Study Of Fuzzy-Ahp Model To Search The Criterion In The Evaluation Of The Best Technical Institutions: A Case Study

Study Of Fuzzy-Ahp Model To Search The

Criterion In The Evaluation Of The Best

Technical Institutions: A Case Study

Debmallya Chatterjee1*, Dr.Bani Mukherjee2

1

Senior Lecturer, Department of Business Administration, Management Institute of Durgapur, Durgapur-713212, India. *Email: [email protected]

2

Associate Professor, Department of Applied Mathematics, ISM Dhanbad, Dhanbad, Jharkhand, India. ABSTRACT

Technical education contributes a major share to the overall education system and plays a vital role in the social and economic development of the nation. A technical Institution was under government undertaken or funded by government agencies before nineties but since nineties onwards, there was a metamorphic turn around the field of education. Thousands of private institution emerged with a business orientation leading to the degradation of quality education. The stakeholders are in a state of utter confusion in the selection of a technical institution for their growth and prosperity. In this paper by fuzzy analytical hierarchy process (fuzzy-AHP) a model is developed to search the criteria’s for the evaluation of best technical institutions, which can tolerate vagueness and uncertainty of human judgment. At the end, a case study is presented to make this model more understandable.

Key words: AHP, Fuzzy-AHP, Triangular fuzzy numbers, Technical education 1. INTRODUCTION

1.1 Background of the study

Technical education in India contributes a major share to the overall education system and plays a vital role in the social and economic development of our nation. In India, technical education is imparted at various levels such as: craftsmanship, diploma, degree, post-graduate and research in specialized fields, catering to various aspects of technological development and economic progress [22]. Since 1991 there is a metamorphic turn around in the field of Technical Education in India. Three new things had emerged; Privatization, Tailor made courses and introduction of Information Technology in the education system. Thousands of Private self financing Technical Institutions have emerged with a business orientation. Few are truly worthy and offering quality education in India but many of them are compromising with the quality. The stakeholders are in a state of utter confusion in choosing a quality Institution for their career development and prosperity. Hundreds of agencies are publishing materials containing contradictory judgments about the Institutions and thus confusing the stakeholders at the highest level. This paper presents a Fuzzy-AHP model to evaluate technical institutions. In this model Triangular Fuzzy numbers are utilized along with Analytical Hierarchy Process for better result.

1.2 Objective of the Study

The primary objective of the present study is to identify the performance measurement indicators for evaluating the best Technical Institutions in India and secondly to develop an appropriate Fuzzy-AHP model for evaluating the Technical Institutions.

1.3 Introduction to Analytic Hierarchy Process

wise comparisons, which result in the determination of factor weights. Finally the alternative with the highest total weighted score is selected as the best alternative [13].

1.4 Fuzzy set and Triangular Fuzzy Number

To deal with vagueness of human thought, Zadeh first introduced the fuzzy set theory, which was oriented to the rationality of uncertainty. A major contribution of fuzzy set theory is its capability of representing vague data. A fuzzy set is a class of objects with a membership function ranging between zero and one. It was specifically designed to mathematically represent uncertainty and vagueness. Fuzzy set theory implements groupings of data with boundaries that are not sharply defined (i.e. fuzzy). Any methodology or theory implementing “crisp” definitions such as classical set theory, arithmetic, and programming, may be “fuzzified” by generalizing the concept of a crisp set to a fuzzy set with blurred boundaries. The benefit of extending crisp theory and analysis methods to fuzzy techniques is the strength in solving real-world problems, which inevitably entail some degree of imprecision in the variables and parameters measured and processed for the application [3]. A triangular fuzzy number (TFN) is the special class of fuzzy number whose membership is defined by three real numbers, expressed as (l, m, u). The triangular membership function is represented as follows [18]. Figure 1.1 displays the structure of a Triangular Fuzzy Number (TFN).

. . . .

,

, . . . .

.

. . . ( 1 )

0 ,

A

x l

l x m

m l

u x

m x u

u m

o t h e r w i s e





 _{ }

 _

  

  _  

  

Fig 1.1 Triangular Fuzzy Number

The operational laws between two triangular fuzzy numbers M1 and M2 are as follows

1 2 1 2 1 2 1 2

1 2 1 2 1 2 1 2

( , , ) . . . ( 2 )

( , , ) . . . ( 3 )

M M l l m m u u

M M l l m m u u

    

 

1.5 Fuzzy- Analytical Hierarchy Process Method (Fuzzy-AHP)

The conventional AHP method is incapable of handling the uncertainty and vagueness involved in the mapping of one's preference to an exact number or ratio. The major difficulty with classical AHP is its inability in mapping human judgments. In recent years it has been observed that due to confusion in decision makers mind probable deviations should be integrated to the decision making process [1]. In Fuzzy-AHP, pair wise comparisons are done using fuzzy linguistic preference scale ranging from 0 to 10. For simplicity, the reciprocal fuzzy numbers are replaced by individual TFN’s in the pair wise comparison matrix [12].

l

m

u

1.6 Organisation of the Study

In order to develop a Fuzzy-AHP decision making model for the evaluation of private technical institutions, the piece of work is organized as follows. In the next section review of existing work is done. Then the methodology is introduced along with the stages of development of the model. After that an empirical study is conducted along wÅŸh its findings. Finally the paper ends with the conclusion.

1.7 . REVIEW OF EXISTING WORK

Among the different methodologies used, it has been observed that Fuzzy-AHP method was used extensively in decision making. The method was used to select the best bridge construction method among the alternatives avoiding the inconsistency there in [12]. In the literature, fuzzy-AHP has been widely used in solving many complicated decision making problems. Fuzzy-AHP and its extensions were developed in selecting the key capabilities in technology management [20]. The fuzzy AHP approach was used in the evaluation of computer integrated manufacturing alternatives. The same approach was used in the selection of the best location for a facility and in the evaluation of catering firms in Turkey [9]. Fuzzy Integrated Analytic Hierarchy Process Approach is used for Selecting Strategic Big-sized R&D Programs in the Sector of Energy Technology Development [17]. It is further used it in Multi-criteria Supplier Evaluation and vendor selection [3, 8]. Many researchers who have studied Fuzzy AHP provided evidences that it shows more efficiency in handling human judgments than the Classical AHP method [5, 6, 7, 10].

2. METHODOLOGY

2.1 Development of Fuzzy-AHP model in multicriteria decision making

2.1.1 Conceptual Hierarchy of Fuzzy –AHP model

Analytical Hierarchy Process starts by laying out the overall hierarchy of the decision making problem. The hierarchy is structured from the top (the overall goal of the problem) through the intermediate levels (criteria and sub-criteria on which subsequent levels depend) to the bottom level (the list of alternatives). Each criterion in the lower level of hierarchy is compared with respect to the criteria in the upper level of hierarchy. The criteria in the same level are compared using pair wise comparison. Fig 2.1 describes the hierarchy of a decision making problem.

Fig 2.1

Hierarchy of the Decision making problem

2.1.2 Fuzzy pair wise comparison method

Once the hierarchy is established, the pair wise comparison evaluation takes place. All the criteria on the same level of the hierarchy are compared to each of the criterion of the preceding (upper) level. A pair wise comparison is

Overall Goal

Criteria 1

Criteria 2

Criteria 3

Criteria 4

performed by using Fuzzy linguistic terms in the scale of 0 – 10 described by the Triangular Fuzzy Numbers in the Table 2.1.

Table 2.1

Fuzzy Importance scale with TFN

Verbal judgment Explanation Fuzzy number

Extremely Un-important (EXUI) A criterion is strongly inferior to another (0, 1, 2) Un-important (UI) A criterion is slightly inferior to another (1, 2.5, 4) Equally Important (EI) Two criteria contribute equally to the

object

(3, 5, 7)

Moderately Important (MI) Judgment slightly favor one criterion over another

(6, 7.5, 9)

Extremely Important (EXI) Judgment strongly favor one criterion over another

(8, 9, 10)

To reflect pessimistic, most likely and optimistic decision making environment, triangular fuzzy numbers with minimum value, most plausible value & maximum value are considered.

Here the fuzzy comparison matrix is defined as

Where is the relative importance of each criteria in Pair wise comparison and are the minimum value, most plausible value & maximum value of the triangular fuzzy number.

To simplify the calculation of element weight the fuzzy pair wise comparison matrix is broken into crisp matrices formed by taking the minimum values, most plausible values & maximum values from the triangular fuzzy numbers. 2.1.3 Generation of Criteria and Sub-Criteria weight

The Normalization of the Geometric Mean (NGM) method (Buckley et al, 1985) is applied to compute weights from the fuzzy pair wise comparison matrices which is given by

1 i n i i i

a

a









, where

1/ 1 n n i ij j

a

a







 









1 2 1

2 1 2

3 1 3 2 3

1

1

....

1

....

...( 4 )

1 ...

..

..

1

n n n n n

a

a

a

a

A

a

a

a

a

_















_

_



























( L , M , U ) a_{i j}  a _{i j} a _{i j} a _{i j}

,

,

,

L M U

i j i j i j

In the above equations,

a

_i is geometric mean of criterion i.

a

_{i j} is the comparison value of criterion i to criterion j.



_i is the ith criterion's weight, where



_i



0

and

1 1 n i i



 



.

For group evaluation, it is required to aggregate evaluator’s opinions into one. Considering the evaluation given by expert

E

_i



(

a

_L( )i

,

a

_M( )i

,

a

_U( )i

)

the aggregate of all experts’ judgments can be calculated using average means

( ) ( ) ( )

1 1 1

1

1

1

,

,

. . . ( 5 )

n n n

i i i

L M U

i i i

A

a

a

a

n



n



n







 















The final weight vector is generated by defuzzyfying the average [11]

( ) ( ) ( )

1 1 1

( )

1

1

1

2

. . . ( 6 )

4

n n n

i i i

L M U

i i i

i

a

a

a

n

n

n

w

  



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_





_

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

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

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





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

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



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 





















The weight of ith sub criteria under kth main criteria is obtained

by( w _k  s _{k i} ) . . . ( 7 )

where

w

_k is the kth main criteria weight and

s

_ki is the weight of ith sub criteria with respect to kth main criteria. 2.1.4 Calculation of overall score for alternatives

Once the weight of criteria, sub criteria are evaluated and are multiplied using equation (3) to obtain global weight of sub criteria, it is required to calculate the overall score of alternatives for their evaluation.

The overall score of mth alternative is obtained by

1

. . . ( 8 )

N

m l m l

l

A

s

a









where

s

_l is the weight of lth sub criteria and

a

_{m l} is the weight of

m

t h alternative with respect to lth sub criteria.

2.2 Identification of Criteria and Sub Criteria for evaluating alternatives

One of the important steps of the proposed model is to determine all the important criteria and their relationship with the decision variables. This step is crucial because the selected criteria and sub criteria can influence the final choice. Here in this project the criteria and sub-criteria are selected based on the format mentioned by National Board of accreditation & through expert’s opinion. The alternatives taken are the private self financing technical institutions of Durgapur, West Bengal, India. The criteria and sub-criteria selected are described in Table 2.2

Criteria Sub Criteria

Campus Infrastructure Hostel, Transport/ canteen/ Internet, Power backup, Security

Faculty Teacher/ Student ratio, Qualification/ Experience of

Faculty, Faculty retention

Student Admission, Academic Result, Placement

Academic Ambience Classroom, Laboratory, Library

Teaching Learning Process Syllabus coverage, Tutorial/ remedial class, Use of Advance Teaching Aid

2.3 Construction of the detailed hierarchy of the problem

The hierarchy is constructed taking all the criteria, sub-criteria and alternatives specific to the research problem. The hierarchy is structured from the top (performance evaluation of technical institutions) through the intermediate levels (main and sub-criteria on which subsequent levels depend) to the bottom level (the list of technical institutions).Figure 2.2 describes the hierarchy in detail.

Figure 2.2

Detailed hierarchy of the problem

3. RESULTS AND DISCUSSIONS

The detail of the steps of Fuzzy-AHP model described in section 2.1.2 to 2.1.4 are explained elaborately using the data collected from experts and the engineering students of Durgapur.

3.1 Illustration of the Fuzzy-AHP model

Once the hierarchy was established and a series of questions were asked to direct pair wise comparisons, each expert performed a pair wise comparison. Hence the main criteria weights from the first e éert’s judgment can be expressed in Table 3.1.

Evaluation of Technical

Inst.

F

Faaccuullttyy C

Caammppuuss

I

Innffrraassttrruuccttuurree

S

Sttuuddeenntt

A

Accaaddeemmiicc A

Ammbbiieennccee

P

Poowweerrbbaacckkuupp S

Seeccuurriittyy H

Hoosstteell

Transport/ canteen

Qualification/ Experience Faculty retention

A

AccaaddeemmiiccRReessuulltt

P

Pllaacceemmeenntt

L

Liibbrraarryy

A

Addvvtteeaacchhiinnggaaiidd

D DIIAATTMM B BCCRREECC

B BCCEETT

T

Teeaacchhiinngg

L

Leeaarrnniinngg

L

Laabboorraattoorryy

S

Suupppplleemmeennttaarryy

P

Prroocceessss

Teacher/ Std ratio

C

Cllaassssrroooomm A

Addmmiissssiioonn

A

Alluummnnii

C

Coo--CCuurrrriiccuullaarr

C

Cuullttuurraallaaccttiivviittyy T

Tuuttoorriiaallccllaassss S

Table 3.1

First Expert’s judgment

Fuzzy weight of Main Criteria

( l m u )

Campus Infrastructure 0.4473 0.2470 0.2291

Faculty 0.5527 0.2808 0.2474

Student 0.0000 0.1777 0.1795

Academic Ambience 0.0000 0.1058 0.1220

Teaching Learning Process 0.0000 0.1132 0.1270 Supplementary Process 0.0000 0.0756 0.0950

Repeating the same procedure for all experts’ judgments following equation (5) in section 2.1.3 the global weights of the main criteria was obtained in Table 3.2

Table 3.2

Global weight of main criteria

Name of the Main Criteria Global weight of main criteria

( l m u )

Campus Infrastructure 0.3620 0.2433 0.2261

Faculty 0.3379 0.2075 0.1935

Student 0.0831 0.1766 0.1800

Academic Ambience 0.1098 0.1565 0.1609

Teaching Learning Process 0.0381 0.1064 0.1191

Supplementary Process 0.0692 0.1096 0.1203

Table 3.3

Sub criteria weights

Weight of sub criteria

Sub criteria ( l m u )

Hostel 0.3880 0.2894 0.2752

Transport/ canteen etc 0.0002 0.2282 0.2299 Power backup 0.2189 0.2966 0.2886

Security 0.1894 0.1858 0.2064

Teacher/ student ratio 0.7133 0.4867 0.4458 Qualification/ exp of faculty 0.1174 0.2755 0.2940 Faculty retention 0.1693 0.2377 0.2602

Admission 0.5000 0.4336 0.3999

Academic result 0.0000 0.2569 0.2766

Placement 0.5000 0.3096 0.3235

Classroom 0.4354 0.3544 0.3291

Laboratory 0.3737 0.3272 0.3131

Library 0.0731 0.1589 0.1755

Syllabus coverage 0.1178 0.1595 0.1823 Tutorial/remedial class 0.5370 0.4621 0.4368 Use of Adv teaching aid 0.1157 0.2367 0.2576 Alumni Association 0.3473 0.3012 0.3056 Cultural activity 0.2807 0.2818 0.2883 Co-curricular activity 0.6033 0.4902 0.4527 Seminar/ workshop 0.1160 0.2280 0.2591

Table 3.4

Global weight of sub-criteria

Global weight of sub criteria

( l m u )

Hostel 0.1405 0.0704 0.0622

Transport/ canteen etc 0.0002 0.0555 0.0520

Power backup 0.0792 0.0722 0.0653

Security 0.0686 0.0452 0.0467

Teacher/ student ratio 0.2410 0.1010 0.0863

Qualification/ exp of faculty 0.0397 0.0572 0.0569

Faculty retention 0.0572 0.0493 0.0504

Admission 0.0415 0.0766 0.0720

Academic result 0.0000 0.0454 0.0498

Placement 0.0415 0.0547 0.0582

Classroom 0.0478 0.0555 0.0530

Laboratory 0.0410 0.0512 0.0504

Library 0.0080 0.0249 0.0282

Syllabus coverage 0.0129 0.0250 0.0293

Tutorial/remedial class 0.0204 0.0492 0.0520

Use of Adv teaching aid 0.0044 0.0252 0.0307

Alumni Association 0.0132 0.0320 0.0364

Cultural activity 0.0194 0.0309 0.0347

Co-curricular activity 0.0417 0.0537 0.0545

Seminar/ workshop 0.0080 0.0250 0.0312

Table 3.5

Weights of alternatives

Weights of the Alternatives

BCREC DIATM BCET

( l m u ) ( l m u ) ( l m u )

Hostel 0.362 0.354 0.349 0.199 0.246 0.266 0.439 0.400 0.385

Transport/ canteen etc 0.271 0.297 0.306 0.490 0.427 0.401 0.239 0.276 0.293

Power backup 0.403 0.380 0.370 0.242 0.272 0.288 0.356 0.347 0.342

Security 0.713 0.474 0.444 0.117 0.246 0.264 0.169 0.281 0.293

Teacher/ student ratio 0.355 0.340 0.343 0.529 0.441 0.412 0.117 0.219 0.245

Qualification/ exp of faculty 0.339 0.340 0.343 0.331 0.330 0.329 0.331 0.330 0.329

Faculty retention 0.426 0.356 0.356 0.574 0.430 0.400 0.000 0.214 0.244

Admission 0.580 0.387 0.365 0.210 0.306 0.317 0.210 0.306 0.317

Academic result 0. 713 0.496 0.447 0.000 0.215 0.246 0.210 0.289 0.307

Placement 0.415 0.386 0.371 0.386 0.367 0.359 0.198 0.248 0.270

Classroom 0.625 0.477 0.434 0.117 0.243 0.267 0.259 0.280 0.299

Laboratory 0.655 0.439 0.405 0.123 0.250 0.275 0.223 0.311 0.320

Library 0.423 0.391 0.376 0.198 0.243 0.263 0.379 0.366 0.361

Syllabus coverage 0.309 0.289 0.278 0.280 0.274 0.269 0.256 0.259 0.258

Tutorial/remedial class 0.556 0.453 0.422 0.155 0.244 0.262 0.289 0.304 0.316

Use of Adv teaching aid 0. 790 0.474 0.432 0.117 0.246 0.270 0.169 0.281 0.299

Alumni Association 0.534 0.452 0.416 0.233 0.274 0.292 0.233 0.274 0.292

Cultural activity 0.167 0.289 0.303 0.667 0.479 0.440 0.167 0.232 0.258

Co-curricular activity 0.333 0.333 0.333 0.333 0.333 0.333 0.333 0.333 0.333

Seminar/ workshop 0.380 0.361 0.353 0.284 0.302 0.309 0.336 0.337 0.338

Table 3.6

Global weights of alternatives

Final Score of the Alternatives

BCREC DIATM BCET

( l m u ) ( l m u ) ( l m u )

Hostel 0.051 0.025 0.022 0.028 0.017 0.017 0.062 0.028 0.024

Transport/ canteen etc 0.002 0.016 0.016 0.002 0.024 0.021 0.002 0.015 0.015

Power backup 0.032 0.027 0.024 0.019 0.020 0.019 0.028 0.025 0.022

Security 0.049 0.021 0.021 0.008 0.011 0.012 0.012 0.013 0.014

Teacher/ student ratio 0.085 0.034 0.030 0.127 0.045 0.036 0.028 0.022 0.021

Qualification/ exp of faculty 0.013 0.019 0.020 0.013 0.019 0.019 0.013 0.019 0.019

Faculty retention 0.024 0.018 0.018 0.033 0.021 0.020 0.000 0.011 0.012

Admission 0.024 0.030 0.026 0.009 0.023 0.023 0.009 0.023 0.023

Academic result 0.000 0.023 0.022 0.000 0.010 0.012 0.000 0.013 0.015

Placement 0.017 0.021 0.022 0.016 0.020 0.021 0.008 0.014 0.016

Classroom 0.030 0.026 0.023 0.006 0.013 0.014 0.012 0.016 0.016

Laboratory 0.027 0.022 0.020 0.005 0.013 0.014 0.009 0.016 0.016

Library 0.003 0.010 0.011 0.002 0.006 0.007 0.003 0.009 0.010

Syllabus coverage 0.004 0.007 0.008 0.004 0.007 0.008 0.003 0.006 0.008

Tutorial/remedial class 0.011 0.022 0.022 0.003 0.012 0.014 0.006 0.015 0.016

Use of Adv teaching aid 0.003 0.012 0.013 0.001 0.006 0.008 0.001 0.007 0.009

Alumni Association 0.007 0.014 0.015 0.003 0.009 0.011 0.003 0.009 0.011

Cultural activity 0.003 0.009 0.011 0.013 0.015 0.015 0.003 0.007 0.009

Co-curricular activity 0.014 0.018 0.018 0.014 0.018 0.018 0.014 0.018 0.018

Seminar/ workshop 0.003 0.009 0.011 0.002 0.008 0.010 0.003 0.008 0.011

Sum of Weights : 0.404 0.385 0.372 0.307 0.316 0.318 0.219 0.294 0.305

3.2 Findings and discussions

From the main and sub-criteria weights in the tables is can be inferred that there exists variation between the priorities of the main and sub criteria mentioned in the model. It is further observed that the priority of the main criteria “Campus Infrastructure” is highest followed by “Faculty”. In case of sub criteria the priority is highest for ”Hostel” under “Campus Infrastructure” , “Teacher student ratio” among “Faculty”, “Admission” and “placement” among “Student”, “Classroom” among “Academic ambience” and “Co-curricular activity” among “Supplementary process”. When it comes to the alternative technical institutions it is found that the Hostel of BCET, Teacher student ratio, Placement of BCREC and Cultural activity of DIATM are the best. Finally from the defuzzified final score of the alternative technical institutions it has been observed that the overall score ofDr. B. C. Roy engineering college is the highest followed by Durgapur Institute of Advanced Technology and Management and Bengal college of engineering and Technology.

4. CONCLUSIONS

Since nineties there is a sea change in the field of Technical Education in India. Lots of Private self financing Technical Institutions have emerged with a business orientation offering readymade courses. Few of them are truly worthy and offering quality education in India but many of them are managing with the quality. The stakeholders are in a state of utter confusion in choosing a quality Institution for their career development and prosperity. Agencies are giving contradictory judgments about the Institutions and thus confusing the stakeholders at the highest level. Previously no attempt was made to generate a model which would help the stakeholders in decision making. This paper presents a Fuzzy-AHP model to overcome stakeholders problem in evaluating Technical Institutions. In this model Triangular Fuzzy numbers are utilized in collecting human judgments through linguistic variables. Further Analytical Hierarchy Process was used in generating criteria weights and sub criteria weights for the evaluation of alternatives. Although for simplicity less number of alternatives are taken but this model can be used in evaluating a number of alternatives. Further this study is not limited to the evaluation of Technical Institutions; rather it can be used in multi-criteria decision making in any field of study.

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