Profit Malmquist Index and Its Global Form in the Presence of the Negative Data in DEA

This paper first introduces the allocative and profit efficiency in the presence of the negative data and then presents a new circular index to measure the productivity change of decision making units (DMUs) for the case that the dataset contains the inputs and/or outputs with the negative values in data envelopment analysis (DEA). The proposed index is decomposed into four components in the two stages. The range directional model (RDM) and the proposed efficiencies are used to compute the proposed index and its components. The interpretations of the components are presented. Finally, a numerical example is organized to illustrate the proposed index and its components at three successive periods of time.


Introduction
The Malmquist productivity index which measures the productivity change over time was introduced by Caves et al. [1].Fare et al. [2] developed the Malmquist productivity index that was based on data envelopment analysis (DEA).The Malmquist index can be applied in many fields [3,4].By using Malmquist index, the productivity growth can be decomposed into the efficiency change and technical change components.The second component measures the shift in the technology frontier.
Pastor and Lovell [5] suggested the global Malmquist productivity index that is circular and can be decomposed into the circular components.The global index and the meta-Malmquist index developed in Portela and Thanassoulis [6].In addition, Hosseinzadeh Lotfi et al. developed the new ideas about Malmquist index in [7,8].
Maniadakis and Thanassoulis [9] assumed the input cost vector is known and suggested the cost Malmquist index.Tohidi et al. [10] extended the cost Malmquist index into the profit Malmquist index that is used when the input cost and the output price vector are available.Then, Tohidi and Razavyan [11] proposed the global profit Malmquist index.The cost Malmquist index was also developed in Tohidi et al. [12] and is called the global cost Malmquist productivity index.
Sometimes in the process of production negative inputs and/or outputs may occur.For example, DMUs may generate undesirable outputs.In such a case, negative values can be considered for these undesirable outputs.Instances of systems with negative inputs and/or outputs are explained in [13,14].The traditional DEA models can be applied to compute the Malmquist index with the nonnegative data.They cannot deal with the negative data.Portela et al. [15] presented an approach named the range directional model (RDM) and solved this problem.They calculated the meta-Malmquist index using the RDM model in Portela and Thanassoulis [16].
To investigate the productivity change of DMUs with the negative data, this paper defines the profit and allocative efficiency and introduces an index to measure the productivity changes of DMUs when some inputs and/or outputs are negative and the costs of inputs and the prices of outputs are available.Then, the proposed index is decomposed into some components in the two stages.The range directional model (RDM) and the proposed efficiencies are used to compute the proposed index and its components.

The Range Directional Model
Portela et al. [15] introduced the RDM to compare DMUs under the negative data.Assume that, in time period  ( = 1, . . ., ) and th unit ( = 1, . . ., ) consumes an input vector    = ( The value of  *  in model ( 1) is an inefficiency measure and the RDM efficiency measure of unit  is RDM  (
This paper uses the definition of profit [13] to compute the global profit and the observed profit of DMU  with the negative data.The observed profit of DMU  under the cost and price vectors of time period  can be calculated as OP where MP  (  ,   ,   ,   ) = max{   −    : (,) ∈   ,   > 0,   > 0} and   = conv{ 1 ∪⋅ ⋅ ⋅∪  } is the global production technology [5].The term OP  (  ,   ) =     −     is the observed profit of (  ,   ) under the common price vectors   and   .The ratio MP  (  ,   ,   ,   )/OP  (  ,   ) in the denominator of PM  is the reciprocal to measure of the profit efficiency introduced in Cooper et al. [13] for (  ,   ) under the common price vectors.This ratio measures the distance between the observed profit OP  (  ,   ) and the common profit boundary and will have a minimum value of 1.A value greater than 1 of PM  index indicates the productivity regress and a value less than 1 implies the productivity progress between  and  + 1.A value of 1 indicates that the productivity remains unchanged.Because of the limitations of the DEA models with the negative inputs and outputs [17,18], we cannot investigate the productivity change of DMUs with the negative data by using tradition Malmquist indices.To this end, the next section proposes an appropriate global profit Malmquist index when some inputs or/and outputs are negative.

The Global Profit Malmquist Index with the Negative Data
In this section we assume that there are some negative inputs and/or negative outputs and evaluate the productivity changes of DMUs by using the PM  index under the VRS technology.We denote the profit of the ideal point of time period  by IP  that the superscript  on IP  indicates the profit is computed under the cost and the price vectors of time .By using the definition of the profit, we will have Now we define the profit and allocative efficiency of a DMU with the negative data.
Definition 1.The measure of the profit efficiency for (  ,   ) under the price vectors of time  is as By using (3) it is clear that the value of PE  (  ,   ,   ,   ) is equal to or less than one.As the profit efficiency is less than 1 it may be because production takes place at the wrong input and/or output mix in light of the input costs and the output prices; this is captured by using the measure of the allocative efficiency which is defined as follows.
Definition 2. Let (  * ,   * ) be the target DMU of (  ,   ) in time period  using RDM.The allocative efficiency of (  ,   ) can be defined as follows: If profit efficiency is less than 1 and it is because production is based on excessive input or shortage output usage, we can capture it by using the RDM efficiency measure that was denoted by RDM  ( where GIP  is the profit of the global ideal point and the superscript  indicates that the profit of this point is computed under the common price vectors   and   .The ratio (MP  (  ,   ,   ,   )−GIP  )/(OP  (  ,   )−GIP  ) is the profit efficiency defined in Definition 1 under the common cost and price vectors.When the PM  index has the value greater than 1, it means that the productivity of unit  has improved from the time  to  + 1.The productivity has declined when the value of the PM  index is below 1 and remains unchanged if PM  = 1.The PM  index is circular and it can be decomposed into four circular components as shown in the next section.

Decomposition of the PM 𝐺 Index
In the first stage the PM  index is decomposed into two PEC  and PTC  as follows: The term outside the brackets in the right-hand side of (7) represents the profit efficiency change (PEC  ) component of the unit under evaluation from  to  + 1 and the term inside the brackets provides the profit frontier shift (the technical change) between the periods  and  + 1 under the VRS production technologies oftwo times t and  + 1 (PTC  ).
The PEC  and PTC  components of the PM  index can themselves be decomposed.
The Decomposition of the   .The PEC  component can be decomposed into two components as follows: where GIP  and GIP +1 are the profit of the global point under the costs and prices of the periods  and  + 1, respectively.The first ratio in the right-hand side of ( 8) is the allocative efficiency change (AEC  ) components of the PM  index; and the second ratio is the RDM within-period-efficiency change (REC  ) of the unit that is under evaluation from  to  + 1; the RDM within-period-efficiency change was introduced by Portela et al. [15].
The Decomposition of   .We can decompose the PTC  into two components as follows: The first term in the above decomposition measures the frontier shift between the VRS frontiers of times  and  + 1 and the technical change (TC  ), along the ray ( +1  ,  +1  ) [16].The second term is a residual price effect (PE  ) part.The numerical values of the components of the PM  index obtained in this section are interpreted in the similar manner as the index itself; a value below 1 indicates regress, greater than 1 indicates progress, and 1 indicates that performance stayed constant.

The Circularity Property of 𝑃𝑀 𝐺 and All of Its Components.
Circularity is a prominent property of the PM  index and all of its components in the presence of the negative data.To show this property, the following theorem is stated.Theorem 3.For every DMU j ( = 1, . . ., ), in three successive periods, = (( In other words, the global profit Malmquist change from period  to  + 2 is the product of the successive global profit Malmquist change from period  to  + 1 and from the period +1 to +2.Similarly, we can show that all of the components are circular.
The above properties show that in the presence of the negative data the PM  index and its components are appropriate indices in the fact that they link in clear way productivity change indices over successive time periods.

The Computation of the Proposed Index and Its Components
This section prepares some DEA models to compute the PM  index and its components.The observed profit OP  (  ,   ) is computed as Similarly, the observed profits denoted by OP  ( +1 ,  +1 ), OP  (  ,   ), and OP +1 ( +1 ,  +1 ) are, respectively, We can compute MP  (  ,   ,   ,   ) for unit  by using the following model: Similarly, we will have The term MP +1 ( +1 ,  +1 ,  +1 ,  +1 ) can be computed by using model (11) after replacing the time period  with  + 1.  1) for the case that the production technology formed from all DMUs observed in all time periods (  ).The proposed model is shown in (13).In fact, the optimal value of model (13) ) iscomputed using model (13) after replacing the time period  with  + 1.
By replacing the values obtained from the above models in the formulations of PM  index and its components, the values of them can be computed.A value greater than 1 of PM  index and its components indicates the regress and a value less than 1 implies the progress between periods  and  + 1.A value equal to 1 indicates that there is not any change between two time periods.

Numerical Example
This section illustrates the property of PM  and its components, for example, the circularity property, using a numerical example.Table 1 shows units DMU1-DMU4 with one input (I) and two outputs (O1 and O2) and their cost (  ) and prices (  1 and   2 ) for 3 successive periods.To compute the PM  indices and their components by using the data in Table 1, suppose that the preferences about the input costs and output prices are available and have been specified by decision-makers as   =   = (1/3) ( = 1, 2, 3).Therefore, the common costs and prices are obtained by using the decision-makers' preferences: By using the data of periods 1, 2, and 3, the inputs and the outputs of the global ideal point will be 2, −1, and 7, respectively.
Table 2 shows PM  , , PEC  , , and TEC  , of all of the DMUs for  = 1, 2,  = 2, 3, and  ̸ = .For instance, the columns 2, 3, and 4 in Table 2 show PM  1,2 , PM  2,3 , and PM  1,3 , respectively.For DMU2, as an example, we have PM G 1,2 = 1.3049 > 1.That is, the productivity of DMU2 in period 2 is more than its productivity in period 1 and hence the productivity has been improved from the time period 1 to 2.

Table 1 :
Input, outputs, and their cost and prices for 3 successive periods.

Table 2 :
The results of the global profit Malmquist, profit efficiency change, and profit technical change.

Table 3 :
The results of the RDM within-period-efficiency and allocative efficiency change.

Table 4 :
The results of the frontier shift between the VRS frontiers and price effect.