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

## 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)

**A**n IFN (Mahapatra and Roy 2009) \( \tilde{A}^{I} \) is

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

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

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

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.

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

- (i)
*Addition*\( \ A^{I} \oplus \tilde{B}^{I} = (a_{1} + b_{1} ,a_{2} + b_{2} ,a_{3} + b_{3} ; a_{{1^{{\prime }} }} + b_{{1^{{\prime }} }} ,a_{2} + b_{2} ,a_{{3^{{\prime }} }} + b_{{3^{{\prime }} }} ) \). - (ii)
*Subtraction*\( \ A^{I} { \ominus }\tilde{B}^{I} = \left( {a_{1} - b_{3} ,a_{2} - b_{2} ,a_{3} - b_{1} ; a_{{1^{{\prime }} }} - b_{{3^{{\prime }} }} ,a_{2} - b_{2} ,a_{{3^{{\prime }} }} - b_{{1^{{\prime }} }} } \right) \). - (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 \( \ A^{I} ,\tilde{B}^{I} > 0 \). - (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

## 3 Parallel DEA model

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.

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

## 4 Intuitionistic fuzzy DEA model

*i*th intuitionistic fuzzy input utilized by the

*k*th DMU; \( \tilde{y}_{rk}^{I} \) is the amount of the

*r*th intuitionistic fuzzy output produced by the

*k*th 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

*i*th intuitionistic fuzzy input and

*r*th intuitionistic fuzzy output of the

*k*th 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

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*

### **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*

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.

## 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.

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) |

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 | | 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 | | 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 | | 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 | | 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 | | 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 | | 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 | | 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 | | 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.

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.

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