Perception based performance analysis of higher education institutions: a soft computing approach

Abstract

In the tertiary education institutions, rankings have started gaining ample attention all over the world. This has created a profound impact on the indian higher education system. As a result of that, in 2015, the government of India announced National institutional ranking framework (NIRF) to rank the indian institutions. NIRF is based on multiple parameters which are evaluated by standardised survey. In this work, a mathematical model, which can handle such multiple parameters to rank higher education institutions (HEIs), has been proposed. In this model, six criteria, named as, student intake, faculty strength, expenditure of the institution, research paper published per faculty, placements and perception, are considered. Since the criterion perception is a qualitative criterion and can not be modelled by classical mathematics, fuzzy rule based inference system is proposed to determine its precise value. Then DEA-Entropy-TOPSIS approach has been employed to rank HEIs. To emphasize the applicability of the proposed method, a numerical illustration is provided. It is asserted that this proposed mathematical model is a unique HEIs ranking approach involving human perception.

1 Introduction

Higher educational institutions are experiencing the highest order of competition globally with an impact of ranking systems like Quacquarelli Symonds (QS) rankings and Times Higher Education (THE) rankings. In 2015, the government of India took a conscious decision to introduce the National Ranking Institutional Framework (NIRF) to encourage Indian higher educational institutions (HEIs) to participate in the indian ranking system as well as to prepare them for global ranking. This framework ranks indian HEIs majorly based on the following important parameters:

• Graduation outcomes.

• Teaching, learning and resources.

• Outreach and inclusivity.

• Perception.

• Research and professional practice.

Under NIRF, the perception parameter was evaluated by an appropriate survey with a large category of Academies, Heads of institutions, HR of institutions, Students, Parents, Members of funding agencies and NGOs etc. It can be observed that above mentioned parameters play a critical role in defining the performance of an institution. Hence, it is necessary to develop a mathematical model, which can handle such multiple criteria/parameters to rank HEIs.

In the last several years, a variety of methods, such as parametric methods (stochastic and frontier analysis, ordinary least square), non-parametric methods (Data Envelopment Analysis (DEA)) and some performance indicators, have been discussed by the researchers to empirically measure the performance of decision making units (DMUs). Among all these methods, DEA (Charnes et al. 1978) has been used widely in evaluating the performance of DMUs as it can handle multiple inputs and outputs.

While each approach, available in the literature, is beneficial in its specific domain, no methodology may be prescribed as the complete solution to the question of ranking. It is noteworthy that there is no methodology available in the literature which handles the criterion of human perception. The criterion ‘perception’, explicitly what a human has a perception about the institute in his different roles, contains uncertainty and vagueness. To cope up with this uncertainty and vagueness, we propose a fuzzy set theory approach, i.e., fuzzy rule based inference system (FIS). After analysing the existing ranking parameters and methodology, this paper suggests a suitable perception based performance ranking framework by applying fuzzy rule based inference system.

In this paper, to measure HEIs efficiency, different criteria viz.; student Intake (SI), faculty strength (FS), expenditure of institutes (EX), research paper published per faculty (PP/FS), placements (PL) and perception (PC) are taken into account. These criteria are categorized into two categories, one is quantitative and the other is qualitative. The criterion ‘perception’ is considered as a qualitative criterion and the rest are quantitative criteria. To determine the precise value of the qualitative criterion ‘perception’, fuzzy rule based inference system (FIS) is introduced. Further, DEA methodology computes efficiency scores of these HEIs, the entropy method is applied for the determination of the weight of each evaluating criterion and TOPSIS approach is utilized to make a final rank of these HEIs.

The rest of the paper is organized as follows: Sect. 2 presents the detail literature survey of DEA and the related work. Section 3 provides the description of the proposed mathematical model for the ranking of HEIs. In Sect. 4, a numerical illustration is provided to validate the proposed approach for the performance evaluation of nine HEIs. Finally, concluding remarks are summarized in Sect. 5.

2 Literture survey

In the last two decades, several studies to measure the efficiency and productivity of universities/institutes have been carried out by using DEA in various countries. Some of these are listed as follows. Johnes and Johnes (1995) applied DEA to investigate the relative performance of UK universities based on the research process. Johnes (1996) examined the possibility of developing new measures for the performance of universities in UK. They developed a technique in the framework of production theory and used multiple regression in order to estimate the relationship between inputs and outputs of universities. In order to develop more understanding about the universities operations, Athanassopoulos and Shale (1997) used DEA to assess the concepts of cost and outcome efficiency.

Further, Flegg et al. (2004) examined the technical efficiency of British universities during the period 1980–1981 and 1992–1993 by applying DEA. Johnes (2006) applied DEA for the purpose of evaluating teaching efficiency by considering the first parameter as the university itself where the student completed his studies and the second parameter as the student himself. Based on their research results, each university was able to determine whether the student’s efforts or the university’s efforts should be increased to enhance their performance.

Later, Avkiran (2008) applied DEA to analyze the relative efficiency on the premise of three performance models namely overall performance, performance on fee-paying enrolments and performance on delivery of educational services. It had been discovered that although the university sector performed well on technical and scale efficiency, however, there was still a scope of improvement in performance on fee-paying enrolments scenario. Abbott and Doucouliagos (2003) used DEA to estimate technical and scale efficiency and ascertained that in spite of the combination of input–output, the efficiency level of Australian universities was recorded high relative to each other.

Worthington and Lee (2008) estimated economies of scale and scope over the period of 1998–2006 by constructing cost efficiency index using the sector benchmark. It was concluded that universities of Southern Cross, Flinders and Ballarat had a higher level of cost efficiency, whereas Royal Melbourne Institute of Technology, James Cook and New South Wales are cost inefficient. Kao and Hung (2008) carried out DEA in order to evaluate the relative efficiency of various academic departments in National Cheng Kung University, Taiwan. Efficiency decomposition helped in the identification of weak areas, where more effort should be accomplished to enhance the efficiency of the departments. Korhonen et al. (2001) proposed a systematic analysis for academic and research performance at universities and research institutes. The efficiency of research units/departments calculated by DEA complemented with the preference statistics of the decision maker.

Abramo et al. (2011) measured the technical efficiency and allocative efficiency of Italian universities research activity by applying DEA methodology. He considered university research staff as input and the impact of their research product as output. Li (2011) analyzed the output efficiency, scale efficiency and deficiency of human resources by using DEA method, and also mentioned the improvement direction. It was observed that most of the universities had low technical efficiency based on the input–output criteria. Kuah and Wong (2011) used a variant of DEA, called joint DEA maximization for performance evaluation. In India, Tyagi et al. (2009) calculated technical, pure technical and scale efficiencies, and identified the reference sets for inefficient departments.

In addition to DEA methodology, Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) (Hwang and Yoon 1981) is also used for ranking and performance evaluation of DMUs. TOPSIS approach ranks DMUs on the basis of two measures. The first measure is the shortest distance from the best option (known as the positive ideal solution) and the second measure is the longest distance from the worst option (known as the negative ideal solution). It can handle large as well as a small number of criteria and is appropriate to use with objective or quantitative data (Shih et al. 2007).

3 Mathematical model

This work proposes a mathematical model/framework to rank HEIs based on the following criteria.

• 1. Student intake (SI)

• 2. Faculty strength (FS)

• 3. Expenditure of institutes (EX)

• 4. Research paper published per faculty (PP/FS)

• 5. Placements (PL)

• 6. Perception (PC)

These criteria are generated by the earlier mentioned parameters. They can be classified into two categories: quantitative criteria (SI, FS, EX, PP/FS, PL) and the qualitative criterion (PC). Due to the imprecision intrinsic to the qualitative criterion ‘perception (PC)’, a soft computing approach based on fuzzy rule based inference system (FIS) is applied to achieve the precise value of this criterion ‘perception’.

3.1 Computation of the criterion ‘perception’

Perception refers to a process by which a person takes in something through his/her senses. It varies from one person to another and obviously, can be considered as imprecise, vague and context dependent. In traditional mathematics, there is no deterministic model available that can handle perception, specifically human perception, and can assign a precise value to it.

3.1.1 Perception parameters

In this section, we propose a methodology based on fuzzy set theory to derive a precise value for the criterion ‘perception (PC)’. This study identifies four major parameters with their sub-parameters to calculate the deterministic value of the criterion ‘perception’ on the basis of expert’s opinion and extensive literature survey. These four parameters, shown in Fig. 1, are as follows:

• 1. Student’s perception (SP)

• 2. Faculty’s perception (FP)

• 3. Research investors’ perception (RIP)

• 4. Public perception (PP)

Fig. 1

Different parameters of the criterion ‘perception’

Table 1 describes these parameters in detail, e.g., a student’s perception about an institute depends, in general, on the institute’s location, its infrastructure and its alumni reputation. Similarly, the other parameters are also explained. In total, this work considers the above mentioned 4 parameters and 14 sub-parameters.

Table 1 Description of major parameters and its sub-parameters

3.1.2 Fuzzy rule based inference system (FIS)

Zadeh (1965) proposed fuzzy set theory which deals with unclear boundaries, represents imprecise concepts and works with linguistic variables. Fuzzy set and fuzzy logic are the powerful mathematical tools to model uncertain, vague and ambiguous non-classical systems. In other words, fuzzy logic has been used as a non-parametric modeling methodology that allows computers to substitute human thinking at somewhat extent and to handle imprecise and uncertain information.

Definition 1

(Fuzzy set) (Zadeh 1965) Let X be the Universal set. A fuzzy set A is defined by a pair {x, ${\mu _{{A}}(x)}$}, $x\in X$. Here ${\mu _A(x)}$: $X \rightarrow [0,1]$ is a membership function which associates each element x in X to a real number in the interval [0, 1].

When the membership function takes values 0 and 1, it shows a complete non-membership and a complete membership respectively. However, when ${\mu _{{A}}(x)}$ gets a value between 0 and 1, it shows that x partially belongs to the fuzzy set A with degree of membership ${\mu _{{A}}(x)}$.

Fuzzy number, which is considered as a special class of a fuzzy set, is defined as follows:

Definition 2

(Fuzzy number) (Ross 2010) A fuzzy set A in ${\mathbb {R}}$ is called a fuzzy number if its membership function is normal, i.e., $\displaystyle \sup _{x\in X} \mu _A(x)=1$ and its membership function satisfies the condition of convexity, i.e., $\mu _A(\lambda x_1+(1-\lambda )x_2)\ge \min [\mu _A(x_1), \mu _(x_2)], \forall x_1,x_2\in X$ and $\forall \lambda \in [0,1]$.

The fuzzy rule based inference system (FIS) is the process to formulate the map from a specified input to an output by using the deep concepts of fuzzy set theory. The FIS model has been applied to a wide variety of real life problems (Lai and Tsai 2009; Zarrazvand and Shojafar 2012, etc.). In this work, FIS has been used to determine a precise value of the HEI’s ranking criterion ‘perception’.

Figure 2 represents FIS in which the fuzzifier maps crisp inputs from X to fuzzy sets over X and the defuzzifier maps the FIS output, i.e., fuzzy sets over Y to crisp outputs in Y. The core of the inference engine in FIS is the set of fuzzy if-then rules. In this research work, fuzzy rules are constructed based on Mamdani FIS. It consists of five steps:

• 1. Determine membership functions for the input fuzzy sets,

• 2. Define a set of fuzzy rules, combine the fuzzified inputs according to the fuzzy rules in order to establish a rule strength by applying fuzzy operator,

• 3. Find the consequence of the rule by combining the output membership function and the rule strength through implication method, and

• 4. Then combine the consequences in order to obtain an output applying aggregation method and finally,

• 5. Defuzzification of the output to receive a crisp output.

Fig. 2

Fuzzy inference system (FIS)

There are various types of fuzzy number/membership function defined in the literature. The most popular membership functions are triangular fuzzy numbers, trapezoidal fuzzy numbers and bell-shaped fuzzy numbers (Zimmermann 1991).

If-Then fuzzy rule R is defined as ‘If premise (antecedent), Then conclusion (consequent)’. Consider that a given fuzzy rule has multiple antecedents, i.e., fuzzy rule R is of type ‘If x is A and/or y is B, Then z is C’. Then the fuzzy operator (and/or) is used to obtain a single number $\alpha _R=\min / \max$$\{\mu _A (x), \mu _B(y)\}$ respectively, where $\alpha _R$ represents the result of the antecedent evaluation. Minimum is the commonly used implication operator, expressed by the equation $\mu _{R}^C(x,z)=\min \{\alpha _R,\mu _C(z)\}$, where the fuzzy set C is consequent in the fuzzy rule R. For n fuzzy rules in FIS, the aggregation operator max is preferred to get the output and then the commonly used defuzzification technique ‘centroid method’ (Ross 2010) is applied to get the crisp value of the output.

In FIS, setting up the rules is the most challenging task. At first, with experts’ knowledge, antecedent and consequent need to be represented as fuzzy sets in the form of linguistic variables. In this work, five important qualitative attributes [Excellent (Excel), VeryGood (VG), Good (G), Average (Avg), Low (L)] are considered for antecedents, i.e., each of the four major parameters (SP, FP, RIP and PP). For simplicity, consequent is also described by the same qualitative attributes. To understand, let us consider one of the rules as follows: ‘If SP is good, FP is VG, RIP is excel and PC is good, Then (according to the expert’s knowledge) PC is VG’. The number of the fuzzy rules in FIS are determined by the membership functions and the count of the inputs. It is important to note that this methodology uses generalized bell-shaped membership functions for the inputs and output as these functions contain successive output and smoothly generates less fuzzy square measure, which made less fuzziness. The fuzzy sets are described as corresponding generalized bell shaped fuzzy number in Table 2. Hence, Fig. 3 shows the bell shaped membership function for sub-parameter ‘student’s perception (SP)’ in Mamdani FIS by using Matlab tool: fuzzy logic designer. Every rule has a unique output defined for every possible set of inputs. Thus, there are $5^4=625$ possible fuzzy rules and are shown in Table 3.

Table 2 Linguistic variables for FIS’s inputs and output, and their corresponding fuzzy numbers

Fig. 3

Membership function of ‘student’s perception (SP)’

Table 3 Inference rules for fuzzy inference system

3.2 Computation of the efficiency scores of each HEI

In this section, DEA technique (Charnes et al. 1978) has been used to measure relative efficiencies of HEIs without prior weights on inputs and outputs. It is measured by the ratio of total weighted output to total weighted input in the ratio form of DEA. This approach reduces the multiple output/multiple input situation to a single virtual input and a single virtual output model (for each HEI). Further, for a specific HEI, the measure of efficiency is obtained as the ratio of this single virtual output to single virtual input and it is a function of the corresponding multipliers. Efficiency of each HEI is determined based on input and output criteria.

Construct a decision matrix consisting of n HEIs and m criteria, with the intersection of each HEI and criterion given as $x'_{ij}$. Then, the normalized decision matrix $(x_{ij})_{(n\times m)}$ is constructed by using the following equation.

$$\begin{aligned} x_{ij} = \frac{x'_{ij}}{\sqrt{\displaystyle \sum \nolimits _{k=1}^{n} x^{'2}_{kj}}}, i=1,2,\ldots ,n; j=1,2,\ldots ,m. \end{aligned}$$

(1)

Here, n is the number of HEIs considered for ranking, $x'_{ij}$ is the value of the jth criterion for the ith HEI in decision matrix and $x_{ij}$ is the value of the jth criterion for the ith HEI in the normalized decision matrix.

All these criteria will be classified into two categories, one is output and the other is input. Consider the number of input as $m_1$ and the number of output as $m_2$, and clearly, the total number of criteria m is $m_1+m_2$.

Based on the number of inputs and outputs, $x_{ij}$ is considered into two different ways, named as $y_{ip}$ and $z_{iq}$. Here, $y_{ip}$ and $z_{iq}$ are values of $p\mathrm{th}$ input and $q\mathrm{th}$ output respectively, for the $i\mathrm{th}$ HEI in the normalized decision matrix.

DEA method solves n number of linear programming problems (Charnes et al. 1978), one for each HEI, as given below:

$$\begin{aligned} \max g_i =\max \bigg (\sum _{q=1}^{m_2} v_q y_{iq} \bigg ) \end{aligned}$$

(2)

subject to,

$$\begin{aligned} \sum _{p=1}^{m_1} u_p x_{ip}= & {} 1, \\ \sum _{q=1}^{m_2} v_q y_{iq} - \sum _{p=1}^{m_1} u_p x_{ip}\le & {} 0,\\ u_p\ge & {} 0, \\ v_q\ge & {} 0, \end{aligned}$$

where $p = 1,\ldots ,m_1;~q = 1,\ldots , m_2; ~i=1,2,\ldots ,n.$

Here, variables $u_p$ and $v_q$ are non-negative weights associated with $p\mathrm{th}$ input and $q\mathrm{th}$ output respectively and $g_i$ is the efficiency score of the $i{\text {th}}$ HEI.

3.3 Computation of the weight of each criterion using entropy method

Entropy method is a measurement of degree of randomness in a system, hence it can be useful in the measurement of effective information provided by the data. When the evaluating objects have a large difference between each other on a particular criterion, the entropy is smaller, which indicates that when the criterion provides more effective information, the weight index is larger. On the contrary, the smaller the difference, larger is the entropy, which shows that the less effective information provided by criterion results in the smaller value of the weight index. When the value of every evaluating object reaches the same value, then the entropy reaches the maximum, which means that this criterion doesn’t provide any useful information in decision making. Hence, those criteria will not be considered for further study.

Let $J_\mathrm{B}$ and $J_\mathrm{NB}$ be sets of beneficial (maximization) criteria and non-beneficial (minimization) criteria, respectively. Then, the $t_{ij}$ matrix will be obtained after normalizing the decision matrix using the following equations:

if $j \in J_\mathrm{B}$:

$$\begin{aligned} t_{ij} = \frac{x_{ij}-\min [x_{ij}]}{\max [x_{ij}]-\min [x_{ij}]}. \end{aligned}$$

(3)

if $j \in J_\mathrm{NB}$:

$$\begin{aligned} t_{ij} = \frac{\max [x_{ij}]-x_{ij}}{\max [x_{ij}]-\min [x_{ij}]}. \end{aligned}$$

(4)

Here, $t_{ij}$ be the data of the $i\mathrm{th}$ HEI on the $j\mathrm{th}$ evaluating criterion, $i=1, \ldots , n$ and $j = 1,\ldots ,m$.

For m criteria and n evaluating objects, the entropy of the $j\mathrm{th}$ evaluating criteria is defined as:

$$\begin{aligned} H_j= & {} -\,k\sum _{i=1}^n f_{ij}\ln (f_{ij}), \end{aligned}$$

(5)

$$\begin{aligned} \text {where } f_{ij}= & {} \frac{t_{ij}}{\displaystyle \sum \nolimits _{i=1}^nt_{ij}} \text { and } \displaystyle k = \frac{1}{\ln (n)}. \end{aligned}$$

(6)

If $f_{ij} = 0$, then $f_{ij} \ln (f_{ij}) = 0$.

The weight of $j\mathrm{th}$ evaluating criteria is defined as:

$$\begin{aligned} \displaystyle w_j = \frac{1-H_j}{m-\displaystyle \sum \nolimits _{r=1}^m H_r}, j=1,2,\ldots ,m, \end{aligned}$$

(7)

where $0 \le w_j \le 1$ and $\displaystyle \sum \nolimits _{j=1}^m w_j = 1$.

3.4 Ranking of HEIs using TOPSIS method

The Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) picks the best option which has the shortest distance from the positive ideal solution and the farthest distance from the negative ideal solution. Option with the maximum benefits and minimum cost of all selected ones is considered as the positive ideal solution and the one with the minimum benefits and maximum cost as the negative ideal solution. Ranking of all the HEIs are assessed on the basis of the relative closeness to the ideal solutions.

In Sect. 3.2, DEA technique shortlists the HEIs with the efficiency score greater than or equal to 1. Let $n'$ be the number of these shortlisted HEIs after DEA methodology. Thus, the obtained decision matrix will be normalized as follows:

$$\begin{aligned} r_{ij} = \frac{x_{ij}}{\sqrt{\displaystyle \sum \nolimits _{i=1}^{n'} x_{ij}^2}}, ~~~i = 1,2,\ldots ,n';~j = 1,2,\ldots ,m. \end{aligned}$$

The weighted normalized decision matrix is developed as:

$$\begin{aligned} v_{ij} = w_j r_{ij} ~~~i = 1,2,\ldots ,n'; ~j = 1,2,\ldots ,m \end{aligned}$$

(8)

and $w_j$ is the weight of the $j\mathrm{th}$ evaluating criteria as computed in Sect. 3.3.

The following formulae helps to find the positive and negative ideal solutions as:

$$\begin{aligned} A^+= & {} \{v_1^+,v_2^+,\ldots ,v_{n'}^+\}\\= & {} \Big \{\Big (\text {Max} (v_{ij}) | j\in J_\mathrm{B} \Big ), \Big (\text {Min} (v_{ij}) | j\in J_\mathrm{NB} \Big ) \Big \}\\ A^-= & {} \{v_1^-,v_2^-,\ldots ,v_{n'}^-\} \\ {}= & {} \Big \{\Big (\text {Min} (v_{ij}) |j\in J_\mathrm{B} \Big ), \Big (\text {Max} (v_{ij}) |j\in J_\mathrm{NB} \Big )\Big \} \end{aligned}$$

where $J_\mathrm{B}$ and $J_\mathrm{NB}$ are sets of beneficial criteria and non-beneficial criteria, respectively.

The positive ideal ($S_i^+$) and negative ideal ($S_i^-$) solutions are calculated as:

$$\begin{aligned}&S_i^+ = \sqrt{\sum _{j=1}^m(v_i^+ - v_{ij})^2}, ~~~i = 1,2,\ldots ,n'. \end{aligned}$$

(9)

$$\begin{aligned}&S_i^- = \sqrt{\sum _{j=1}^m(v_i^- - v_{ij})^2}, ~~~i = 1,2,\ldots ,n'. \end{aligned}$$

(10)

Further, the closeness coefficient ($C_i$) is computed as:

$$\begin{aligned} C_i = \frac{S_i^-}{S_i^- + S_i^+}, ~~~i = 1,2,\ldots ,n', \end{aligned}$$

(11)

where $0< C_i < 1$.

Now, HEIs are ranked according to the decreasing order of the values of closeness coefficient $C_i$. In other words, rank 1 is assigned corresponding to the highest value of closeness coefficient whereas the lowermost rank is assigned to the one with lowest value of closeness coefficient.

4 Numerical illustration

In this section, a numerical illustration is provided to explain the proposed methodology for ranking the HEIs. For this purpose, nine HEIs are considered and are assessed based on criteria, i.e. student Intake (SI), faculty strength (FS), expenditure of institutes (EX), research paper published per faculty (PP/FS), placements (PL) and perception (PC).

The qualitative criterion ‘perception’ is computed by applying FIS and is shown in Fig. 4. Henceforth, Table 4 shows the computed value of ‘perception’ criterion for all these HEIs by applying Mamdani FIS.

Among the selected six criteria, four criteria, i.e., SI, FS, EX, PC are considered as inputs and two criteria, i.e., PP/FS, PL as outputs. Thus, on the data provided at NIRF website, the decision matrix is generated for the performance evaluations of nine HEIs and is shown in Table 5. Then, Table 6 presents its normalized decision matrix.

Fig. 4

Simulation results of fuzzy logic inference system

Table 4 Value of perception criterion

Table 5 Decision matrix $(x'_{ij})_{n\times m}$

Table 6 Normalized decision matrix $(x_{ij})_{n\times m}$

Further, efficiency scores of these HEIs are calculated by the DEA methodology, exhibited in Table 7. It is observed that among these 9 HEIs, only five HEIs’ efficiency score are greater than or equal to 1. Consequently, these HEIs are shortlisted to analyze their performance. Further, entropy method has been employed as described in Sect. 3.3. By using Eqs. (3) and  (4), $t_{ij}$ matrix is constructed for $i=1,\ldots ,5$ and $j=1,\ldots , 6$ and then, entropy for each criterion $H_j$, shown in Table 8, is determined by using Eq. (5). Further, the weight for each criterion has been determined by using Eq. (7) and results are shown in Table 9.

Table 10 exhibits weighted normalized decision matrix by using Eq. (8), where $v_{ij}$ denotes the value of jth criterion ($j = 1, 2, \ldots , 6$) with respect to i${\text {th}}$ HEI ($i = 1, 2, \ldots , 5$). Table 11 shows the results of corresponding separation measures ($S_i^+$ and $S_i^-$) and closeness coefficient values ($C_i$) for all 5 HEIs which are calculated by using Eqs. (9)–(11), respectively.

Table 7 Efficiency score of each HEIs

Table 8 Entropy matrix ($H_{j}$, $j=1,\ldots , m$)

Table 9 Weight matrix ($w_{j}$, $j=1,\ldots , m$)

Table 10 Weighted normalized matrix  ($v_{ij}$)

Table 11 Separation measures ($S_i^+$ & $S_i^-$) and closeness coefficient ($C_i$) values

This study reveals that ranking of HEIs is $\text { inst. } 1> \text { inst. } 4>\text { inst. } 5> \text { inst. } 2 >\text { inst. } 9.$ With this numerical illustration, it is concluded that inst. 1 is performing well among all other institutes, whereas other institutes need improvement in the respective criterion. It is clearly visible by analyzing all five HEIs on the basis of individual criterion, why inst. 1 is best performed and inst. 9 is the least performed. From Table 11, it is observed that for beneficial criterion, highest value indicates the best performance. In the contrary, for non-beneficial criteria, lowest value indicates the best performance. Institution 1 is performing best in SI, PP/FS and PL criteria, while Institution 9 is performing worst in FS, PC and PP/FS. It also matters that how far are they from ideal solution.

Further, it can be concluded from Table 11, that performances of HEIs can be improved by paying attention to individual evaluating criterion. Inst. 2 requires attention in the case of SI and EX. In the same way, inst. 9 requires more attention in the case of PP/FS, PL and FS. Remaining four HEIs (Institute 3, Institute 6, Institute 7 and Institute 8) which were not considered in TOPSIS after DEA method, need special attention in every criterion, to improve their overall performance.

5 Conclusion and future work

In this paper, a novel mathematical model for HEIs ranking has been proposed. This model is based on mainly six criteria, namely, student Intake (SI), faculty strength (FS), expenditure of institutes (EX), research paper published per faculty (PP/FS), placements (PL) and perception (PC). Among all these criteria, ‘perception’ is the only qualitative criterion, and therefore, can be precisely quantified by fuzzy rule based inference system. Further, DEA-Entropy-TOPSIS approach has been implemented to rank the chosen HEIs. It can be concluded that the derived model, which is comprehensive in nature and novel, can be applied to a relative class or group of institutions irrespective of their fields to ascertain their performance and rank them accordingly.