Common Weights in DEA Models with Reduced Singular Value Decomposition

Hassan Naseri,S.Esmaeil Najafi,Abbas Saghaei

doi:10.6339/jds.201804_16(2).0010

Abstract

In DEA framework there are many techniques for finding a common set of efficient weights depend on inputs and outputs values in a set of peer DecisionMaking Units (DMUs). In a lot of papers, has been discussed multiple criteria decision-making techniques and multiple objective-decision criteria for modeling. We know the objective function of a common set of weights is defined like an individual efficiency of one DMU with a basic difference: "trying to maximize the efficiency of all DMUs simultaneously, with unchanged restrictions". An ideal solution for a common set of weights can be the closest set to the derived individual solution of each DMU. Now one question can be: "are the closest set and minimized set, which is found in most of the techniques, are different?" The answer can be: "They are different when the variance between the generated weights of a specific input (output) from n DMUs is big". In this case, we will apply Singular Value Decomposition (SVD) such that, first, the degree of importance weights for each input (output) will be defined and found, then, the Common Set of Weights (CSW) will be found by the closest set to these weights. The degree of importance values will affect the CSW of each DMU directly.

Highlights

Data Envelopment Analysis (DEA), which is a non-parametric method for measuring the efficiency of a set of peer DecisionMaking Units (DMUs), was first introduced into the Operations Research (OR) literature by Charnes, Cooper, Rhodes (CCR, 1978)
Singular Value Decomposition (SVD) is a method for identifying and ordering the dimensions along which data points exhibit the most variation. This ties into the third way of viewing SVD, which is that once we have identified where the most variation is, it's possible to find the best approximation of the original data points using fewer dimensions
The main difference between this criterion with the others is the difference between the common weights for each DMU, because in this method first a whole weight is derived and this quantity will be multiplied by each individual degree of importance which is found in the first run

Summary

1-Introduction

Data Envelopment Analysis (DEA), which is a non-parametric method for measuring the efficiency of a set of peer DMUs (such as firms or public sectors), was first introduced into the Operations Research (OR) literature by Charnes, Cooper, Rhodes (CCR, 1978). For example in an assumed CCR-Input Oriented, if for each input (output), n weights are derived by n repetitions on an optimization model (as will be described in (I)), in a special case, probably, the produced variance of n input (output) weights can be big and this variance has the principal effect on the results of a CSW technique, during minimizing the distances. If a CCR-Input Oriented case is applied for hDMU in a set of n+1 DMUs as will be illustrated in (IV) the result would be a CSW such that the common weights are the closest weights to the importance individual efficiency weights related to the inputs (outputs).

GU are orthonormal eigenvectors of VVT and UUT

Global Effects

4-Results

5-Conclusions