# INTERVAL LEFT AND RIGHT RETURNS TO SCALES.

Byline: Somayeh Sargolzaei and Ali Payan

ABSTRACT: One of the discussions of data envelopment analysis (DEA) is the returns to scale. Regarding the fact that in the real life issue we usually face undetermined data the study of phenomena with determined data does not seem rational. In th is regard considering the data as interval has special importance. Therefore the purpose of this study is to extend the methods of determining the types of left and right returns to scales in DEA into inexact interval concepts. In this study we extend the represented models by Eslami and Khoveyni [1] in determining the types of left and right returns to scales at the presence of the Interval Data. First the efficient units are evaluated in pessimistic and optimistic situations. Then we determine the types of left and right returns to scales at the two situations on the efficient units. Finally an example is provided to further explain the method.

Keywords: Data Envelopment Analysis Left and Right Returns to Scales Interval Data Interval Efficiency

INTRODUCTION

Charles et al. [2] extended Farrell's view [3] on the evaluation of performance and represented a model which is able to measure the efficiency of decision making units (DMUs) with multiple inputs and outputs. The presented method became known as data envelopment analysis (DEA). Banker et al. [4] presented another model that performed the analysis of units through the assumption of variable returns to scale which is known as the BCC model.

Returns to scale represents the link between changes in inputs and outputs of a system. One of the potentials of the DEA is the use of different models corresponding to different returns to scale and measuring returns to scale of units. Constant returns to scale means that any multiplication of input produces the same multiplied output. CCR model [2] supposes constant returns to scale for DMUs. Therefore the small and big units are compared together. Variable returns to scale means that any multiplication of input produces the same more or less multiplied output. Banker et al. model (BCC model) assumes variable returns to scale (upward stable or downward). Banker [5] estimated returns to scale of a unit using the optimal solution of the CCR model. Banker and Thrall [6] estimated returns to scale of a DMU through solving dual model of BCC and gaining

In these methods returns to scale of the extreme efficient units cannot be estimated. Golany and Yu [7] presented a method that estimated left and right returns to scales of efficient units and estimated the returns to scale of inefficient DMU units through picturing them on the efficient frontier.

One of the research topics in returns to scale is left and right return to scale. Right returns to scale is analyzed by the feature of hyperplane that passes the unit which is defined by increasing the level of changes in the inputs and outputs of the unit and left returns to scale is analyzed by the feature of hyperplane that passes the unit which is defined by decreasing the level of changes in the inputs and outputs of the unit. Golany and Yu [7] when studying the theory and analyzing return on scale in DEA presented models through which it is possible to determine the left and right returns to scales. Of course the mentioned research is not always feasible for all units. Jahanshahloo and Soleimani-damaneh [8] presented the basic concepts and methods of calculating returns to scale with BCC model. Khodabakhshi et al. [9] proved theorems for determining left and right returns to scales being in the design of models.

Eslami and Khoveyni [1] presented a method that measures left and right returns to scales for all efficient units. One of the advantages of the proposed method is to determine the types of left and right returns to scales is that the proposed method is performable for all target DMUs while the method of Golany and Yu [7] is not always performable. Another advantage of Eslami and Khoveyni [1] is that it is possible to measure the left and right returns to scale values of efficient DMUs.

The term inaccurate data in DEA means that the inputs and outputs cannot be precisely obtained due to uncertainty. The only thing we know is that they are all defined within the upper bound and lower bound given by intervals. Copper et al. [10] discussed about interval data and considered the combination of ordinal data and interval data as inaccurate data and presented the IDEA (Inaccurate DEA). Thus the efficiency should be inaccurate. Despotis et al. [11] achieved the interval efficiency through interval data. Here with the inspiration from Eslami and Khoveyni [1] who presented a method in determining the types of left and right returns to scales we will extend their method at the presence of interval data.

This paper is arranged as follows: In section 2 we introduce the methods of determining the types of left and right returns to scale. Next after reviewing the data envelopment analysis of interval data the left and right returns to scales at the presence of interval data is introduced. Finally an example will be provided to explain the proposed method. The final section is related to the conclusion and suggestions for further studies.

EXAMPLE

In this example we suppose 5 decision making units with 2 inputs and 2 outputs.

Based on the interval data the efficiency of the interval is by obtained models (4) and (5) listed in Table (1). In order to determine the left and right returns to scales in the pessimistic mode only DMU 5 is analyzed because this decision making unit is efficient in the upper and lower bounds. Also in order to determine the left and right returns to scale in the optimistic mode al DMUs are analyzed except DMU 4 . The results are listed in Table (2) and (3).

CONCLUSION

In this article first through considering Death interval data the BCC multiplier model is extended into optimistic and pessimistic modes by the help of which the interval efficiency of the units is obtained. Thus the models of left and right returns to scales by Eslami and Khoveyni [1] are extended into optimistic and pessimistic modes. The results of the example indicate different types of left and right returns to scale at the presence of interval data. The measurement of values of returns to scale with interval data are the future studies of the writers. Also the presented model in this study is analyzable for sequential stochastic and fuzzy data.

Table 1. Interval data and interval efficiency

###DMU###x1###x2###y1###y2###Efficiency

###1###1215###0.210.48###138144###2122###0.221

###2###1017###0.100.70###143159###2835###0.2271

###3###412###0.160.35###157198###2129###0.8231

###4###1922###0.120.19###158181###2125###0.4450.907

###5###1415###0.060.09###157161###2840###11

Table 2. Interval right returns to scale models (6) and (7)

###DMU###t

###l u

###t###

###l###u

###Optimistic###Pessimistic###Interval

###RTS###RTS###Right RTS

###1###2.406E+9###1.001###-###DRS###-

###2###1.00E+10###1.001###-###DRS###-

###3###1.00E+10###1.001###-###DRS###-

###4###-###-###-

###5###0.168 1.00E+10###1.0011.001###IRS###DRS###-

Table 3. Interval left returns to scale Models (8) and (9)

###DMU###t t

###l u

###l u

###Optimistic###Pessimistic###Interval Left

###RTS###RTS###RTS

###1###1.000###1.0000E-6###-###IRS###-

###2###1.001###1.0000E-6###-###IRS###-

###3###1.001###1.0000E-6###-###IRS###-

###4###-###-###-

###5###[- 0.733 - 1.000 ]###1.0000E-6 1.0000E-6###IRS###IRS###IRS

REFERENCES

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