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Self-assessment of parallel network systems with intuitionistic fuzzy data: a case study

  • Zahra Ameri
  • Shib Sankar SanaEmail author
  • Reza Sheikh
Methodologies and Application

Abstract

In hospital management, it is often observed that each decision making unit is under different and uncontrollable conditions. Consequently, comparing the units to each other cannot necessarily identify ineffectiveness of the units. This paper suggests the manager of a hospital to implement self-assessment technique for measuring the efficiency. The objective of this paper is to measure the efficiency of parallel system in intuitionistic fuzzy environment by introducing self-assessment technique which is the best type of evaluation where the maximum stability of the conditions is considered. The proposed model evaluates the performance of system and processes and determines the reasons of inefficiency in order to reduce the risk of lack of information about decision and deal with vague and complex conditions in real world.

Keywords

Self-assessment Parallel network systems Intuitionistic fuzzy parallel DEA 

1 Introduction

Any organization for proper management should use scientific patterns for performance evaluation in order to measure the amount of effort and results of its function. Although there are many ways to evaluate the performance of an organization, they do not show the degree of improvement in each unit during successive operational periods according to productive resources received in terms of efficiency and ranking. For instance, one of the most important tools for evaluating the performance of an organization proposed by Charnes et al. (1978) is data envelopment analysis (DEA) technique that shows the organization’s situation at a specific time period. Numerical size of efficiency index will change by changing time interval. If we want to specify the performance of an organization in different time intervals, performance model should be designated to demonstrate the performance of the organization at the time interval. Although the use of DEA technique to evaluate several units together has been highly regarded, its application for a unit during different times provides self-assessment capability.

Regardless of the consideration of time variations in performance evaluation, by applying network DEA (NDEA) models, we can identify reasons of efficiency and inefficiency that help managers to find out the optimal decisions. For the first time, Färe (1991) took the operations of the internal processes into account for measuring the system efficiency. The parallel structure is one of the basic structures in a network system. Castelli et al. (2004) proposed hierarchical structures where each system was constructed of consecutive stages of parallel processes. Kao (2009) developed a slack-based measure (SBM) model to decompose the inefficiency slack of the system to the sum of those of the processes in parallel systems. Kao (2012) proposed a parallel DEA model where the system efficiency measured from this model can be decomposed into efficiencies of the processes. Kao (2014) measured efficiency of a general multistage system. The basic idea of efficiency decomposition in Kao (2014) is to transform the general multistage system into a series structure, where each stage on the series includes a parallel structure.

The conventional parallel DEA model deals with crisp inputs and outputs. However, necessity characteristic of a real environment is uncertainty. In fact, many of the data are not crisp and should be assessed by experts in an environment of uncertainty. Fuzzy or subjective evaluations are important tools to measure numerical values. In fuzzy set (Zimmermann 1996), non-membership (rejection) degree of an element equal is one minus its membership (acceptance) degree. Kao and Lin (2012) proposed a parallel DEA model with fuzzy data where the system’s fuzzy efficiency measured from this model can be decomposed into fuzzy efficiencies of the processes. Sometimes in measuring fuzzy criteria, there may be imprecise matters for decision maker team. Therefore, we consider the information which is vague or insufficient so that the sum of the membership and non-membership degrees of an element is less than one and remaining of degree is hesitation. Hajiagha et al. (2013) proposed an intuitionistic fuzzy DEA model and provided its application to finance and credit institutions with the outputs presented in the form of intuitionistic fuzzy. Puri and Yadav (2015) developed an intuitionistic fuzzy DEA model in banking sector in India where the subjective information of two inputs, namely labor and operating expenses, was represented by triangular intuitionistic fuzzy numbers (TIFNs).

Moreover, none of the above studies have considered the parallel DEA model in intuitionistic fuzzy environments. Therefore, a model is needed to integrate network structures and intuitionistic fuzzy data simultaneously which will provide more complete and comprehensive information for decision makers or managers of business industries.

Puri and Yadav (2015) introduced a conventional data envelopment analysis model in intuitionistic fuzzy environment, whereas our proposed model introduced a parallel data envelopment analysis model in intuitionistic fuzzy environment. Researchers like Hajiagha et al. (2013) and Puri and Yadav (2015) represented intuitionistic fuzzy DEA models. They neglected internal structure in measuring efficiency. Kao and Hwang (2008) measured the efficiency of a series system when all of the inputs and outputs are crisp numbers. Kao (2012) proposed a parallel DEA model where data possess crisp essence. Kao (2014) measured efficiency of a general multistage system in crisp environment. In this paper, we have extended crisp parallel DEA to intuitionistic fuzzy parallel DEA and applied in real-life problem. This paper shows parallel network systems in uncertainty environments and measures the performance of these types of systems under conditions of uncertainty. Therefore, data are collected in the form of intuitionistic fuzzy numbers. But in various applications and engineering scenarios there will be a need to “defuzzify” the intuitionistic fuzzy results. In other words, we may eventually find a need to convert the intuitionistic fuzzy results to crisp results. We have used relationships of Grzegorzewski (2003) for defuzzification of the numbers for data envelopment analysis parameters. Another aspect of innovation in this study is the use of self-assessment method in measuring performance. Self-assessment is a comprehensive review, systematic and regular activities and achievements of the organization over time. Self-assessment of a decision making unit (DMU) due to more stable conditions is far more efficient than evaluating it with other decision making units (DMUs). If analysis with others provides capability of ranking, self-assessment causes analysis of current and previous positions and provides development and promotion conditions. By using this technique, we can identify best and worse time interval and analyze its reasons. As far as the authors’ knowledge goes, such type of parallel data envelopment analysis model in intuitionistic fuzzy environment has not yet been studied in real-life case study.

The rest of this paper is unfolded as follows: Sect. 2 presents the preliminaries. Section 3 explains a parallel DEA model to calculate the system and process crisp efficiencies. Section 4 describes the IFDEA model to calculate the system efficiency in intuitionistic fuzzy environments. Section 5 proposes the extension of crisp parallel DEA to intuitionistic fuzzy parallel DEA for measuring the system and process efficiencies in intuitionistic fuzzy environments. Section 6 shows an application of the proposed approach in the hospital’s department. Section 7 presents managerial implication proposed model. Finally, Sect. 8 describes the conclusions.

2 Preliminaries

In this section, we explain the concepts of intuitionistic fuzzy set (IFS), intuitionistic fuzzy number (IFN) and triangular intuitionistic fuzzy number (TIFN) (Mahapatra and Roy 2009), arithmetic operations on TIFNs (Nagoorgani and Ponnalagu 2012) and expected value of IFNs (Grzegorzewski 2003).

2.1 Intuitionistic Fuzzy Set (IFS)

If \( X \) be a fixed set, then an intuitionistic fuzzy set \( \ A^{I} \) in \( X \) is defined as: \( \ A^{I} = \left\{ {x,\mu_{{\ A^{I} }} \left( x \right),v_{{\ A^{I} }} \left( x \right):x \in X} \right\} \). Here, \( \mu_{{\ A^{I} }} :X \to \left[ {0,1} \right] \) and \( v_{{\ A^{I} }} :X \to \left[ {0,1} \right] \) with \( 0 \le \mu_{{\ A^{I} }} \left( x \right) + v_{{\ A^{I} }} \left( x \right) \le 1, \forall x \in x \) represent the degrees of membership and non-membership, respectively. Further, \( \pi_{{\ A^{I} }} \left( x \right) = 1 - \mu_{{\ A^{I} }} \left( x \right) - v_{{\ A^{I} }} \left( x \right), \forall x \in X \) is a hesitation degree of \( x \) to the set \( \ A^{I} \). It is clear that \( 0 \le \pi_{{\ A^{I} }} \left( x \right) \le 1, \forall x \in X \).

2.2 Intuitionistic Fuzzy Number (IFN)

An IFN (Mahapatra and Roy 2009) \( \tilde{A}^{I} \) is
  1. (i)

    Normal, i.e., there is any \( x_{0} \in R \) such that \( \mu_{{\ A^{I} }} \left( {x_{0} } \right) = 1 \) (so \( v_{{\ A^{I} }} \left( {x_{0} } \right) = 0 \)(.

     
  2. (ii)

    Convex for \( \mu_{{\ A^{I} }} \left( x \right) \), i.e., \( \mu_{{\ A^{I} }} \left( {\lambda x_{1} + \left( {1 - \lambda } \right)x_{2} } \right) \ge \hbox{min} \left( {\mu_{{\ A^{I} }} \left( {x_{1} } \right),\mu_{{\ A^{I} }} \left( {x_{2} } \right)} \right), \forall x_{1} ,x_{2} \in R,\lambda \in \left[ {0,1} \right] \).

     
  3. (iii)

    Concave for \( v_{{\ A^{I} }} \left( x \right) \), i.e., \( v_{{\ A^{I} }} \left( {\lambda x_{1} + \left( {1 - \lambda } \right)x_{2} } \right) \le max \left( {v_{{\ A^{I} }} \left( {x_{1} } \right),v_{{\ A^{I} }} \left( {x_{2} } \right)} \right), \forall x_{1} ,x_{2} \in R,\lambda \in \left[ {0,1} \right] \).

     
An IFN can be denoted by \( (a_{1} ,a_{2} ,a_{3} ,a_{4} ;b_{1} ,b_{2} ,b_{3} ,b_{4} ) \) with the following membership function \( \mu_{{\ A^{I} }} \) and non-membership function \( v_{{\ A^{I} }} \)
$$ \mu_{{\ A^{I} }} \left( x \right) = \left\{ {\begin{array}{*{20}l} {f_{A} \left( x \right),} \hfill & {a_{1} \le x < a_{2} ,} \hfill \\ {1,} \hfill & {a_{2} \le x \le a_{3} ,} \hfill \\ {g_{A} \left( x \right),} \hfill & {a_{3} < x \le a_{4} ,} \hfill \\ {0,} \hfill & {{\text{otherwise}},} \hfill \\ \end{array} } \right. $$
and
$$ v_{{\ A^{I} }} \left( x \right) = \left\{ {\begin{array}{*{20}l} {h_{A} \left( x \right),} \hfill & {b_{1} \le x < b_{2} ,} \hfill \\ {0,} \hfill & {b_{2} \le x \le b_{3} ,} \hfill \\ {k_{A} \left( x \right), } \hfill & {b_{3} < x \le b_{4} ,} \hfill \\ {1,} \hfill & {{\text{otherwise}},} \hfill \\ \end{array} } \right. $$

The functions \( f_{A} \) and \( k_{A} \) are non-decreasing functions and the functions \( g_{A} \) and \( h_{A} \) are non-increasing functions.

2.3 Triangular Intuitionistic Fuzzy Number (TIFN)

An intuitionistic fuzzy number (IFN) \( \ A^{I} \) is a TIFN (Mahapatra and Roy 2009) if the following \( \mu_{{\ A^{I} }} \left( x \right) \) and \( v_{{\ A^{I} }} \left( x \right) \) are the membership and non-membership functions, respectively.

$$ \mu_{{\ A^{I} }} \left( x \right) = \left\{ {\begin{array}{*{20}l} {\frac{{x - a_{1} }}{{a_{2} - a_{1} }},} \hfill & {a_{1} < x \le a_{2} ,} \hfill \\ {1,} \hfill & {x = a_{2} ,} \hfill \\ {\frac{{x - a_{3} }}{{a_{2} - a_{3} }},} \hfill & {a_{2} \le x < a_{3} ,} \hfill \\ {0,} \hfill & {{\text{otherwise}},} \hfill \\ \end{array} } \right. $$
and
$$ v_{{\ A^{I} }} \left( x \right) = \left\{ {\begin{array}{*{20}l} {\frac{{x - a_{2} }}{{a_{{1^{{\prime }} }} - a_{2} }},} \hfill & {a_{{1^{{\prime }} }} < x \le a_{2} ,} \hfill \\ {0,} \hfill & {x = a_{2} ,} \hfill \\ {\frac{{x - a_{2} }}{{a_{{3^{{\prime }} }} - a_{3} }},} \hfill & {a_{2} \le x < a_{{3^{{\prime }} }} ,} \hfill \\ {1,} \hfill & {{\text{otherwise}},} \hfill \\ \end{array} } \right. $$
This TIFN is represented by \( \ A^{I}_{\text{TIFN}} = (a_{1} ,a_{2} ,a_{3} ;a_{{1^{\prime } }} ,a_{2} ,a_{{3^{\prime } }} ) \) where \( a_{1}^{{\prime }} \le a_{1} \le a_{2} \le a_{3} \le a_{3}^{{\prime }} \) and shown in Fig. 1.
Fig. 1

Membership and non-membership functions of TIFN

A TIFN \( \ A^{I} = (a_{1} ,a_{2} ,a_{3} ;a_{{1^{\prime } }} ,a_{2} ,a_{{3^{\prime } }} ) \) is a triangular fuzzy number (TFN) \( \ A = (a_{1} ,a_{2} ,a_{3} ) \) if \( a_{1} = a_{{1^{{\prime }} }} , a_{3} = a_{{3^{{\prime }} }} \) and \( v_{{\ A^{I} }} \left( x \right) = 1 - \mu_{{\ A^{I} }} \left( x \right),\; \pi_{{\ A^{I} }} \left( x \right) = 0,\;\forall x \in R \), and it is a real number \( a \) if \( a_{{1^{{\prime }} }} = a_{1} = a_{2} = a_{3} = a_{{3^{{\prime }} }} = a \).

2.4 Arithmetic operations on TIFNs

If \( \ A^{I} = \left( {a_{1} ,a_{2} ,a_{3} ;a_{1}^{{\prime }} ,a_{2} ,a_{3}^{{\prime }} } \right) \) and \( \tilde{B}^{I} = \left( {b_{1} ,b_{2} ,b_{3} ;b_{1}^{{\prime }} ,b_{2} ,b_{3}^{{\prime }} } \right) \) be two TIFNs, then (Nagoorgani and Ponnalagu 2012) the following properties hold.
  1. (i)

    Addition\( \ A^{I} \oplus \tilde{B}^{I} = (a_{1} + b_{1} ,a_{2} + b_{2} ,a_{3} + b_{3} ; a_{{1^{{\prime }} }} + b_{{1^{{\prime }} }} ,a_{2} + b_{2} ,a_{{3^{{\prime }} }} + b_{{3^{{\prime }} }} ) \).

     
  2. (ii)

    Subtraction\( \ A^{I} { \ominus }\tilde{B}^{I} = \left( {a_{1} - b_{3} ,a_{2} - b_{2} ,a_{3} - b_{1} ; a_{{1^{{\prime }} }} - b_{{3^{{\prime }} }} ,a_{2} - b_{2} ,a_{{3^{{\prime }} }} - b_{{1^{{\prime }} }} } \right) \).

     
  3. (iii)

    Multiplication\( \ A^{I} \otimes \tilde{B}^{I} = (a_{1} b_{1} ,a_{2} b_{2} ,a_{3} b_{3} ;a_{{1^{{\prime }} }} b_{{1^{{\prime }} }} ,a_{2} b_{2} ,a_{{3^{{\prime }} }} b_{{3^{{\prime }} }} ) \) for \( \ A^{I} ,\tilde{B}^{I} > 0 \).

     
  4. (iv)

    Scalar multiplication For \( k \in R \),

    $$ k\ A^{I} = \left\{ {\begin{array}{*{20}l} {(ka_{1} ,ka_{2} ,ka_{3} ;ka_{{1^{{\prime }} }} ,ka_{2} ,ka_{{3^{{\prime }} }} )} \hfill & {k > 0} \hfill \\ {(ka_{3} ,ka_{2} ,ka_{1} ;ka_{{3^{{\prime }} }} ,ka_{2} ,ka_{{1^{{\prime }} }} )} \hfill & {k < 0} \hfill \\ \end{array} } \right. $$
     

2.5 Expected value of IFNs

The expected interval (Grzegorzewski 2003) of an IFN \( \ A^{I} = (a_{1} ,a_{2} ,a_{3} ,a_{4} ;b_{1} ,b_{2} ,b_{3} ,b_{4} ) \) is a crisp interval \( EI\left( {\ A^{I} } \right) \) given by
$$ {\text{EI}}\left( {\ A^{I} } \right) = \left[ {E_{*} \left( {\ A^{I} } \right), E^{*} \left( {\ A^{I} } \right)} \right], $$
(1)
where
$$ \left\{ {\begin{array}{*{20}c} {E_{*} \left( {\ A^{I} } \right) = \frac{{b_{1} + a_{2} }}{2} + \frac{1}{2}\mathop \smallint \limits_{{b_{1} }}^{{b_{2} }} h_{A} \left( x \right){\text{d}}x - \frac{1}{2}\mathop \smallint \limits_{{a_{1} }}^{{a_{2} }} f_{A} \left( x \right){\text{d}}x} \\ {E^{*} \left( {\ A^{I} } \right) = \frac{{a_{3} + b_{4} }}{2} + \frac{1}{2}\mathop \smallint \limits_{{a_{3} }}^{{a_{4} }} g_{A} \left( x \right){\text{d}}x - \frac{1}{2}\mathop \smallint \limits_{{b_{3} }}^{{b_{4} }} k_{A} \left( x \right){\text{d}}x} \\ \end{array} } \right.. $$
(2)
The expected value (Grzegorzewski 2003) of an IFN is given as
$$ {\text{EV}}\left( {\ A^{I} } \right) = \left( {E_{*} \left( {\ A^{I} } \right) + E^{*} \left( {\ A^{I} } \right)} \right)/2. $$
(3)
If \( \ A^{I} = (a_{1} ,a_{2} ,a_{3} ;a_{{1^{\prime } }} ,a_{2} ,a_{{3^{\prime } }} ) \) be a TIFN, then by using (1), we have
$$ {\text{EI}}\left( {\ A^{I} } \right) = \left[ {E_{*} \left( {\ A^{I} } \right), E^{*} \left( {\ A^{I} } \right)} \right] = \left[ {\frac{{a_{{1^{\prime } }} + 2a_{2} + a_{1} }}{4},\frac{{a_{3} + 2a_{2} + a_{{3^{\prime } }} }}{4}} \right] $$
(4)
and by using (3), we have
$$ {\text{EV}}\left( {\ A^{I} } \right) = \left( {a_{{1^{{\prime }} }} + a_{1} + 4a_{2} + a_{3} + a_{{3^{{\prime }} }} } \right)/8. $$
(5)

3 Parallel DEA model

The parallel structure is one of the network structures corresponding to the systems constructed by number of parallel processes (shown in Fig. 2) where each process \( p,\left( {p = 1, \ldots ,q} \right), \) utilizes crisp inputs \( X_{i}^{\left( p \right)} , i \in I^{\left( p \right)} , \) to product crisp outputs \( Y_{r}^{\left( p \right)} , r \in O^{\left( p \right)} . \) The sums of the inputs \( X_{i}^{\left( p \right)} \) and outputs \( Y_{r}^{\left( p \right)} \) for all \( q \) processes are the system inputs \( \mathop \sum \nolimits_{p = 1}^{q} X_{ij}^{\left( p \right)} = X_{ij} \) and system outputs \( \mathop \sum \nolimits_{p = 1}^{q} Y_{rj}^{\left( p \right)} = Y_{rj} \) for each \( {\text{DMU}}_{j} \). For measuring the system and process crisp efficiencies in a parallel structure with crisp inputs and outputs, Kao (2012) proposed the following model:
$$ \begin{aligned} & \hbox{max} \, E_{k} = \mathop \sum \limits_{r = 1}^{s} u_{rk} Y_{rk} \\ & {\text{s}}.{\text{t}}:\mathop \sum \limits_{i = 1}^{m} v_{ik} X_{ik} = 1, \\ & \mathop \sum \limits_{{r \in O^{\left( p \right)} }} u_{rk} Y_{rj}^{\left( p \right)} - \mathop \sum \limits_{{i \in I^{\left( p \right)} }} v_{ik} X_{ij}^{\left( p \right)} \le 0,p = 1, \ldots ,q,j = 1, \ldots , n, \\ & u_{rk} ,v_{ik} \ge \varepsilon ,r = 1, \ldots ,s,i = 1, \ldots ,m. \\ \end{aligned} $$
(6)
Fig. 2

Parallel production system, where \( {\text{DMU}}_{k} \) has \( q \) production

This model is called parallel DEA model which measures system crisp efficiency and its process crisp efficiencies in a unified framework. This parallel model means that the system efficiency is calculated as the main objective and process efficiencies as its components.

By solving the above model, we can calculate system and its process crisp efficiencies as follows:
$$ E_{k} = \frac{{\mathop \sum \nolimits_{r = 1}^{s} u_{rk}^{*} Y_{rk} }}{{\mathop \sum \nolimits_{i = 1}^{m} v_{ik}^{*} X_{ik} }} = \mathop \sum \limits_{r = 1}^{s} u_{rk}^{*} Y_{rk} $$
(7)
$$ E_{k}^{\left( p \right)} = \frac{{\mathop \sum \nolimits_{{r \in O^{\left( p \right)} }} u_{rk}^{*} Y_{rk}^{\left( p \right)} }}{{\mathop \sum \nolimits_{{i \in I^{\left( P \right)} }} v_{ik}^{*} X_{ik}^{\left( p \right)} }},p = 1, \ldots ,q. $$
(8)

The weight associated with process \( p \) can be defined as: \( w^{\left( p \right)} = \frac{{\mathop \sum \nolimits_{{i \in I^{\left( P \right)} }} v_{ik}^{*} X_{ik}^{\left( p \right)} }}{{\mathop \sum \nolimits_{i = 1}^{m} v_{ik}^{*} X_{ik} }}. \)

As a result, the average of the \( q \) process crisp efficiencies weighted by \( w^{\left( p \right)} \) is:
$$ \mathop \sum \limits_{p = 1}^{q} w^{\left( p \right)} E_{k}^{\left( P \right)} = \mathop \sum \limits_{p = 1}^{q} \left[ {\left( {\frac{{\mathop \sum \nolimits_{{i \in I^{\left( P \right)} }} v_{ik}^{*} X_{ik}^{\left( p \right)} }}{{\mathop \sum \nolimits_{i = 1}^{m} v_{ik}^{*} X_{ik} }}} \right)\left( {\frac{{\mathop \sum \nolimits_{{r \in O^{\left( p \right)} }} u_{rk}^{*} Y_{rk}^{\left( p \right)} }}{{\mathop \sum \nolimits_{{i \in I^{\left( P \right)} }} v_{ik}^{*} X_{ik}^{\left( p \right)} }}} \right)} \right] = \mathop \sum \limits_{p = 1}^{q} \left( {\frac{{\mathop \sum \nolimits_{{r \in O^{\left( p \right)} }} u_{rk}^{*} Y_{rk}^{\left( p \right)} }}{{\mathop \sum \nolimits_{i = 1}^{m} v_{ik}^{*} X_{ik} }}} \right) = \frac{{\mathop \sum \nolimits_{r = 1}^{s} u_{rk}^{*} Y_{rk} }}{{\mathop \sum \nolimits_{i = 1}^{m} v_{ik}^{*} X_{ik} }}. $$
(9)
In a parallel structure, the system efficiency represents the aggregate efficiency of all of its processes if the operation of each process is taken into consideration.

4 Intuitionistic fuzzy DEA model

Suppose there are \( n \) congruent DMUs \( \left( {{\text{DMU}}_{j} ; j = 1, \ldots ,n} \right) \). The \( {\text{DMU}}_{j} \) converts \( m \) intuitionistic fuzzy inputs \( \tilde{x}_{ij}^{I} , i = 1, \ldots ,m \) into \( s \) intuitionistic fuzzy outputs \( \tilde{y}_{rj}^{I} ,r = 1, \ldots ,s \), where \( \tilde{x}_{ij}^{I} = \left( {x_{1}^{ij} ,x_{2}^{ij} ,x_{3}^{ij} ;x_{{1^{{\prime }} }}^{ij} ,x_{2}^{ij} ,x_{{3^{{\prime }} }}^{ij} } \right) \) and \( \tilde{y}_{rj}^{I} = \left( {y_{1}^{rj} ,y_{2}^{rj} ,y_{3}^{rj} ;y_{{1^{{\prime }} }}^{rj} ,y_{2}^{rj} ,y_{{3^{{\prime }} }}^{rj} } \right) \) with \( x_{1}^{ij} > 0,\forall i,j \) and \( y_{1}^{rj} > 0,\forall r,j \) are positive TIFNs. In order to calculate the system of intuitionistic fuzzy efficiency, Puri and Yadav (2015) have developed the following model:
$$ \begin{aligned} & \hbox{max} \;\tilde{E}_{k}^{I} = \mathop \sum \limits_{r = 1}^{s} \tilde{u}_{rk}^{I} \otimes \tilde{y}_{rk}^{I} \\ & {\text{s}}.{\text{t}}: \mathop \sum \limits_{i = 1}^{m} \tilde{v}_{ik}^{I} \otimes \tilde{x}_{ik}^{I} = \tilde{1}^{I} , \\ & \mathop \sum \limits_{r = 1}^{s} \tilde{u}_{rk}^{I} \otimes \tilde{y}_{rj}^{I} { \ominus }\mathop \sum \limits_{i = 1}^{m} \tilde{v}_{ik}^{I} \otimes \tilde{x}_{ij}^{I} \le \tilde{0}^{I} ,j = 1, \ldots ,n, \\ & \tilde{u}_{rk}^{I} \ge \varepsilon ,r = 1, \ldots , s, \\ & \tilde{v}_{ik}^{I} \ge \varepsilon ,i = 1, \ldots ,m,\varepsilon > 0, \\ \end{aligned} $$
(10)
where \( \tilde{E}_{k}^{I*} \) is the optimal value of the \( {\text{DMU}}_{k} . \) In the above model, \( \tilde{x}_{ik}^{I} \) is the amount of the ith intuitionistic fuzzy input utilized by the kth DMU; \( \tilde{y}_{rk}^{I} \) is the amount of the rth intuitionistic fuzzy output produced by the kth DMU; \( \tilde{v}_{ik}^{I} = \left( {v_{1}^{ik} ,v_{2}^{ik} ,v_{3}^{ik} ;v_{1}^{ik} ,v_{2}^{ik} ,v_{3}^{ik} } \right) \) and \( \tilde{u}_{rk}^{I} = \left( {u_{1}^{rk} ,u_{2}^{rk} ,u_{3}^{rk} ;u_{1}^{rk} ,u_{2}^{rk} ,u_{3}^{rk} } \right) \) are intuitionistic fuzzy weights for the ith intuitionistic fuzzy input and rth intuitionistic fuzzy output of the kth DMU, respectively, and are represented by TIFNs; and \( k \left( {k = 1, \ldots ,n} \right) \) presents the D under evaluation. By applying model (10), we can only calculate system intuitionistic fuzzy efficiency and, thus, cannot calculate process intuitionistic fuzzy efficiencies.

5 Proposed intuitionistic fuzzy parallel DEA (IFPDEA) with self-assessment model

The existing network DEA models for measuring efficiencies of system and process are limited to crisp and fuzzy data. However, in real-world problems, some inputs and outputs in parallel structures may possess intuitionistic fuzzy essence instead of crisp or fuzzy. The intuitionistic fuzzy parallel DEA model measures the efficiency of a system and its processes in terms of intuitionistic fuzzy efficiency when some observations are intuitionistic fuzzy numbers and the system has parallel structure. Intuitionistic fuzzy efficiency for parallel production systems is calculated via the following IFPDEA model:
$$ \begin{aligned} & \hbox{max} \;\tilde{E}_{k}^{I} = \mathop \sum \limits_{r = 1}^{s} \tilde{u}_{rk}^{I} \otimes \tilde{y}_{rk}^{I} \\ & {\text{s}}.{\text{t}}:\mathop \sum \limits_{i = 1}^{m} \tilde{v}_{ik}^{I} \otimes \tilde{x}_{ik}^{I} = \tilde{1}^{I} , \\ & \mathop \sum \limits_{{r \in O^{\left( p \right)} }} \tilde{u}_{rk}^{I} \otimes \tilde{y}_{rj}^{I\left( P \right)} { \ominus }\mathop \sum \limits_{{i \in I^{\left( p \right)} }} \tilde{v}_{ik}^{I} \otimes \tilde{x}_{ij}^{I\left( P \right)} \le \tilde{0}^{I} ,p = 1, \ldots ,q,j = 1, \ldots ,n, \\ & \tilde{u}_{rk}^{I} \ge \varepsilon ,r = 1, \ldots ,s, \\ & \tilde{v}_{ik}^{I} \ge \varepsilon ,i = 1, \ldots ,m,\varepsilon > 0. \\ \end{aligned} $$
(11)
In self-assessment of parallel systems with intuitionistic fuzzy inputs and outputs, assume that the performance of \( n \) cutting times from a DMU \( \left( {{\text{DMU}}_{j} ; j = 1, \ldots ,n} \right) \) with \( q \) production processes is measured. The \( p \) production process utilizes \( m \) intuitionistic fuzzy inputs \( \tilde{x}_{ij}^{I\left( P \right)} , i = 1, \ldots ,m,p = 1, \ldots ,q \) to produce \( s \) intuitionistic fuzzy outputs \( \tilde{y}_{rj}^{I\left( P \right)} ,r = 1, \ldots ,s,p = 1, \ldots ,q \); and \( k \left( {k = 1, \ldots ,n} \right) \) presents the cutting time under evaluation.
If \( \left( {\tilde{u}_{rk}^{I*} ; \tilde{v}_{ik}^{I*} } \right) \) be an optimal solution of model (11), then the system and process intuitionistic fuzzy efficiencies can be calculated as follows:
$$ \tilde{E}_{k}^{I} = \frac{{\mathop \sum \nolimits_{r = 1}^{s} \tilde{u}_{rk}^{I*} \otimes \tilde{y}_{rk}^{I} }}{{\mathop \sum \nolimits_{i = 1}^{m} \tilde{v}_{ik}^{I*} \otimes \tilde{x}_{ik}^{I} }} = \mathop \sum \limits_{r = 1}^{s} \tilde{u}_{rk}^{I*} \otimes \tilde{y}_{rk}^{I} $$
(12)
$$ \tilde{E}_{k}^{I\left( p \right)} = \frac{{\mathop \sum \nolimits_{{r \in O^{\left( p \right)} }} \tilde{u}_{rk}^{I*} \otimes \tilde{y}_{rk}^{I\left( p \right)} }}{{\mathop \sum \nolimits_{{i \in I^{\left( P \right)} }} \tilde{v}_{ik}^{I*} \otimes \tilde{x}_{ik}^{I\left( p \right)} }},p = 1, \ldots ,q . $$
(13)

If we define the weight associated with process \( p \) as the aggregate intuitionistic fuzzy input \( \tilde{w}^{I\left( p \right)} = \frac{{\mathop \sum \nolimits_{{i \in I^{\left( P \right)} }} \tilde{v}_{ik}^{I*} \otimes \tilde{x}_{ik}^{I\left( p \right)} }}{{\mathop \sum \nolimits_{i = 1}^{m} \tilde{v}_{ik}^{I*} \otimes \tilde{x}_{ik}^{I} }}, \) then the average of the \( q \) process intuitionistic fuzzy efficiencies is the system intuitionistic fuzzy efficiency described as follows.

Proposition 1

The system intuitionistic fuzzy efficiency of \( q \) processes with the parallel structure is a weighted average of the \( q \) process intuitionistic fuzzy efficiencies.

Proof

Assuming \( \tilde{w}^{I\left( p \right)} = \frac{{\mathop \sum \nolimits_{{i \in I^{\left( P \right)} }} \tilde{v}_{ik}^{I*} \otimes \tilde{x}_{ik}^{I\left( p \right)} }}{{\mathop \sum \nolimits_{i = 1}^{m} \tilde{v}_{ik}^{I*} \otimes \tilde{x}_{ik}^{I} }},\;\tilde{E}_{k}^{I} = \mathop \sum \limits_{p = 1}^{q} \tilde{w}^{I\left( p \right)} \tilde{E}_{k}^{I\left( P \right)} ,\; \mathop \sum \limits_{p = 1}^{q} \tilde{w}^{I\left( p \right)} = 1, \;\tilde{w}^{I\left( p \right)} \ge 0,p = 1, \ldots ,q. \) Observe that
$$ \sum\limits_{p = 1}^{q} {\tilde{w}^{I\left( p \right)} \tilde{E}_{k}^{I} } = \sum\limits_{p = 1}^{q} {\left[ {\left( {\frac{{\mathop \sum \nolimits_{{i \in I^{\left( P \right)} }} \tilde{v}_{ik}^{I*} \otimes \tilde{x}_{ik}^{I\left( p \right)} }}{{\mathop \sum \limits_{i = 1}^{m} \tilde{v}_{ik}^{I*} \otimes \tilde{x}_{ik}^{I} }}} \right)\left( {\frac{{\mathop \sum \nolimits_{{r \in O^{\left( p \right)} }} \tilde{u}_{rk}^{I*} \otimes \tilde{y}_{rk}^{I\left( p \right)} }}{{\mathop \sum \nolimits_{{i \in I^{\left( P \right)} }} \tilde{v}_{ik}^{I*} \otimes \tilde{x}_{ik}^{I\left( p \right)} }}} \right)} \right]} = \sum\limits_{p = 1}^{q} {\left( {\frac{{\mathop \sum \nolimits_{{r \in O^{\left( p \right)} }} \tilde{u}_{rk}^{I*} \otimes \tilde{y}_{rk}^{I\left( p \right)} }}{{\mathop \sum \nolimits_{i = 1}^{m} \tilde{v}_{ik}^{I*} \otimes \tilde{x}_{ik}^{I} }}} \right)} = \frac{{\mathop \sum \nolimits_{r = 1}^{s} \tilde{u}_{rk}^{I*} \otimes \tilde{y}_{rk}^{I} }}{{\mathop \sum \nolimits_{i = 1}^{m} \tilde{v}_{ik}^{I*} \otimes \tilde{x}_{ik}^{I} }}. $$
(14)
On the other hand, \( \sum\nolimits_{p = 1}^{q} {\tilde{w}^{I\left( p \right)} } = \sum\nolimits_{p = 1}^{q} {\frac{{\mathop \sum \nolimits_{{i \in I^{\left( P \right)} }} \tilde{v}_{ik}^{I*} \otimes \tilde{x}_{ik}^{I\left( p \right)} }}{{\mathop \sum \nolimits_{i = 1}^{m} \tilde{v}_{ik}^{I*} \otimes \tilde{x}_{ik}^{I} }} = \frac{{\mathop \sum \nolimits_{p = 1}^{q} \mathop \sum \nolimits_{{i \in I^{\left( P \right)} }} \tilde{v}_{ik}^{I*} \otimes \tilde{x}_{ik}^{I\left( p \right)} }}{{\mathop \sum \nolimits_{i = 1}^{m} \tilde{v}_{ik}^{I*} \otimes \tilde{x}_{ik}^{I} }} = \frac{{\mathop \sum \nolimits_{{i \in I^{\left( P \right)} }} \mathop \sum \nolimits_{p = 1}^{q} \tilde{v}_{ik}^{I*} \otimes \tilde{x}_{ik}^{I\left( p \right)} }}{{\mathop \sum \nolimits_{i = 1}^{m} \tilde{v}_{ik}^{I*} \otimes \tilde{x}_{ik}^{I} }} = \frac{{\mathop \sum \nolimits_{i = 1}^{m} \tilde{v}_{ik}^{I*} \otimes \tilde{x}_{ik}^{I} }}{{\mathop \sum \nolimits_{i = 1}^{m} \tilde{v}_{ik}^{I*} \otimes \tilde{x}_{ik}^{I} }} = 1} \) and clearly \( \tilde{w}^{I\left( p \right)} = \frac{{\mathop \sum \nolimits_{{i \in I^{\left( P \right)} }} \tilde{v}_{ik}^{I*} \otimes \tilde{x}_{ik}^{I\left( p \right)} }}{{\mathop \sum \nolimits_{i = 1}^{m} \tilde{v}_{ik}^{I*} \otimes \tilde{x}_{ik}^{I} }} \ge 0 \) since all \( \tilde{v}_{ik}^{I*} \ge 0 \) and all \( \tilde{x}_{ik}^{I\left( p \right)} \ge 0 \).

Proposition 2

The optimal value of model (11) is not greater than the optimal value of model (10), namely, the intuitionistic fuzzy efficiency score of a DMU in a parallel structure is less than or equal to its intuitionistic fuzzy efficiency score using conventional intuitionistic fuzzy model.

Proof

Assume an arbitrary feasible solution of model (11), \( \left( {\tilde{v}_{ik}^{{I{ \star }}} ,\tilde{u}_{rk}^{{I{ \star }}} } \right) \); thus, it satisfies the relative constraints as follows:
$$ \mathop \sum \limits_{{r \in O^{\left( p \right)} }} \tilde{u}_{rk}^{{I{ \star }}} \otimes \tilde{y}_{rj}^{I\left( P \right)} { \ominus }\mathop \sum \limits_{{i \in I^{\left( p \right)} }} \tilde{v}_{ik}^{{I{ \star }}} \otimes \tilde{x}_{ij}^{I\left( P \right)} \le 0, p = 1, \ldots ,q,j = 1, \ldots , n. $$
If we sum up the above set of constraint over the processes index of \( p, \) we get the following equalities:
$$ \mathop \sum \limits_{p = 1}^{q} \mathop \sum \limits_{{r \in O^{\left( p \right)} }} \tilde{u}_{rk}^{{I{ \star }}} \otimes \tilde{y}_{rj}^{I\left( P \right)} { \ominus }\mathop \sum \limits_{p = 1}^{q} \mathop \sum \limits_{{i \in I^{\left( p \right)} }} \tilde{v}_{ik}^{{I{ \star }}} \otimes \tilde{x}_{ij}^{I\left( P \right)} \le 0,j = 1, \ldots ,n \Rightarrow $$
$$ \mathop \sum \limits_{{r \in O^{\left( p \right)} }} \mathop \sum \limits_{p = 1}^{q} \tilde{u}_{rk}^{{I{ \star }}} \otimes \tilde{y}_{rj}^{I\left( P \right)} { \ominus }\mathop \sum \limits_{{i \in I^{\left( p \right)} }} \mathop \sum \limits_{p = 1}^{q} \tilde{v}_{ik}^{{I{ \star }}} \otimes \tilde{x}_{ij}^{I\left( P \right)} \le 0,j = 1, \ldots ,n \Rightarrow $$
$$ \mathop \sum \limits_{r = 1}^{s} \tilde{u}_{rk}^{{I{ \star }}} \otimes \tilde{y}_{rj}^{I} { \ominus }\mathop \sum \limits_{i = 1}^{m} \tilde{v}_{ik}^{{I{ \star }}} \otimes \tilde{x}_{ij}^{I} \le 0,j = 1, \ldots ,n. $$

That is, the constraint set of model (10). Note that other constraint set and objective function are the same in both models. This shows \( \left( {\tilde{v}_{ik}^{{I{ \star }}} , \tilde{u}_{rk}^{{I{ \star }}} } \right) \) is a feasible solution of model (10) and therefore \( \tilde{E}_{k}^{I} \left( {11} \right) \le \tilde{E}_{k}^{I} \left( {10} \right) \), that is, a conventional intuitionistic fuzzy efficiency of a DMU is not less than its intuitionistic fuzzy efficiency in a parallel structure.

Model (11) can transform to the following form:
$$ \begin{aligned} & \hbox{max} \;\tilde{E}_{k}^{I} = \mathop \sum \limits_{r = 1}^{s} \left( {u_{1}^{rk} ,u_{2}^{rk} ,u_{3}^{rk} ;u_{1}^{rk} ,u_{2}^{rk} ,u_{3}^{rk} } \right) \otimes \left( {y_{1}^{rk} ,y_{2}^{rk} ,y_{3}^{rk} ;y_{1}^{rk} ,y_{2}^{rk} ,y_{3}^{rk} } \right) \\ & {\text{s}}.{\text{t}}: \mathop \sum \limits_{i = 1}^{m} \left( {v_{1}^{ik} ,v_{2}^{ik} ,v_{3}^{ik} ;v_{1}^{ik} ,v_{2}^{ik} ,v_{3}^{ik} } \right) \otimes \left( {x_{1}^{ik} ,x_{2}^{ik} ,x_{3}^{ik} ;x_{1}^{ik} ,x_{2}^{ik} ,x_{3}^{ik} } \right) = \left( {1,1,1;1,1,1} \right), \\ & \quad \mathop \sum \limits_{r = 1}^{s} \left( {u_{1}^{rk} ,u_{2}^{rk} ,u_{3}^{rk} ;u_{1}^{rk} ,u_{2}^{rk} ,u_{3}^{rk} } \right) \otimes \left( {y_{1}^{rj\left( p \right)} ,y_{2}^{rj\left( p \right)} ,y_{3}^{rj\left( p \right)} ;y_{1}^{rj\left( p \right)} ,y_{2}^{rj\left( p \right)} ,y_{3}^{rj\left( p \right)} } \right) \\ & \quad { \ominus }\mathop \sum \limits_{i = 1}^{m} \left( {v_{1}^{ik} ,v_{2}^{ik} ,v_{3}^{ik} ;v_{1}^{ik} ,v_{2}^{ik} ,v_{3}^{ik} } \right) \otimes \left( {x_{1}^{ij\left( p \right)} ,x_{2}^{ij\left( p \right)} ,x_{3}^{ij\left( p \right)} ;x_{1}^{ij\left( p \right)} ,x_{2}^{ij\left( p \right)} ,x_{3}^{ij\left( p \right)} } \right) \le \left( {0,0,0;0,0,0} \right),p \\ & \quad = 1, \ldots ,q,j = 1, \ldots ,n, \\ & \left( {u_{1}^{rk} ,u_{2}^{rk} ,u_{3}^{rk} ;u_{1}^{rk} ,u_{2}^{rk} ,u_{3}^{rk} } \right) \ge \varepsilon ,r = 1, \ldots ,s,\left( {v_{1}^{ik} ,v_{2}^{ik} ,v_{3}^{ik} ;v_{1}^{ik} ,v_{2}^{ik} ,v_{3}^{ik} } \right) \ge \varepsilon ,i = 1, \ldots ,m,\varepsilon > 0. \\ \end{aligned} $$
(15)
By using the rules presented in Sect. 2.4, model (15) transforms to the following form:
$$ \begin{aligned} & \hbox{max} \;\tilde{E}_{k}^{I} = \left( {\mathop \sum \limits_{r = 1}^{s} u_{1}^{rk} y_{1}^{rk} ,\mathop \sum \limits_{r = 1}^{s} u_{2}^{rk} y_{2}^{rk} ,\mathop \sum \limits_{r = 1}^{s} u_{3}^{rk} y_{3}^{rk} ;\mathop \sum \limits_{r = 1}^{s} u_{{1^{{\prime }} }}^{rk} y_{{1^{{\prime }} }}^{rk} ,\mathop \sum \limits_{r = 1}^{s} u_{2}^{rk} y_{2}^{rk} ,\mathop \sum \limits_{r = 1}^{s} u_{{3^{{\prime }} }}^{rk} y_{{3^{{\prime }} }}^{rk} } \right) \\ & {\text{s}}.{\text{t}}:\left( {\mathop \sum \limits_{i = 1}^{m} v_{1}^{ik} x_{1}^{ik} ,\mathop \sum \limits_{i = 1}^{m} v_{2}^{ik} x_{2}^{ik} ,\mathop \sum \limits_{i = 1}^{m} v_{3}^{ik} x_{3}^{ik} ;\mathop \sum \limits_{i = 1}^{m} v_{{1^{{\prime }} }}^{ik} x_{{1^{{\prime }} }}^{ik} ,\mathop \sum \limits_{i = 1}^{m} v_{2}^{ik} x_{2}^{ik} ,\mathop \sum \limits_{i = 1}^{m} v_{{3^{{\prime }} }}^{ik} x_{{3^{{\prime }} }}^{ik} } \right) = \left( {1,1,1;1,1,1} \right), \\ & \left( {\mathop \sum \limits_{r = 1}^{s} u_{1}^{rk} y_{1}^{rj\left( p \right)} - \mathop \sum \limits_{i = 1}^{m} v_{3}^{ik} x_{3}^{ij\left( p \right)} ,\mathop \sum \limits_{r = 1}^{s} u_{2}^{rk} y_{2}^{rj\left( p \right)} - \mathop \sum \limits_{i = 1}^{m} v_{2}^{ik} x_{2}^{ij\left( p \right)} ,\mathop \sum \limits_{r = 1}^{s} u_{3}^{rk} y_{3}^{rj\left( p \right)} - \mathop \sum \limits_{i = 1}^{m} v_{1}^{ik} x_{1}^{ij\left( p \right)} ,\mathop \sum \limits_{r = 1}^{s} u_{{1^{{\prime }} }}^{rk} y_{{1^{{\prime }} }}^{rj\left( p \right)} } \right. \\ & \quad \quad \left. { - \mathop \sum \limits_{i = 1}^{m} v_{{3^{{\prime }} }}^{ik} x_{{3^{{\prime }} }}^{ij\left( p \right)} ,\mathop \sum \limits_{r = 1}^{s} u_{2}^{rk} y_{2}^{rj\left( p \right)} - \mathop \sum \limits_{i = 1}^{m} v_{2}^{ik} x_{2}^{ij\left( p \right)} ,\mathop \sum \limits_{r = 1}^{s} u_{{3^{{\prime }} }}^{rk} y_{{3^{{\prime }} }}^{rj\left( p \right)} - \mathop \sum \limits_{i = 1}^{m} v_{{1^{{\prime }} }}^{ik} x_{{1^{{\prime }} }}^{ij\left( p \right)} } \right) \le \left( {0,0,0;0,0,0} \right), p \\ & \quad = 1, \ldots ,q, j = 1, \ldots ,n, \\ & u_{{3^{{\prime }} }}^{rk} \ge u_{3}^{rk} \ge u_{2}^{rk} \ge u_{1}^{rk} \ge u_{{1^{{\prime }} }}^{rk} \ge \varepsilon , r = 1, \ldots ,s, \\ & v_{{3^{{\prime }} }}^{ik} \ge v_{3}^{ik} \ge v_{2}^{ik} \ge v_{1}^{ik} \ge v_{{1^{{\prime }} }}^{ik} \ge \varepsilon ,i = 1, \ldots ,m,\varepsilon > 0. \\ \end{aligned} $$
(16)
Model (16) is a parallel intuitionistic fuzzy linear programming problem (LPP) which, by using an expected value discussed in Sects. 25, transforms into the follow parallel crisp LPP:
$$ \begin{aligned} & \hbox{max} \;{\text{EV}}\left( {\tilde{E}_{k}^{I} } \right) = {\text{EV}}\left( {\mathop \sum \limits_{r = 1}^{s} u_{1}^{rk} y_{1}^{rk} ,\mathop \sum \limits_{r = 1}^{s} u_{2}^{rk} y_{2}^{rk} ,\mathop \sum \limits_{r = 1}^{s} u_{3}^{rk} y_{3}^{rk} ;\mathop \sum \limits_{r = 1}^{s} u_{{1^{\prime } }}^{rk} y_{{1^{\prime } }}^{rk} ,\mathop \sum \limits_{r = 1}^{s} u_{2}^{rk} y_{2}^{rk} ,\mathop \sum \limits_{r = 1}^{s} u_{{3^{\prime } }}^{rk} y_{{3^{\prime } }}^{rk} } \right) \\ & {\text{s}}.{\text{t}}:{\text{EV}}\left( {\mathop \sum \limits_{i = 1}^{m} v_{1}^{ik} x_{1}^{ik} ,\mathop \sum \limits_{i = 1}^{m} v_{2}^{ik} x_{2}^{ik} ,\mathop \sum \limits_{i = 1}^{m} v_{3}^{ik} x_{3}^{ik} ;\mathop \sum \limits_{i = 1}^{m} v_{{1^{\prime } }}^{ik} x_{{1^{\prime } }}^{ik} ,\mathop \sum \limits_{i = 1}^{m} v_{2}^{ik} x_{2}^{ik} ,\mathop \sum \limits_{i = 1}^{m} v_{{3^{\prime } }}^{ik} x_{{3^{\prime } }}^{ik} } \right) \\ & \quad = {\text{EV}}\left( {1,1,1;1,1,1} \right),{\text{EV}}\left( {\mathop \sum \limits_{{r \in O^{\left( p \right)} }} u_{1}^{rk} y_{1}^{rj\left( p \right)} - \mathop \sum \limits_{{i \in I^{\left( P \right)} }} v_{3}^{ik} x_{3}^{ij\left( p \right)} , \mathop \sum \limits_{{r \in O^{\left( p \right)} }} u_{2}^{rk} y_{2}^{rj\left( p \right)} - \mathop \sum \limits_{{i \in I^{\left( P \right)} }} v_{2}^{ik} x_{2}^{ij\left( p \right)} ,} \right. \\ & \quad \quad \mathop \sum \limits_{{r \in O^{\left( p \right)} }} u_{3}^{rk} y_{3}^{rj\left( p \right)} - \mathop \sum \limits_{{i \in I^{\left( P \right)} }} v_{1}^{ik} x_{1}^{ij\left( p \right)} ,\mathop \sum \limits_{{r \in O^{\left( p \right)} }} u_{{1^{\prime } }}^{rk} y_{{1^{\prime } }}^{rj\left( p \right)} - \mathop \sum \limits_{{i \in I^{\left( P \right)} }} v_{{3^{\prime } }}^{ik} x_{{3^{\prime } }}^{ij\left( p \right)} , \mathop \sum \limits_{{r \in O^{\left( p \right)} }} u_{2}^{rk} y_{2}^{rj\left( p \right)} - \mathop \sum \limits_{{i \in I^{\left( P \right)} }} v_{2}^{ik} x_{2}^{ij\left( p \right)} , \\ & \quad \quad \left. {\mathop \sum \limits_{{r \in O^{\left( p \right)} }} u_{{3^{\prime } }}^{rk} y_{{3^{\prime } }}^{rj\left( p \right)} - \mathop \sum \limits_{{i \in I^{\left( P \right)} }} v_{{1^{\prime } }}^{ik} x_{{1^{\prime } }}^{ij\left( p \right)} } \right) \le {\text{EV}}\left( {0,0,0;0,0,0} \right), p = 1, \ldots ,q,j = 1, \ldots ,n, \\ & u_{{3^{\prime } }}^{rk} \ge u_{3}^{rk} \ge u_{2}^{rk} \ge u_{1}^{rk} \ge u_{{1^{\prime } }}^{rk} \ge \varepsilon ,r = 1, \ldots ,s, \\ & v_{{3^{\prime } }}^{ik} \ge v_{3}^{ik} \ge v_{2}^{ik} \ge v_{1}^{ik} \ge v_{{1^{\prime } }}^{ik} \ge \varepsilon ,i = 1, \ldots ,m,\varepsilon > 0. \\ \end{aligned} $$
(17)
By using the expected value of TIFNs, model (17) transforms to the following parallel crisp LPP:
$$ \begin{aligned} & \hbox{max} \;E_{k}^{I} = 1/8\left( {\mathop \sum \limits_{r = 1}^{s} (u_{{1^{{\prime }} }}^{rk} y_{{1^{{\prime }} }}^{rk} + u_{1}^{rk} y_{1}^{rk} + 4u_{2}^{rk} y_{2}^{rk} + u_{3}^{rk} y_{3}^{rk} + u_{{3^{{\prime }} }}^{rk} y_{{3^{{\prime }} }}^{rk} )} \right) \\ & {\text{s}}.{\text{t}}: \mathop \sum \limits_{i = 1}^{m} \left( {v_{{1^{{\prime }} }}^{ik} x_{{1^{{\prime }} }}^{ik} + v_{1}^{ik} x_{1}^{ik} + 4v_{2}^{ik} x_{2}^{ik} + v_{3}^{ik} x_{3}^{ik} + v_{{3^{{\prime }} }}^{ik} x_{{3^{{\prime }} }}^{ik} } \right) = 8, \\ & \mathop \sum \limits_{{r \in O^{\left( p \right)} }} \left( {u_{{1^{{\prime }} }}^{rk} y_{{1^{{\prime }} }}^{rj\left( p \right)} + u_{1}^{rk} y_{1}^{rj\left( p \right)} + 4u_{2}^{rk} y_{2}^{rj\left( p \right)} + u_{3}^{rk} y_{3}^{rj\left( p \right)} + u_{{3^{{\prime }} }}^{rk} y_{{3^{{\prime }} }}^{rj\left( p \right)} } \right) \\ & \quad - \mathop \sum \limits_{{i \in I^{\left( P \right)} }} \left( {v_{{1^{{\prime }} }}^{ik} x_{{1^{{\prime }} }}^{ij\left( p \right)} + v_{1}^{ik} x_{1}^{ij\left( p \right)} + 4v_{2}^{ik} x_{2}^{ij\left( p \right)} + v_{3}^{ik} x_{3}^{ij\left( p \right)} + v_{{3^{{\prime }} }}^{ik} x_{{3^{{\prime }} }}^{ij\left( p \right)} } \right) \le 0, p = 1, \ldots ,q,j = 1, \ldots ,n, \\ & u_{{3^{{\prime }} }}^{rk} \ge u_{3}^{rk} \ge u_{2}^{rk} \ge u_{1}^{rk} \ge u_{{1^{{\prime }} }}^{rk} \ge \varepsilon , r = 1, \ldots ,s, \\ & v_{{3^{{\prime }} }}^{ik} \ge v_{3}^{ik} \ge v_{2}^{ik} \ge v_{1}^{ik} \ge v_{{1^{{\prime }} }}^{ik} \ge \varepsilon , i = 1, \ldots ,m,\varepsilon > 0. \\ \end{aligned} $$
(18)
And, Eqs. (12) and (13) transform to the following form:
$$ \begin{aligned} E_{k}^{I} & = \frac{{\mathop \sum \nolimits_{r = 1}^{s} (u_{{1^{{\prime }} }}^{rk} y_{{1^{{\prime }} }}^{rk} + u_{1}^{rk} y_{1}^{rk} + 4u_{2}^{rk} y_{2}^{rk} + u_{3}^{rk} y_{3}^{rk} + u_{{3^{{\prime }} }}^{rk} y_{{3^{{\prime }} }}^{rk} )}}{{\mathop \sum \nolimits_{i = 1}^{m} (v_{{1^{{\prime }} }}^{ik} x_{{1^{{\prime }} }}^{ik} + v_{1}^{ik} x_{1}^{ik} + 4v_{2}^{ik} x_{2}^{ik} + v_{3}^{ik} x_{3}^{ik} + v_{{3^{{\prime }} }}^{ik} x_{{3^{{\prime }} }}^{ik} )}} \\ & = 1/8\left( {\mathop \sum \limits_{r = 1}^{s} (u_{{1^{{\prime }} }}^{rk} y_{{1^{{\prime }} }}^{rk} + u_{1}^{rk} y_{1}^{rk} + 4u_{2}^{rk} y_{2}^{rk} + u_{3}^{rk} y_{3}^{rk} + u_{{3^{{\prime }} }}^{rk} y_{{3^{{\prime }} }}^{rk} )} \right) \\ \end{aligned} $$
(19)
$$ E_{k}^{I\left( p \right)} = \frac{{\mathop \sum \limits_{{r \in O^{\left( p \right)} }} \left( {u_{{1^{{\prime }} }}^{rk} y_{{1^{{\prime }} }}^{rj\left( p \right)} + u_{1}^{rk} y_{1}^{rj\left( p \right)} + 4u_{2}^{rk} y_{2}^{rj\left( p \right)} + u_{3}^{rk} y_{3}^{rj\left( p \right)} + u_{{3^{{\prime }} }}^{rk} y_{{3^{{\prime }} }}^{rj\left( p \right)} } \right)}}{{\mathop \sum \limits_{{i \in I^{\left( P \right)} }} \left( {v_{{1^{{\prime }} }}^{ik} x_{{1^{{\prime }} }}^{ij\left( p \right)} + v_{1}^{ik} x_{1}^{ij\left( p \right)} + 4v_{2}^{ik} x_{2}^{ij\left( p \right)} + v_{3}^{ik} x_{3}^{ij\left( p \right)} + v_{{3^{{\prime }} }}^{ik} x_{{3^{{\prime }} }}^{ij\left( p \right)} } \right)}},p = 1, \ldots ,q $$
(20)
which are the system and process crisp efficiencies, respectively.

6 Case study

Hospital department is an example of a parallel network system. Although many studies are available about hospital efficiency analyses (see Garcia-Lacalle and Martin 2010; Rosko and Mutter 2011; Nedelea and Fannin 2013; Chowdhury and Zelenyuk 2016), there are no studies that measure efficiency of hospital and its sections when some inputs and outputs are intuitionistic fuzzy numbers. It should be mentioned that the use of intuitionistic fuzzy approach to control uncertainty has better performance in terms of management in the real world and provides forecasts close to reality for organizations’ decision makers.

In this study, a DMU is a cutting time from a fixed hospital. We consider eight cutting times in the hospital. Each cutting time is called part, and each part has four processes called sections. In total, there are 32 sections. For each intuitionistic fuzzy or crisp input/output, the amount of a part is the sum of its sections. Imam Khomeini hospital, which is one of the biggest hospitals of Iran, is interested in the intuitionistic fuzzy efficiency of each part and its sections.

In this study, we use the intuitionistic fuzzy parallel DEA model (11) to measure the efficiency of the eight hospital parts as well as 32 sections in intuitionistic fuzzy environments. The intuitionistic fuzzy parallel DEA model (11) used in this study requires information on hospital outputs and inputs. The choice of inputs and outputs was guided by the previous literature (Chowdhury and Zelenyuk 2016). Management of the hospital have considered three inputs that include the number of physicians \( \left( {X_{1} } \right) \), the number of nurses \( \left( {X_{2} } \right) \) and the number of beds \( \left( {X_{3} } \right) \) and four outputs included bed occupancy percentage (BOP) \( \left( {Y_{1} } \right) \), satisfaction about physician performance \( \left( {Y_{2} } \right) \), satisfaction about nurse performance \( \left( {Y_{3} } \right) \) and satisfaction about environmental conditions \( \left( {Y_{4} } \right) \). In hospital, three outputs, namely satisfaction about physician performance, satisfaction about nurse performance and satisfaction about environmental conditions, are estimated as intuitionistic fuzzy numbers due to unavailability at the time of assessment. The data are shown in Table 1.
Table 1

Input and output data of Imam Khomeini hospital

Periods

Input

Output

\( X_{1} \)

\( X_{2} \)

\( X_{3} \)

\( Y_{1} \)

\( Y_{2} \)

\( Y_{3} \)

\( Y_{4} \)

Spring 2015 (part 1)

33

70

103

215.5

(133,149,163;124,149,169)

(240,268,295;220,268,307)

(70,79,93;49,79,108)

 Internal Section

12

24

47

61.45

(40,43,47;37,43,48)

(65,68,77;58,68,81)

(48,50,59;32,50,67)

 General Surgery Section

10

15

24

32.22

(26,30,33;24,30,35)

(61,68,72;58,68,74)

(8,10,12;6,10,13)

 Pediatrics Section

7

17

12

43.63

(22,28,31;20,28,33)

(42,51,58;37,51,61)

(9,11,13;7,11,15)

 CCU Section

4

14

20

75.20

(45,48,52;43,48,53)

(72,81,88;67,81,91)

(5,8,9;4,8,13)

Summer 2015 (part 2)

37

71

108

212.74

(131,156,169;119,156,199)

(136,157,187;116,157,217)

(126,141,151;114,141,162)

 Internal Section

14

24

50

59.01

(80,97,102;75,97,124)

(35,40,42;22,40,59)

(23,27,29;21,27,32)

 General Surgery Section

10

16

24

36.23

(13,17,20;10,17,24)

(40,47,65;38,.47,73)

(29,32,35;24,32,38)

 Pediatrics Section

8

16

13

38.58

(10,12,14;9,12,16)

(43,50,57;41,50,60)

(19,22,24;17,22,28)

 CCU Section

5

15

21

78.92

(28,30,33;25,30,35)

(18,20,23;15,20,25)

(55,60,63;52,60,64)

Autumn 2015 (part 3)

40

70

114

203.57

(129,154,167;109,154,190)

(209,224,256;199,224,274)

(71,87,104;61,87,118)

 Internal Section

15

26

50

79.23

(30,34,36;28,34,40)

(145,149,152;143,149,154)

(26,30,39;22,30,43)

 General Surgery Section

12

15

26

25.96

(31,40,45;18,40,57)

(34,35,41;32,35,48)

(15,18,21;13,18,23)

 Pediatrics Section

8

16

13

53.02

(45,50,54;42,50,56)

(12,20,38;9,20,45)

(14,20,22;11,20,26)

 CCU Section

5

13

25

45.36

(23,30,32;21,30,37)

(18,20,25;15,20,27)

(16,19,22;15,19,26)

Winter 2016 (part 4)

40

71

118

244.89

(160,176,191;143,176,212)

(210,225,242;190,225,257)

(190,201,214;179,201,222)

 Internal Section

13

25

52

45.56

(3,7,10;2,7,17)

(14,16,25;12,16,28)

(32,35,37;30,35,39)

 General Surgery Section

14

16

27

48.32

(81,84,86;80,84,87)

(60,67,71;50,67,73)

(51,53,56;48,53,58)

 Pediatrics Section

9

18

13

61.45

(28,35,39;18,35,50)

(39,42,44;36,42,46)

(45,48,51;43,48,53)

 CCU Section

4

12

26

89.56

(48,50,56;43,50,58)

(97,100,102;92,100,110)

(62,65,70;58,65,72)

Spring 2016 (part 5)

44

73

121

283.92

(232,246,257;213,246,266)

(485,497,512;476,497,529)

(189,207,244;157,207,257)

 Internal Section

15

27

51

49.26

(76,80,85;64,80,89)

(250,252,254;247,252,255)

(69,75,98;58,75,102)

 General Surgery Section

15

15

29

78.23

(35,40,42;32,40,43)

(78,80,83;75,80,95)

(87,95,100;70,95,105)

 Pediatrics Section

10

18

16

82.11

(52,54,57;50,54,59)

(82,85,92;81,85,94)

(14,17,23;12,17,25)

 CCU Section

4

13

25

74.32

(69,72,73;67,72,75)

(75,80,83;73,80,85)

(19,20,23;17,20,25)

Summer 2016 (part 6)

42

82

126

239.84

(212,232,244;196,232,254)

(513,524,533;495,524,550)

(233,243,255;199,243,264)

 Internal Section

12

30

55

52.56

(22,32,36;18,32, 37)

(178,180,182;176,180,184)

(52,56,58;49,56,60)

 General Surgery Section

15

16

30

32.45

(47,50,52;46,50,53)

(81,84,87;79,84,89)

(89,90,94;82,90,96)

 Pediatrics Section

9

21

17

89.25

(87,90,93;78,90,99)

(238,240,241;226,240,251)

(24,27,32;22,27,36)

 CCU Section

6

15

24

65.58

(56,60,63;54,60,65)

(16,20,23;14,20,26)

(68,70,71;46,70,72)

Autumn 2016 (part 7)

50

90

136

258.8

(258,283,294;240,283,301)

(227,240,253;217,240,265)

(61,72,78;51,72,87)

 Internal Section

15

32

53

65.14

(59,64,66;46,64,68)

(24,29,31;22,29,35)

(16,19,20;14,19,22)

 General Surgery Section

17

19

35

84.11

(35,44,46;34,44,48)

(128,129,130;124,129,131)

(9,11,12;8,11,13)

 Pediatrics Section

10

21

22

35.24

(45,54,58;42,54,60)

(20,22,24;19,22,25)

(19,22,24;15,22,27)

 CCU Section

8

18

26

74.31

(119,121,124;118,121,125)

(55,60,68;52,60,74)

(17,20,22;14,20,25)

Winter 2017 (part 8)

54

96

140

228.34

(194,221,233;181,221,248)

(308,331,342;298,331,351)

(147,158,165;139,158,173)

 Internal Section

14

36

54

56.85

(19,22,25;16,22,29)

(17,20,25;13,20,32)

(12,14,16;9,14,19)

 General Surgery Section

20

23

38

48.25

(79,88,90;76,88;95)

(74,86,89;71,86,89)

(115,118,119;113,118,122)

 Pediatrics Section

12

20

21

54.68

(31,32,34;27,32;36)

(8,12,13; 7,12,14)

(3,6,8;2,6,9)

 CCU Section

8

17

27

68.56

(65,79,84;62,79;88)

(209,213,215;207,213,216)

(17,20,22;15,20,23)

Table 2 shows the results of self-assessment, where the second column is the efficiency score solved from model (10), the third column is the efficiency score by using model (10) when each section considered as independence DMU, and the fourth column is the efficiency score solved from model (11). Due to the use of the expected value method for converting intuitionistic fuzzy efficiency to crisp efficiency, efficiency score of the intuitionistic fuzzy parallel DEA model has a similar interpretation with efficiency score of crisp parallel DEA model.
Table 2

Efficiencies of hospital and its sections in intuitionistic fuzzy environment

Periods

Part efficiency (IFDEA model)

Section efficiency (IFDEA model)

Intuitionistic fuzzy parallel DEA (IFPDEA) model

\( E_{k}^{\left( p \right)} \)

\( w^{\left( p \right)} \)

\( w^{\left( p \right)} E_{k}^{\left( p \right)} \)

Spring 2015 (part 1)

1

 

0.53

1

0.53

 Internal Section

 

0.48

0.37

0.41

0.15

 General Surgery Section

 

0.47

0.36

0.22

0.08

 Pediatrics Section

 

0.69

0.62

0.17

0.11

 CCU Section

 

1

0.93

0.20

0.19

Summer 2015 (part 2)

0.89

 

0.58

1

0.58

 Internal Section

 

0.76

0.63

0.41

0.26

 General Surgery Section

 

0.52

0.44

0.22

0.10

 Pediatrics Section

 

0.64

0.42

0.17

0.07

 CCU Section

 

1

0.78

0.20

0.15

Autumn 2015 (part 3)

0.79

 

0.50

1

0.50

 Internal Section

 

0.57

0.40

0.40

0.16

 General Surgery Section

 

0.59

0.46

0.22

0.10

 Pediatrics Section

 

0.83

0.76

0.18

0.14

 CCU Section

 

0.54

0.54

0.20

0.10

Winter 2016 (part 4)

1

 

0.66

1

0.66

 Internal Section

 

0.26

0.22

0.41

0.09

 General Surgery Section

 

1

0.91

0.23

0.21

 Pediatrics Section

 

1

1

0.16

0.16

 CCU Section

 

1

1

0.20

0.20

Spring 2016 (part 5)

1

 

0.80

1

0.80

 Internal Section

 

0.98

0.75

0.40

0.30

 General Surgery Section

 

1

0.97

0.22

0.21

 Pediatrics Section

 

1

0.67

0.19

0.13

 CCU Section

 

1

0.82

0.19

0.16

Summer 2016 (part 6)

1

 

0.77

1

0.77

 Internal Section

 

0.62

0.48

0.42

0.20

 General Surgery Section

 

1

1

0.25

0.25

 Pediatrics Section

 

1

1

0.15

0.15

 CCU Section

 

1

0.93

0.18

0.17

Autumn 2016 (part 7)

1

 

0.57

1

0.57

 Internal Section

 

0.37

0.37

0.37

0.14

 General Surgery Section

 

0.72

0.63

0.23

0.14

 Pediatrics Section

 

0.58

0.44

0.20

0.09

 CCU Section

 

1

1

0.20

0.20

Winter 2017 (part 8)

0.81

 

0.52

1

0.52

 Internal Section

 

0.30

0.16

0.38

0.06

 General Surgery Section

 

1

1

0.26

0.26

 Pediatrics Section

 

0.58

0.34

0.18

0.06

 CCU Section

 

1

0.76

0.18

0.14

Self-assessment of hospital department by applying model (11) shows that of the eight parts, none is efficient. The one with the biggest efficiency is cutting time of spring 2016 (part 5); its efficiency of 0.80 is attributed to four sections. Certainly, in this cutting time, management has properly used resources. Some cutting times do not have efficient section. The reason why all sections of a cutting time can be inefficient is because the sections of a cutting time are compared with sections of all other cutting times. The rankings of the eight parts are, in sequence, Spring 2016, Summer 2016, Winter 2016, Summer 2015, Autumn 2016, Spring 2015, Winter 2017 and Autumn 2015.

In self-assessment of hospital sections, the 32 sections can be treated as independent DMUs to calculate their efficiency by applying the IFDEA model (10). The results are shown in the third column of Table 2. They are very consistent with those calculated from the intuitionistic fuzzy parallel DEA model (11); only the section efficiencies measured by IFDEA model (10) are greater than or equal to those measured from the intuitionistic fuzzy parallel DEA model (11).

The second column of Table 2 shows self-assessment of hospital department by using model (10). This model neglects the requirement that each process should have an aggregated output which is smaller than the aggregated input. By applying model (10), of the eight parts, five parts are efficient.

7 Managerial implications

Hospital managers’ awareness about the level of effectiveness and identifying reasons of efficiency and inefficiency will enable them to take actions in the event of observation on poor performance relative to the previous period which will improve the condition of the hospital. Therefore, it is required that managers of each sector have an evaluation about the performance of their sector in each period. This self-assessment will allow managers to compare the performance of each section in different time periods.

The first section (Internal Section) in the last four periods has been a downward trend with Delta growth Changes 0.18. Based on researches implemented in this case, it was found that the first section is experiencing a downward trend due to the lack of utilization of technology and the development of new knowledge in the medical field in the last four periods.

The \( \left( {n - 1} \right) \)th root of the product of the difference of both successive numbers is the growth trend of \( n \) number. The numbers in the second row represent the difference between two consecutive numbers.
The efficiency trend of the second section (General Surgery Section) in the first six periods has been an upside trend with Delta growth Changes 0.07 due to the use of experienced personnel and the use of up-to-date equipment.
The diagram of each section in eight periods is shown in Figs. 3, 4, 5 and 6 which clearly show the upside and downward trend of each section in different time periods.
Fig. 3

Change in trend in the efficiency of the Internal Section in eight consecutive periods

Fig. 4

Change in trend in the efficiency of the General Surgery Section in eight consecutive periods

Fig. 5

Change in trend in the efficiency of Pediatrics Section in eight consecutive periods

Fig. 6

Change in trend in the efficiency of CCU Section in eight consecutive periods

Considering the results of this paper, it is suggested that policy makers and managers of hospital apply the following suggestions to improve hospital performance. The proposed model in this paper can be used to determine the efficient DMUs and inefficient DMUs of hospital departments, when some inputs and outputs are intuitionistic fuzzy in nature. The inefficient DMUs must be carefully examined, and the causes of inefficiency must be identified and resolved. Effective DMUs should be considered as patterns for inefficient DMUs. In the case of parallel systems, there is a need to create an accurate managerial system for coordinating sections. The empirical findings can be used as useful guidance for policy makers and managers of hospital to improve the hospital system.

The use of self-assessment technique has led to a comparison of the hospital in different time periods, which helps managers to identify the efficient and inefficient intervals.

8 Conclusions

Regarding the substantive differences in organizations, self-assessment is considered as one of the best assessment methods. Organizations can cut their activities in different time horizons and compare their performance at any point in time with other times and discover the reasons of their efficiency and inefficiency. In this paper, we developed an IFPDEA self-assessment model to measure the efficiency of the parallel systems at the time interval when some inputs and outputs are intuitionistic fuzzy in nature. For this type of system, the conventional IFDEA model is not appropriate, because it treats the system as a whole, while the IFPDEA model of this paper considers the operations of the internal processes in measuring the intuitionistic fuzzy efficiency. To ensure the validity of the proposed model, we have considered the self-assessment problem of eight cutting times from a hospital with three intuitionistic fuzzy outputs.

Parallel and series structures are two basic structures for network systems. A complicated system can be represented by a parallel system of series processes or a series system of parallel processes (Kao 2012). Therefore, the applicability of the proposed approach to more complex non-parallel structures is a subject for further research.

The current model is under the assumption of constant return to scale (CRS). This model may be modified to measure efficiency of parallel network structure by variable return to scale (VRS) model in future research. This paper measures the efficiency of a parallel system in intuitionistic fuzzy environment, while various other types of uncertainty are other areas for further studies.

Notes

Compliance with ethical standards

Conflict of interest

The authors do hereby declare that there is no conflict of interests of other works regarding the publication of this paper.

Human and animal rights

This article does not contain any studies with human participants or animals performed by any of the authors.

References

  1. Castelli L, Pesenti R, Ukovich W (2004) DEA-like models for the efficiency evaluation of hierarchically structured units. Eur J Oper Res 154(2):465–476MathSciNetzbMATHGoogle Scholar
  2. Charnes A, Cooper WW, Rhodes E (1978) Measuring the efficiency of decision making units. Eur J Oper Res 2(6):429–444MathSciNetzbMATHGoogle Scholar
  3. Chowdhury H, Zelenyuk V (2016) Performance of hospital services in Ontario: DEA with truncated regression approach. Omega 63:111–122Google Scholar
  4. Färe R (1991) Measuring Farrell efficiency for a firm with intermediate inputs. Academia Economic Papers 19(2):329–340Google Scholar
  5. Garcia-Lacalle J, Martin E (2010) Rural vs urban hospital performance in a ‘competitive’public health service. Soc Sci Med 71(6):1131–1140Google Scholar
  6. Grzegorzewski P (2003) Distances and orderings in a family of intuitionistic fuzzy numbers. In: Paper presented at the EUSFLAT Conf, pp 223–227Google Scholar
  7. Hajiagha SRH, Akrami H, Kazimieras Zavadskas E, Hashemi SS (2013) An intuitionistic fuzzy data envelopment analysis for efficiency evaluation under uncertainty: case of a finance and credit institution. E a M: Ekonomie a Management 161:128–137Google Scholar
  8. Kao C (2009) Efficiency measurement for parallel production systems. Eur J Oper Res 196(3):1107–1112MathSciNetzbMATHGoogle Scholar
  9. Kao C (2012) Efficiency decomposition for parallel production systems. J Oper Res Soc 63(1):64–71Google Scholar
  10. Kao C (2014) Efficiency decomposition for general multi-stage systems in data envelopment analysis. Eur J Oper Res 232(1):117–124MathSciNetzbMATHGoogle Scholar
  11. Kao C, Hwang S-N (2008) Efficiency decomposition in two-stage data envelopment analysis: an application to non-life insurance companies in Taiwan. Eur J Oper Res 185(1):418–429zbMATHGoogle Scholar
  12. Kao C, Lin P-H (2012) Efficiency of parallel production systems with fuzzy data. Fuzzy Sets Syst 198:83–98MathSciNetzbMATHGoogle Scholar
  13. Mahapatra G, Roy T (2009) Reliability evaluation using triangular intuitionistic fuzzy numbers arithmetic operations. World Acad Sci Eng Technol 3(2):350–357MathSciNetGoogle Scholar
  14. Nagoorgani A, Ponnalagu K (2012) A new approach on solving intuitionistic fuzzy linear programming problem. Appl Math Sci 6(70):3467–3474MathSciNetzbMATHGoogle Scholar
  15. Nedelea IC, Fannin JM (2013) Analyzing cost efficiency of critical access hospitals. J Policy Model 35(1):183–195Google Scholar
  16. Puri J, Yadav SP (2015) Intuitionistic fuzzy data envelopment analysis: an application to the banking sector in India. Expert Syst Appl 42(11):4982–4998Google Scholar
  17. Rosko MD, Mutter RL (2011) What have we learned from the application of stochastic frontier analysis to US hospitals? Med Care Res Rev 68:75S–100SGoogle Scholar
  18. Zimmermann HJ (1996) Fuzzy set theory and its applications, 4th edn. Kluwer, BostonzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Industrial Engineering and Management, Shahrood University of TechnologyShahroodIran
  2. 2.Kishore Bharati Bhagini Nivedita CollegeBehalaIndia

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