Development of intuitionistic fuzzy integrated super-efﬁciency SBM model

: Slack based measure (SBM) model determines the performance efﬁciency of homo-geneous decision making units (DMUs) and also determines the efﬁcient and inefﬁcient DMUs. super-efﬁciency SBM (SESBM) model determines the efﬁciency of efﬁcient DMUs. Guo et al. [7] proposed an integrated super-efﬁciency SBM (ISESBM) model which determines the efﬁciency and super-efﬁciency of DMUs using one model. In conventional ISESBM, the data is crisp. But it ﬂuctuates in the real world applications. Such data can take the form of fuzzy / interval / intuitionistic fuzzy (IF) numbers. In this paper, we propose an IF ISESBM (IFISESBM) model using expected value of intuitionistic fuzzy numbers (IFNs). Finally, a health sector application of the proposed model is presented with two IF inputs and two IF outputs.


Introduction
Slack based measure (SBM) model which is non-radial measure model based on slacks proposed by Tone et al. [12] determines the efficiencies, efficient and inefficient decision making units (DMUs). Tone et al. [13] proposed super-efficiency SBM (SESBM) model to determine the efficient DMUs. Du et al. [6] extended the SESBM model to additive SESBM model. In this model, firstly efficient DMUs are determined and then additive SESBM model is applied to determine the efficiencies of efficient DMUs. Guo et al. [7] proposed a model to determine the efficiencies and super efficiencies using one model which is known as integrated SESBM (ISESBM) model. Intuitionistic fuzzy set (IFS) theory proposed by Atanassov [2] is an extension of fuzzy set theory and have been found to be more useful to deal with vagueness/fluctuation. The IFS considers both the acceptance value and rejection value of an element such that the sum of both values is less than one, i.e., it may have hesitation. Since its invention/inception, the IFS theory has received more and more attention and has been used in a wide range of applications, such as, reliability [11], logic programming [3], decision making [8], medical diagnosis [4], pattern recognition [5]. Puri and Yadav [10] proposed IF optimistic-pessimistic DEA models to determine the efficiencies of DMUs in optimistic and pessimistic situations.
In this paper, community health centers (CHCs) have been taken to determine the efficiencies and slacks. Since the managers or other authorities reorganize facilities: non-medical staff and medical staff time to time, the fluctuation occurs in non-medical staff and medical staff, i.e., fluctuation in input data. Patients leave the hospital due to insufficient resources such as available beds for hospitalization and lackness to provide appropriate care etc. Hence fluctuation occurs in output data. Due to this, in order to deal with fluctuation in CHCs, the input data and output data are taken as intuitionistic fuzzy numbers (IFNs). So, fluctuation in input data and output data at hospital level can be well taken as IFN. In this paper, we extend ISESBM model to intuitionistic fuzzy ISESBM (IFISESBM) model using expected value of intuitionistic fuzzy numbers.
The rest of the paper is organized as follows: Section 2 presents preliminaries required to develop the model. Section 3 presents the proposed IFISESBM model. Section 4 presents an application to the health sector to illustrate the proposed model. Section 5 concludes the findings of this paper.

Intuitionistic fuzzy number (IFN)
x ∈ R} be an IFS with its membership function µÃI and nonmembership function νÃI , where R is the set of real numbers. ThenÃ I is called an IFN [9] if the following conditions hold: where a m is the mean value ofÃ I ; a m − a l and a u − a m are the left and right hand spreads of membership function µÃI respectively; a m − a l and a u − a m are the left and right hand spreads of hesitation function πÃI (x) respectively; g 1 and h 1 are piecewise continuous, strictly increasing and strictly decreasing functions in [a l , a m ) and (a m , a u ] respectively; g 2 and h 2 are piecewise continuous, strictly decreasing and strictly increasing functions in [a l , a m ) and (a m , a u ] respectively. Its graphical representation is given in Figure 1.
Its graphical representation is given in Figure 2.

Expected interval (EI) of a TIFN
The EI of TIFNÃ I = (a l , a m , a u ; a l , a m , a u ) is defined as follows: The expected value (EV) of a TIFNÃ I = (a l , a m , a u ; a l , a m , a u ) is defined as follows:

Model 1
where s − ijo and s + rjo are the inefficiency slacks, t + ijo and t − rjo are the super-efficiency slacks. In ISESBM model, firstly the super-efficiency slacks are determined and then inefficiency slacks are determined. Let s − * ijo , s + * rjo , t + * ijo and t − * rjo be the optimal values of s − ijo , s + rjo , t + ijo and t − rjo respectively. The posterior efficiency (PE) of Model 1 is defined as follows [7] If ξ * jo > 1, then DM U jo is ISESBM efficient and if ξ * jo ≤ 1, then DM U jo is ISESBM inefficient [7].

Intuitionistic fuzzy ISESBM model
In conventional ISESBM the input data and output data are crisp values. But in the real world applications, these data may have intuitionistic fuzzy values [1]. Therefore, we have taken IF input-output data as TIFNs. Letx I ij andỹ I rj be the ith IF input and rth IF output respectively for DM U j . Then we have the following model (Model 2) The posterior efficiency (PE) of Model 2 is defined as follows Let the IF input and IF output be TIFNs: (ỹ I rj ) = (y l rj , y m rj , y u rj ; y l rj , y m rj , y u rj ), (ỹ I rjo ) = (y l rjo , y m rjo , y u rjo ; y l rjo , y m rjo , y u rjo ). Then we get Model 3.

Model 3
(y l rj , y m rj , y u rj ; y l rj , y m rj , y u rj )µ jo = (y l rjo , y m rjo , y u rjo ; y l rjo , y m rjo , y u rjo Taking EV of TIFNs, we get Model 4 from Model 3.

Model 4
min EV (ξ jo ) = EV (y l rj , y m rj , y u rj ; y l rj , y m rj , y u rj )µ jo = EV (y l rjo , y m rjo , y u rjo ; y l rjo , y m rjo , y u rjo We get Model 5 from Model 4 using expected value of TIFNs, which is given in Definition 7.

Conclusion
The real world applications data have some degrees of fluctuations. To deal with such data, we have considered them as TIFNs. In this paper, we extended integrated super-efficiency SBM model to intuitionistic fuzzy integrated super effciency SBM (IFISESBM) model. IFISESBM model determines the efficiencies and super efficiencies of DMUs using one model in IF environment. To ensure the validity of the proposed models, we have considered the performance of CHCs with two IF inputs and two IF outputs (Tables 1 and 2). PIFISESBM model is more effective for real world applications. PIFISESBM model also determines the efficient and inefficient CHCs. These efficiencies and input-output slacks provide extra information to the decision maker. This paper has some limitations. The proposed models are studied under the SBM model. The uncertainty in this paper is limited to TIFNs. We plan to extend these models to the other DEA models and also plan to use the trapezoidal IFNs and interval valued intuitionistic fuzzy sets to determine the efficiencies of real world applications.