Multicriteria decision analysis with minimum information: combining DEA with MAVT

Abstract

In this paper we use some basic principles from data envelopment analysis (DEA) in order to extract the necessary information for solving a multicriteria decision analysis (MCDA) problem. The proposed method (enhanced alternative cross-evaluation, ACE+) is appropriate when either the decision maker is unwilling (or hardly available) to provide information, or there are several decision makers, each one supporting his/her own option. It is similar to the AXE method of Doyle, but it goes one step further: each alternative uses its most favourable weights (as in AXE) and its most favourable value functions in order to perform a self evaluation, according to multi attribute value theory (MAVT). These self-evaluations are averaged in order to derive the overall peer-evaluation for each alternative. The minimum information required from the decision maker is to define the weight interval for each criterion. Beside the peer evaluation and the final rating of the alternatives, the method provides useful conclusions for the sensitivity analysis of the results.

Introduction

In multiple criteria decision analysis (MCDA) problems a number of alternatives have to be evaluated and compared using several criteria. The aim of MCDA is to provide support to the decision maker (DM) in the process of making the choice between alternatives. In other words, MCDA techniques help the DM to articulate his/her preferences in a complex decision making environment. A pre-requisite in most MCDA methods is that the DM is able to provide the necessary information. However, in several situations the DM is hardly available or unwilling to express his/her preferences or, furthermore, there is not an unequivocal DM but a group of stakeholders that each one supports his/her own alternative. In these situations the available information in order to perform the multicriteria analysis is limited or ambiguous. In order to fill the information gap, or to reach a consensus in the latter case, information that emerges from the data set itself can be utilized. In the present work, the suggested way to extract this information is by using the basic ideas of Data Envelopment Analysis (DEA).

DEA origins are found in the late 70s [1]. DEA deals with the evaluation of the performance of objects (Decision Making Units or DMU in DEA's terminology) using the transformation process of several inputs into several outputs. Relying on a technique based on Linear Programming and without having to introduce any subjective or economic prices (weights, costs, etc), DEA provides a measure of efficiency of each object. Its basic aim is to separate efficient from non-efficient DMUs and to indicate for each non-efficient DMU its efficient peers. During the last 25 years thousands of papers related to DEA have been published in OR/MS journals.

MCDA and DEA have been receiving considerable attention in the OR/MS literature. The problem tackled by DEA is one which may equally well be approached using MCDA. Indeed, in common with many MCDA approaches, DEA incorporates a process of assigning weights to criteria. However, despite having much in common, the two fields have developed almost entirely independently to each other [2], [3], [4], [5]. It is only in the last decade the success of DEA in the area of performance evaluation together with the formal analogies existing between DEA and MCDA (which become clear replacing DMU with alternatives, outputs with maximization criteria and inputs with minimization criteria) have lead some authors to propose to use DEA as a tool for MCDA [6], [7], [8]. Among the first steps towards this co-operation, was the notion of cross-efficiency in DEA, originated by Sexton et al. [9] and further developed by Doyle and Green [10]. The basic aim was to increase the discriminatory power of DEA. In cross-efficiency each one of the alternatives is evaluated using the most favourable set of weights (self evaluation). In the same time, the rest of the alternatives are evaluated using the above set of weights. This process is repeated for all the alternatives resulting in a square matrix of evaluations with the diagonal elements being the self evaluation scores and the off-diagonal elements the peer evaluation scores of the alternatives. The final score for each alternative is obtained as the corresponding column average of this matrix. As Doyle and Green state “$\dots$ cross efficiency, with its intuitive interpretation as peer appraisal, has less of the arbitrariness of additional constraints, and has more of the right connotations of a democratic process, as opposed to authoritarianism (externally imposed weights) or egoism (self-appraisal)” [10]. These results where further elaborated by Doyle [11] in order to develop a MCDA method called alternative cross-evaluation (AXE) in its work with the intriguing title “Multiattribute Choice for the Lazy Decision Maker: Let the Alternatives Decide!”.

The notion of cross-efficiency has been emerged as an alternative DEA approach in order to increase the relative discriminating power. As a consequence, it uses DEA's mathematical representation and it is confined in linear relationships. Hence, in MCDA terminology, the value functions of the criteria are assumed linear and the only variables in the Linear Programming (LP) formulation are the relative weights. In the current work we extend the notion of cross efficiency in order to allow the value functions in each criterion to be non-linear. It is accomplished with the incorporation in the model of a specially developed non-linear value functions with one parameter. The model turns inevitably to non-linear with the parameters of each criterion's value function being the second set of decision variables (the first is the set of weights). The proposed method is named enhanced alternative cross-evaluation (ACE+) and allows for a fruitful insight in the problem as it can extract intra-criterion (value functions) and inter-criterion (weights) information. Moreover, it maintains the objective and democratic character of cross-efficiency that makes it attractive in minimum information situations.

The method can be used when the information from the DM in a MCDA problem is rather limited and we want to obtain a better insight in the problem or, even more, to solve the problem adequately under these minimum information conditions. It must be noted that in relative decision situations the usual, naïve approach is to assign equal weights to the criteria, assume linear value functions and use an additive aggregation function. Using more sophisticated methods like ACE+ the underlying information in the alternatives and criteria can be better exploited leading to more robust decisions. In other words, ACE+ is not only used for the evaluation of the alternatives under minimum information conditions, but it also “produces” information that help the decision maker to better perceive the potential of each alternative in the specific decision context.

The rest of the paper is organized as follows: In Section 2 the main topics of DEA and cross-efficiency are briefly reminded. In Section 3, we describe the suggested method (ACE+) in detail. In Section 4 we present the application of the method with an educational example making comparisons with other approaches. Namely, 14 countries of the European Union are compared by means of seven eco-efficiency indicators. Finally, in Section 5 the most important conclusions and issues for further research are presented.