Multicriteria decision analysis with minimum information: combining DEA with MAVT
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 “ 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.
Section snippets
Data envelopment analysis and cross-efficiency
The efficiency of a many-input, many-output decision making unit (DMU) may be defined as a weighted sum of its outputs divided by a weighted sum of its inputs (Eq. (1)).where is the rth output of the DMU k, is the ith input of the DMU k, is the weight of the rth output is the weight of the ith input and the efficiency of kth DMU. This, so called “engineering ratio” is the most popular of a number of alternative measures of efficiency.
General description and differences with AXE
ACE+ can be considered as an attempt to enhance the capabilities of AXE. It is designed to be used with even less information than AXE. In AXE the value function of each criterion is considered to be fixed and given by the DM. In ACE+ the value functions are adjustable, letting each alternative use the most favourable value function during its self appraisal. In other words, the self-evaluation procedure, which is the cornerstone of cross-evaluation, is extended to comprise not only weight
Numerical example
The example that is used in order to demonstrate the method concerns the environmental evaluation of 14 countries of the European Union (we exclude Luxemburg— in most cross-national comparisons in EU—cause it acts as an outlier in the data set). The cross-country comparison based on multiple indices is a common task that requires an objective evaluation procedure with little information by the decision maker. Moreover, the relative information must be extracted by the data, as usually there are
Concluding remarks
The situations where a multicriteria evaluation of alternatives has to be performed based on limited information about the preferences of the DM are not rare. The self and peer evaluation of the alternatives, as it was inspired by DEA methodology, is a good remedy in such situations where the relevant information must be extracted from the data. One of the first approaches in this direction was AXE method that uses the cross-evaluation of alternatives. In the suggested method ACE+ we are going
Acknowledgements
The authors should like to thank the two anonymous referees for their stimulating comments that enrich the paper.
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2018, European Journal of Operational ResearchCitation Excerpt :While such a sensitivity analysis can provide valuable insights, it is limited to varying one (weighting) parameter at a time. Consequently, many researchers and practitioners in the field of decision analysis have proposed approaches for investigating the impact of varying several preference parameters at a time (Bertsch, Treitz, Geldermann, & Rentz, 2007; Butler, Jia, & Dyer, 1997; Insua & French, 1991; Jessop, 2011; Jessop, 2014; Jiménez, Mateos, & Ríos-Insua, 2005; Mateos, Jiménez, & Ríos-Insua, 2006; Matsatsinis & Samaras, 2001; Mavrotas & Trifillis, 2006; Mustajoki, Hämäläinen, & Salo, 2005; Scholten et al., 2015). Several studies have tried to link social concerns and energy system planning models, often by employing a combination of energy system analysis and MCDA tools (for reviews see Mardani et al., 2017; Ribeiro, Ferreira, & Araújo, 2011; Wang et al., 2009; Zhou, Ang, & Poh, 2006).
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2018, Technological Forecasting and Social ChangeCitation Excerpt :For example, MCDA methods can be used to find non-radial targets for the DMUs (Korhonen and Syrjänen, 2004) and to incorporate decision makers' preferences into DEA (e.g. Halme et al., 1999; Gouveia et al., 2008; Almeida and Dias, 2012). On the other hand, when information regarding the decision makers' preferences is limited or ambiguous, DEA can provide necessary information to allow multicriteria analyses to be performed (e.g. Mavrotas and Triffilis, 2006). Other studies exploring the combination of MCDA and DEA to develop business performance evaluation models include Athanassopoulos (1998), Chen and Chen (2007), Giokas and Pentzaropoulos (2008), Jyoti et al. (2008), Tseng et al. (2009), Yang et al. (2009), Baležentis and Baležentis (2011), Chitnis and Vaidya (2016), Bagherikahvarin and De Smet (2016).
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