorcet method and the Borda method.

Determination of criteria weights by applying the Kemeni median

Displaying ranking as binary relations in a matrix form creates a possibility of introducing the measure of distance between pairs of rankings, such as the Kemeni distance.

The resulting ranking closest to all individual rankings, and with a minimum total distance of all the individual rankings, is the Kemeni median.

The Kemeni median is a unique resultant ranking that is neutral, concordant, Condorset’s and, from the all Arrow’s conditions, it does not meet only the requirement of independence.

This paper briefly shows a heuristic algorithm (Литвак, 1982.) which is simpler to implement than a combinatorial algorithm but meets the needs of this paper.

Heuristic algorithm to determine the Kemeni median

The starting point for the calculation of the Kemeni median is the lost matrix. The elements of the lost matrix are the sum of distances of all the individual rankings to the selected ranking.

The task of calculating the Kemeni median is identical to calculating, based on the lost matrix, the minimum summary distance.

The heuristic algorithm calculation of the Kemeni median is conducted in several iterations and it finishes by checking the optimality of the obtained resultant ranking.

An example of the application of selected methods

The expert group consisting of eighteen experts performed a ranking of eight criteria with the goal of determining the criteria weights.

Criteria weights based on individual ranks were obtained by applying the selected conversion functions ranks in weights.

The aggregation of individual weights (ranks) was done by using mathematical methods.

The relative weight ratio of the value of different methods of transformation criteria ranks into weights is the same regardless of the chosen aggregation method.

The order of values of criteria weights ​​is the same for all methods of transformation ranks into weights, except for the geometric weight method in which there is a substitution between two criteria ranks.

The median rank method does not always give a strict group criteria orders.

In the given example, the Kemeni median, the Borda and the Condorcet method give the same rank orders. This result can be regarded as an exception rather than a rule.

Conclusion

Methods for determining the criteria weights based on criteria ranks belong to the subjective approach to determining criteria weights.

Although the methods of determining the criteria weights have a great effect on the value of the obtained criteria weights, there is no consensus on the best method of determining the criteria weights, or the conditions of its application. If it is necessary to emphasize the intensity of the weight differences,,the reciprocal weights method can be applied or the rank order centroid method. The linear weights method with a variable slope can be applied if the criteria are approximately equal in their importance.

The aggregation of individual criteria ranks can be performed in several ways. The author recommends the use of the Kemeni median.