Productivity Center and Son Using the Malmquist Global Index

Subject Areas : International Journal of Data Envelopment Analysis

Esmael Mombini ¹ (Department of mathematics, Ramhormoz Branch, Islamic Azad University)

Received: 2024-02-14 Accepted : 2024-07-13 Published : 2024-10-01

Keywords: Data envelopment Analysis , progres, Improvements , regressions , cost Efficiency , Profitability , Malmquist Global Index ,

Abstract :

Data Envelopment Analysis (DEA) is a Nonparametric method for measuring and does not need to have and calculate the production function, which is often difficult to calculate. In this article, we evaluate the units under investigation in terms of cost and production efficiency in several time periods and the progress(Improvements) or regression of each unit.for this purpose,we use the method based on solving linear programing models using Malmquist productivity index .Data envelopment analysis is a non-parametric method for measuring the performance of decision making units.Finally, by designing and solving a numerical example, we emphasize and test the applicability of the material presented in this article.

References:

[1] Ahangari, T., Rostamy - Malkhalifeh, M. Measurement of profit inefficiency in presence of interval data using the directional distance function. Received 16 March 2016; accepted 26 May 2016.
[2] Allahyar, M. & Rostamy-Malkhalifeh, M. (2014). An improved approach for estimating returns to scale in DEA, Bulletin of the Malaysian Mathematical Sciences Society, (2) 37 (4) 1185 - 1194.
[3] Charnes A, Cooper ww and Rhodse, 1978. “Measuring the efficiency of decision making unit”, European Journal of Operation Research: 429 - 444.
[4] Emrouznejad, A. Yang, G-L. A framework for measuring global malmquist - Leunberger productivity index with CO2 emissions on Chinese manufacturing industries. Energy, 115, (2016), 840 - 856.
[5] Fare and Grosskopf S, Norris M, Zhang Z. 1994. productivity growth, technical progress and efficiency change in industrialized countries”, American economic Review, 84 (1): 66 - 83.
[6] Farrell, M.J., 1957. The measurement of productive efficiency. Journal of the Royal statistical society, series A 120: 253 - 281.
[7] FengHe. Xiaoning Xu. Rong Chen. Na Zhang. Sensitivity and stability analysis in DEA with bounded uncertainty. Springer-Verlag. Berlin Hiedelberg 2015.
[8] Kazutoshi, A. Minamide, M. Kazuyuki, S. Shi, J. Monotonivity of minimum distance inefficiency measures for Data Envelopment Analysis. European Journal of Operational Research… (2016) 1 - 12.
[9] Lee, C.Y., & Johnson, A.L., (2014), Proactive data envelopment analysis: Effective production and capacity expansion in stochastic environments operational research, 232, 537 - 545.
[10] Maryam Allahyar, Mohsen Rostamy-malkhalifeh. Negative data in data envelopment Analysis: Efficiency analysis and estimating returns to sale computers & Industrial Engineering. (2015. 50360-8352 (15)… 41 - 8.
[11] Mehdi toloo,Nazila Aghayi,Mohsen Rostamy-Malkhalifeh.Measuring Overall Profit Efficiency with interval data[J].Applied Mathematics and Computation,2008,201:640-649.
[12] M. Khodabakhshi, Y. Gholami, H. Kheirollahi, An additive model approach for estimating returns to scale in imprecise data envelopment Analysis, original Research Article Applied Mathematical Modelling. 34 (2010) 1247 - 1257.
[13] Pejman Peykani, Emran Mohammadi, Ali Emrouznejad, Mirsaman Pishvaee, Mohsen Rostamy-Malkhalifeh. Fuzzy Data Envelpment Analysis: An Adjustable Approach Expert Systems With Applications (2019) 50957-4174 (19) 30439 - 7.
[14] Pejman Peykani,Emran Mohammadi,Mohsen Rostamy-Malkhalifeh,Farhad Hosseinzadeh Lotfi.Fuzzy Data Envelopment Analysis Approach for ranking of stocks with an Application to Tehran stock Exchange.Advances in mathematical finonce &Applications ,4(1),(2019),31-43.
[15] R.Banker,A.Charnes,W.W.Cooper,SomeModels for estimating technical and scale inefficiencies in data
[16] Tohidi, G. Razavyan, S. Tohidnia, S. A global cost molmquist productivity index using data envelopment analysis, Journal of the operational Research Society, (2012). 63, 72 - 78.

Full-Text:



 

Available online at http://ijdea.srbiau.ac.ir

 

Int. J. Data Envelopment Analysis (ISSN 2345-458X)

 

Vol. 11, No. 4, Year 2023 Article ID IJDEA-00422, Pages 31-37

Research Article

 

 

International Journal of Data Envelopment Analysis                                                                            Science and Research Branch (IAU)

 

 

 

Progress and retrogression of costs Profitability by global Malmquist index

Esmaeil Mombini¹

1 Department of Mathematics, Ramhormoz branch, Islamic Azad University, Ramhormoz, Iran.

 

Received 24 March 2023, Accepted 19 July 2023 

 

Abstract

Data Envelopment Analysis (DEA) is a nonparametric method for measuring and does not need to have and calculate the production function, which is often difficult to calculate. In this article, we evaluate the units under investigation in terms of cost and production efficiency in several time periods and the progress (Improvements) or regression of each unit for this purpose, we use the method based on solving linear programing models using Malmquist productivity index. Data envelopment analysis is a non-parametric method for measuring the performance of decision-making units. Finally, by designing and solving a numerical example, we emphasize and test the applicability of the material presented in this article.

 

Keywords: Data envelopment Analysis (DEA), Progress, Improvements, Regressions, Cost

efficiency, Profitability, Malmquist Global Index.

---

[1]  Corresponding author: Email: [e-mail]

 

1.       Introduction

The concept of cost efficiency is that Measuring the ability of a DMU to produce outputs at the lowest input cost, Farrell began in 1957. He defined cost efficiency as the minimum cost to actual cost [6].

Lee and Johnson proposed in 2014 a short-term capacity planning approach [9]. Rostamy and Ahangari (2016) investigated earnings inefficiency with interval data [1]. Kazutoshi Ando and his colleagues (2016) measured the minimum inefficiency gap in data envelopment analysis [8]. Rostamy and his colleagues start working on inefficiency in 2019 . In 2015, Feng He and his colleagues examined the stability and sensitivity radius for uncertain boundaries [7]. Allahyar and Rostamy reviewed the performance analysis for negative data and performance-based scalability in 2015 very interesting work on economic efficiency was done fuzzily. In 2019, Rahmani and many other colleagues conducted activities and research with fuzzy data [13,14].

In 2019, Peykani and Mohammadi proposed a domain-oriented size model of the feasibility of ranking warehouses in the presence of negative and non-linear data [13,14].

In 2019 Akhlaghi and Rostamy used integer linear programming to determine the most efficient decision-making unit. Another key issue in data envelopment analysis is the Malmquist Productivity Index. The Malmquist Index was first introduced in 1953 as a qualitative index in the analysis of consumer inputs by Stan Malmquist and became well-known among researchers.

Malmquist Index has some drawbacks despite being widely used, including when linear programming techniques. For the calculation of the index, the invariance occurs which is a fundamental problem in the data structure. And sufficient data were obtained to resolve the problem, [15]. Pastor and Lovell showed that the source of the problem is the properties of adjacent step technologies in the index building [5].

However, each of the Malmquist index does not solve all three problems. Then a new index built from all data at all stages and periods of all manufacturers, its technology was presented as the Global Malmquist that has a circular feature and a It measures the magnitude of productivity change. The Malmquist Global Index can solve all three problems by specifying a fixed boundary. In 2012, Tohidi et al. used the Malmquist Global in cost estimation. [16] Emrooznejad and Yang used global Malmquist to investigate carbon dioxide emissions. [4] Tohidi and Razavyan used Malmquist Global Index in 2012 to calculate the profit of decision-making units. Not all articles present and research on the cost productivity of economic units were reviewed.

 

2.       Basic Model in Data Envelopment Analysis

A nonparametric method for measuring the performance of decision-making units (DMUs) is Data Envelopment Analysis (DEA). Charnes, Cooper, and Rhodes provided the first linear models for this measurement (CCR model) [3].

The original CCR model only used fixed-rate (CRS) technology, developed by Banker et al.

If n is the decision unit () that uses m to produce the output S and the vectors and are the inputs and outputs, respectively. Therefore, ,. Input-based technical performance for  is listed below.

                             (1)

 

2.1.                      Cost efficiency

Consider the problem of minimizing the cost of a company with inputs to produce outputs and cost vector . We obtain the minimum total cost of producing the output vector .

                               (2)

Cost efficiency is the ratio of the minimum cost to the real cost.

                (3)

 

2.2.                      Calculating cost productivity improvement

Following the implementation of the CCR economic performance model, we seek to determine whether different decision makers have improved, regressed or unchanged in terms of their performance over different time periods. Accordingly, using the Malmquist Index, the performance improvement or regression of the units is analyzed. Malmquist index measurement requires calculation of distance functions.

 To solve these functions, we can use the linear programming method of comprehensive data analysis. In this regard, for each decision-making unit, according to the Malmquist Index between two time periods t and t + 1, four distance functions must be calculated, which in turn requires solving four linear programming problems. Assuming constant-scale returns (assumed by Fare, Graskov, Norris, and Zhang in their analysis) [12].

Four issues will be addressed and solved for cost efficiency.

                           (4)

(Unit in time L – border in time K)

 

Cost effectiveness:

                                   (5)

We can calculate the Malmquist Index for the cost efficiency of  using the following equation.

            (6)

·        Indicates the improvement in cost efficiency of DMU0 from time period t to time period t + 1.

·       Indicates the recovery of DMU0 cost efficiency from time period t to time period t + 1.

·       Indicates the unchanged cost efficiency of DMU0 from time period t to time period t + 1.

 

2.3.                      Calculating cost productivity improvement and regression with the Malmquist Global Index

If the units are evaluated over time periods , in the Malmquist Global Method, we examine all units in one place and the cost efficiency model for  at time k is given below. Be. In this section, each DMU is compared to itself over different time periods, and it has progressed in two periods, such as p and q.

   (7)

And the cost efficiency comes from the following relationship:

                                  (8)

If we consider the productivity of  at time p compare to time q, we consider the solution of the model K = p, q and two problems arise and are solved. The Malmquist Global Index is written as follows:

                             (9)

·       If  Let the unit at time p have progressed to q

·        If the unit at p has regressed to q.

·        Let unit be unchanged at p relative to q.

 

3.       Numerical examples

 

For example, we consider 10 decision-making units with three inputs and two outputs over the three time periods which are listed in Table 1. Table 2 shows the cost, income, and profit efficiencies for all units over the three time periods.

Tables 3,4 &5 show the proposed models for calculate the improvement and regression of units with the Malmquist index from periods  to  to  and  to .

 

 

 

Table 1: The inputs and outputs data in three-time period for ten DMUs

 

 

 

 

Table 2: Cost Efficiency in three-time period for ten DMUs

1	6	5	4		7	3	6		4	2	6		3	4	3		3	4	3		5	4	5		6	1	7		3	2	6		4	4	5		3	5
7	2	5	6		4	3	5		6	4	5		4	1	7		6	6	3		4	6	4		5	4	2		3	4	3		7	5	6		3	4
4	5	4	1		3	6	2		3	3	2		3	7	4		7	5	6		4	5	7		3	6	5		3	6	5		6	7	4		6	5
8	6	3	4		5	3	4		1	6	3		4	5	6		5	7	7		3	4	5		6	5	4		6	5	3		5	6	7		4	4
4	5	7	6		4	6	7		5	3	4		6	4	5		7	8	3		5	4	6		7	3	6		4	5	3		6	7	4		5	6

 

 

Table 3: Cost productivity improvement and regression with the Malmquist Index ( to )

Cost				DMU
0/673	0/618	0/750	1
0/335	0/632	0/760	2
0/579	1	0/765	3
0/571	0/912	0/660	4
0/693	0/608	0/531	5
0/772	0/708	1	6
0/523	1	0/593	7
0/275	0/678	1	8
1	0/570	0/828	9
1	0/647	0/928	10

CMPIO	DMU
0/98	1
0/84	2
0/08	3
0/31	4
1/11	5
0/28	6
0/76	7
4/88	8
8/86	9
1/10	10

 

Table 4:  Cost productivity improvement and regression with the Malmquist Index ( to )

CMPIO	DMU
32/48	1
2/92	2
0/89	3
8/10	4
0/28	5
3/38	6
0/41	7
0/88	8
1/33	9
1/86	10

 

 

4.       Conclusion

 

Calculating the Malmquist Productivity Index is a very useful and applicable system for determining the progress and regress of a system and helps managers in future decisions and policies of the system and leads to improved productivity. The Global Malmquist Productivity Index helps managers track the progress and remediation of a system or organization over two time periods, across time periods, and improve decision making and policy making. And better evaluate the efficiency of a unit.

 

 

 

 

 

 

References

 

[1] Ahangari, T., Rostamy - Malkhalifeh, M. Measurement of profit inefficiency in presence of interval data using the directional distance function. Received 16 March 2016; accepted 26 May 2016.

[2] Allahyar, M. & Rostamy-Malkhalifeh, M. (2014). An improved approach for estimating returns to scale in DEA, Bulletin of the Malaysian Mathematical Sciences Society, (2) 37 (4) 1185 - 1194.

[3] Charnes A, Cooper ww and Rhodse, 1978. “Measuring the efficiency of decision-making unit”, European Journal of Operation Research: 429 - 444.

[4] Emrouznejad, A. Yang, G-L. A framework for measuring global Malmquist - Lupberger productivity index with CO2 emissions on Chinese manufacturing industries. Energy, 115, (2016), 840 - 856.

[5] Fare and Grosskopf S, Norris M, Zhang Z. 1994. productivity growth, technical progress and efficiency change in industrialized countries”, American economic Review, 84 (1): 66 - 83.

[6] Farrell, M.J., 1957. The measurement of productive efficiency. Journal of the Royal statistical society, series A 120: 253 - 281.

[7] FengHe. Xiaoning Xu. Rong Chen. Na Zhang. Sensitivity and stability analysis in DEA with bounded uncertainty. Springer-Verlag. Berlin Hiedelberg 2015.

[8] Kazutoshi, A. Minamide, M. Kazuyuki, S. Shi, J. Monotonivity of minimum distance inefficiency measures for Data Envelopment Analysis. European Journal of Operational Research… (2016) 1 - 12.

[9] Lee, C.Y., & Johnson, A.L., (2014), Proactive data envelopment analysis: Effective production and capacity expansion in stochastic environments operational research, 232, 537 - 545.

[10] Maryam Allahyar, Mohsen Rostamy-malkhalifeh. Negative data in data envelopment Analysis: Efficiency analysis and estimating returns to sale computers & Industrial Engineering. (2015. 50360-8352 (15) … 41 - 8.

[11] Mehdi toloo,Nazila Aghayi, Mohsen Rostamy-Malkhalifeh. Measuring Overall Profit Efficiency with interval data[J]. Applied Mathematics and Computation,2008,201:640-649.

[12] M. Khodabakhshi, Y. Gholami, H. Kheirollahi, an additive model approach for estimating returns to scale in imprecise data envelopment Analysis, original Research Article Applied Mathematical Modelling. 34 (2010) 1247 - 1257.

[13] Pejman Peykani, Emran Mohammadi, Ali Emrouznejad, Mirsaman Pishvaee, Mohsen Rostamy-Malkhalifeh. Fuzzy Data Envelpment Analysis: An Adjustable Approach Expert Systems with Applications (2019) 50957-4174 (19) 30439 - 7.

[14] Pejman Peykani,Emran Mohammadi, Mohsen Rostamy-Malkhalifeh, Farhad Hosseinzadeh Lotfi. Fuzzy Data Envelopment Analysis Approach for ranking of stocks with an Application to Tehran stock Exchange. Advances in mathematical finonce &Applications ,4(1), (2019),31-43.

[15] R.Banker, A.Charnes, W.W.Cooper, Some Models for estimating technical and scale inefficiencies in data

[16] Tohidi, G. Razavyan, S. Tohidnia, S. A global cost Malmquist productivity index using data envelopment analysis, Journal of the operational Research Society, (2012). 63, 72 - 78.