Malmquist Productivity Index In Ratio Data Envelopment Analysis

The selection of advanced production technologies and the use of data envelopment analysis (DEA) are important complex organizational decisions. The basic DEA models are designed for certain data, while inputs and outputs of real world problems are ratio. This is because the use of specific data is not possible or practical, and appropriately responsive to the given area. This article developed a model capable of computing the Malmquist Productivity Index (MPI) aiming to determine the progress or regression of the units under evaluation in the presence of such data. The proposed model was then applied to a numerical example, and the results were addressed.


Introduction
Organizational performance analysis is of the most significant areas in the science of management, which can be addressed in different ways.DEA is an appropriate non-parametric method for ratio evaluation of decision-making units (DMUs) with multiple inputs and outputs [1], [2].Over the past decades, developing countries have begun to make different economic-social plans to improve the lives of their people.An approach to achieve these objectives is taking a step towards industrialization, which requires accessibility to modern technologies.The measured technologies of DMUs related to the efficiency of the production technology boundary.
Production and hosting by ISPACS GmbH.

Data Envelopment Analysis and Decision Science
A question in the field of DEA is that whether or not the ratio measures are acceptable in input and output data.These measures are seen in many DEA programs [3], [4].Today, the DEA method is used in combination with other methods, such as the MPI, to measure efficiency and productivity.
Productivity measurement of research organizations, and analysis and comparison of their performance have a significant role in their efficiency and productivity improvement.It provides the context for future organizational planning.The concept of productivity is a very important property of open systems.The importance of productivity is high enough to be considered as the main objective of every system.The MPI was first introduced by Malmquist in 1953 as a general index to be used in the analysis of the consumption of inputs [5].Fare et al. [6] combined the Farrell's measures of efficiency and Cave et al.'s measures of productivity [7], and expressed the MPI as the result of the multiplication of efficiency changes by technological changes.In fact, this index maintains that the regression or progress of a unit, under evaluation, from one period to the next depends on not only its efficiency changes during these two assumed periods, but also technological changes of the population, or boundary shift of the evaluated population in the same periods.

Preliminaries and notations
Our main focus is to develop a model to use ratio data in technologies that have clear economic interpretations.In fact, our focus is to develop ratio VRS and CRS models (R-CRS and R-VRS) that allows the inclusion of ratio measures for inputs and outputs without any change in data.For the production technology T, assume that I = {1,2, … , m} and O = {1,2, … , s} are input and output sets, respectively.Assume that   ∈  and   ∈  are certain input and output sets (certain measures), respectively.In addition, assume that   =  −   and   =  −   are complementary input and output sets (ratio measures), respectively.The elements of T are DMUs expressed as follows: (, ) = (  ,   ,   ,   ) (2.1) Where,  ∈ ℝ +  and  ∈ ℝ +  are the input and output vectors;   ،  ،  and   are sub-vectors related to   ،  ،  and   , respectively; (  ,   ) stands for DMUs, while j ∈ J = {1,2, … , n}.
To define R-CRS technology with ratio measures, it should be said that each ratio measure changes in response to the ratio scaling of certain measures with α≥0.Accordingly, ratio measures have been categorized in different types.
Based on the types of ratio measures, assume that P, U, D, and F are the indices of following four types of ratio measures: constant, downward ratio, upward ratio, ratio.For example, I F and O F are constant ratio input and output sets, respectively, and X F and Y F are corresponding vectors.
In classic models of DEA, upper bound is not imposed for inputs and outputs, which certainly cannot be applied to ratio measures.In ratio measures, the upper bound for inputs and outputs are as follows: Similar to Equation (2.2), the upper bounds are expressed as   = (  ,   ,   ,   ) and   = (  ,   ,   ).
According to the model provided in Section 4, p and q are constants ratio measures for ratio inputs and outputs.

Malmquist Productivity Index with Ratio Data
As it was mentioned in Section 1, the MPI expresses that the regression and progress of a unit from one period to the next depends on not only its efficiency changes during these two periods, but also technological changes of the population, or boundary shift of the evaluated population in the same period.
In other words, regression or progress of a unit depends on not only the performance of the unit itself, but also the performance of the population under evaluation.Therefore, the MPI is defined as follows: MPI = TEX × FS ( are the input and output vectors of the j th DMU ( = 1,2, … , ) in period t+1.According to the MPI, the productivity growth of the O th DMU ( = 1,2, … , ) in period t+1 ratio to t is obtained as follows: In fact, we intended to measure the regression and progress of DMUo within two periods, t and t+1.
The following formula was used to compute efficiency changes between two periods, t and t+1: (   ,   ) (3.4) The following formula was used to compute technological changes (shift of population boundary) between two periods, t and t+1: There are three possibilities in the MPI, if: 1)  0 ,  0 > 1 : A progress occurs in period t+1 ratio to t.

Performance calculation of R-CRS
In DEA programs, only their ratio forms are available for the inputs and outputs.Classical DEA models are developed for certain data; Whereas, ratio data are inevitable in real world problems.In this section, a new approach was used to evaluate the efficiency of the DMUs in the presence of ratio data in two different periods.According to the efficiency value obtained from the evaluated models in Equations (4.8)-(4.11), the MPI value was obtained by the placement in Equation (3.7).

Numerical examples
Table 1 shows a hospital in two periods (the first and second years).In each period, the first column belongs to certain inputs (costs), the second column belongs to certain outputs (treated patients), and the third column belongs to ratio outputs (treatment period).We intended to measure the efficiency of hospitals when the ratio outputs are constants.According to According to Table 2, the last column shows the MPI based on the efficiency of seven DMUs in two periods, t and t+1.According to Section 3, DMU1 with MPI>1 shows a progress in period t+1 ratio to t th .In addition, DMUs 2-7 with MPI<1 show regression in period t+1 ratio to the t th period.

Conclusion
The DEA analyses are usually accepted with certain (absolute) data that represents the values of the inputs and outputs of the DMUs.However, in some cases, the use of ratio data is inevitable.There is a fast growing use of DEA for the measurement of ratio efficiency of the production units.This article addressed the MPI in the presence of ratio data.To this end, the degree of progress and regression of the units in two periods were evaluated and calculated according to the model in the provided example.

Table 2 ,
the MPI was measured for 7 hospitals.

Table 2 :
MPI for seven hospitals in two periods