Malmquist productivity index** **evaluates the efficiency change over time as mentioned by Färe, Grosskopf, & Margaritis, (2011). However, Malmquist productivity index literature has been uneven with some authors assuming constant returns to scale and others allowing for variable returns to scale. This article will present the interpretations of productivity index from *Malmquist Data Envelopment Analysis (DEA)*.

## The dataset for *Malmquist DEA*

The image represents the data collected and put in MS Excel from the annual reports of three hospitals; Apollo Hospital, Fortis Hospital, and Wockhardt Hospital. Moreover, use the same format from MS Excel for pasting into Notepad and make the job easier.

The image for the exported data from MS Excel presented underneath for Malmquist productivity index.

## Instruction file

Input-oriented mode of * constant return to scale (CRS)*–

*Malmquist DEA*changed to output oriented. Furthermore, change the code from “0” to “1”. All other coding is kept the same. Follow the image below for reference.

Quick Note: Use either of VRS and

model, both has no influence overCRSMalmquist DEA. In addition bothvariable returns to scale (VRS)andused to calculate distance changes. Henceforth, bothCRSVRSandgives equal results (Coelli, 2008)CRS

## Findings for Malmquist productivity index

The results are divided into three parts;

- distance summaries,
- Malmquist year-wise index summaries and
- annual and firm mean productivity index.

The analysis showed that four distances measurement or technical efficiencies done, T-1, T, T+1 under the * CRS *technical efficiency (te) and

**VRS**technical efficiency (te) (Coelli, 2008). T-1 shows the technical efficiency of the previous year here being 2013-2014, t is the technical efficiency for the current year which is 2014-2015 and T+1 is the corresponding year which is 2015-2016.

Lastly, a predefined statement saying “Note that t-1 in year 1 and t+1 in the final year not defined”, means if from the image T-1 column of year 1 shows 0.000 value and the same in T+1 for year 3. All the values calculated based on the previous year values. This means that, for the year one, T-1 will be the year 2012-2013 and since the value is not available so, the technical efficiency values are 0.000. Similarly, in case if year 3, T+1 is 2016-2017 and T is 2015-2016, so the value for T+1 is not available and hence the technical efficiency values are 0.000. From this, it implicates that, for year 1, T is the year 2013-2014, year 2 has T for 2014-2015 and year 3, T is for the year 2015-2016.

## Distance summaries (input-oriented)

Again, for year one or for the year 2013-2014, * CRS *model shows technical efficiency for hospital 1 and 3 (Apollo and Wockhardt Hospital) and the

**VRS**technical efficiency shows the same results, 1.000. However, hospital 2, in

*model shows TE of 0.739 or the hospital needs to improve its outputs or decrease its inputs by 29.1% to become technically efficient. In addition,*

**CRS****VRS**TE, hospital 2 has TE of 0.911, indicating increasing returns to scale for Fortis hospital. Similarly, in the second year, T (2014-2015) shows that Fortis hospital has inefficiency value of 0.588, or the hospital needs to improve its output or decrease its input by 41.2%.

On the contrary, hospital 1 and 3 showed efficiency value of 1.000. However, in case of **VRS** TE all the hospitals show the technical efficiency of 1.000. Again for year 3 or 2015-2016, the * CRS *TE (T) shows hospital 2 to be inefficient with a value of 0.572, which means that the hospital needs to either improve its output or decrease its input by 42.8%. While in case of

**VRS**TE all the hospitals show the technical efficiency of 1.000.

## Distance summaries (output-oriented)

Similarly in case of the output-oriented *Malmquist DEA*, the year 2013-2014, * CRS *model shows technical efficiency for hospital 1 and 3 (Apollo and Wockhardt Hospital) of 1.000 and also for the year 2014-2015 and 2015-2016. However hospital 2, in

*model shows TE of 0.739 (2013-2014), 0.588 (2014-2015) and 0.572 (2015-2016). Contrasting the results from the input-oriented*

**CRS***Malmquist DEA*, all the findings are the same except for values of efficacy on the basis

**VRS**model showed, 0.796 for the year 2013-2014, but again the preseeding years showed an efficiency of 1.000.

The graph below shows the variation of results on the basis of the * CRS* TE of the three hospitals.

In this article, the main aim was to interpret and contrast the results from input and output oriented * CRS*–

*Malmquist DEA*. In the next article, we will discuss and interpret the Malmquist index summary.

#### References

- Färe, R., Grosskopf, S., & Margaritis, D. (2011). Malmquist productivity indexes and DEA. In
*International Series in Operations Research and Management Science*(Vol. 164, pp. 127–149). https://doi.org/10.1007/978-1-4419-6151-8_5. - Coelli, T. J. (2008). A Guide to DEAP Version 2.1: A Data Envelopment Analysis (Computer) Program.
*CEPA Working Papers*, 1–50. Retrieved from https://absalon.itslearning.com/data/ku/103018/publications/coelli96.pdf.

MALMQUIST index summaries interpretations from DEA |

### Avishek Majumder

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