Piecewise Linear Virtual Inputs/Outputs in Interval DEA

Piecewise Linear Virtual Inputs/Outputs in Interval DEA

Yiannis G. Smirlis (IT & Infrastructure Division, University of Piraeus, Piraeus, Greece) and Dimitris K. Despotis (Department of Informatics, University of Piraeus, Piraeus, Greece)
DOI: 10.4018/joris.2013040103
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Abstract

A recent development in data envelopment analysis (DEA) concerns the introduction of a piece-wise linear representation of the virtual inputs and/or outputs as a means to model situations where the marginal value of an output (input) is assumed to diminish (increase) as the output (input) increases. Currently, this approach is limited to crisp data sets. In this paper, the authors extend the piece-wise linear approach to interval DEA, i.e. to cases where the input/output data are only known to lie within intervals with given bounds. The authors also define appropriate interval segmentations to implement the piece-wise linear forms in conjunction with the interval bounds of the input/output data and the authors propose a new models, compliant with the interval DEA methodology. They finally illustrate their developments with an artificial data set.
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Dea Models With Non-Linear Partial Value Functions

Consider the following input-oriented CCR DEA model (multiplier form) with n DMUs, m inputs and s outputs, where yrj denotes the level of the output r (r =1,…, s) produced by the DMU j (j=1,…,n), xij denotes the level of the input i (i=1,…, m) consumed by the DMU j and the variables u=(ur, r=1,…,s) and v=(vi, i=1,…,m) are the unknown weights attached to the outputs and the inputs respectively:

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