A FUZZY INTERVAL BASED MULTI-CRITERIA HOMOGENEOUS GROUP DECISION MAKING TECHNIQUE: AN APPLICATION TO AIRPORTS RANKING PROBLEM

: This paper aims to develop and introduce a fuzzy Interval based Multi-criteria homogeneous Group Decision making technique (IMGD) to make appropriate decision under fuzzy environment. In fuzzy multi-criteria group decision making process, a group of decision makers often considers several subjective criteria for ranking a set of alternatives. Due to vague and imprecise information, decision makers generally utilize linguistic variables which are mandatorily converted into triangular or trapezoidal fuzzy numbers. The total process then becomes very complex and time consuming. The current investigation advocates fuzzy intervals instead of triangular or trapezoidal fuzzy numbers for simplification of the complex situation and ease of calculation. In this method, fuzzy intervals of performance ratings and weights assessed by homogeneous group decision makers under subjective criteria are converted into first mean fuzzy intervals then into normalized crisp numbers. The normalized crisp performance ratings and normalized crisp weights are combined together to determine initially individual contribution and then into aggregate contribution to each alternative for final ranking and selection of the alternative. The new model is demonstrated with an application to airports ranking and selection problem for better clarification and verification. The outcome of the proposed is validated with the results obtained by well-known existing MCDM techniques. The analysis shows that the proposed method is applicable, useful and effective for appropriate decision making under fuzzy MCDM environment.


Introduction
The ranking and selection procedure of airports in general involves multiple alternatives, multiple conflicting subjective criteria and a group of experts.Therefore multi criteria decision making techniques are required to employ for finding the ranking order of airports.The decision making procedures becomes hard while decision is to make under fuzzy environments considering many subjecting criteria.It becomes more complex while multiple decision makers' perception are required to incorporate in the decision.An airport ranking problem may be different from the others by type and nature of criteria, involvement of decision makers and number of alternative.Past researchers have suggested several techniques for ranking and selection of airports.A detailed literature survey on airport ranking and selection proposed and suggested by past researchers is carried out and presented in this section.Zhao et al. (2019) chooses civil airport site considering bird economical preservation by expert base selection in Dalian, China.Hammad et al. (2017) proposed a model for multi-objective optimization.Mixed integer linear programming model has been applied for solving a bi-level program.Fu et al. (2016) investigated the relocation case of a china airport considering the perspective towards risk of bird strike.Markov chain is applied to analyze the birds' flying procedure and to frame new algorithm in estimating bird strikes with aircraft.
Merkisz-Guranowska (2016) developed a method having multiple layers for solving the location selection problem on airport that involves three methods to allow significant progress than the already available approaches.The initial method extends the problem adding up criteria and constructing genetic algorithm.The second method applies theory of fuzzy set whereas the third method advocates the proposed Min approach.Yang et al. (2016) extended ease of access for indicator to airports by transportation on surface and airside.The relation between airport size and scale of aircraft network is determined by structural equation model.Bo et al. (2014) obtained the advantage of the multiple phase layer fuzzy logic approach which has the ability to remove evaluation disenchantment due to nonsensicalness in the brainpower of human being while multidirectional changes occurs.Bao (2014) solved the dilemma of airport location selection as a MADM problem and the perception of expert group were expressed with crisp decision matrix affecting the TFN to elucidate preference of the decision makers and built a numerical approach to evaluate, rank and select alternatives.Yang et al. (2014) applied a pair of ranking techniques to assess and ranking airport locations.The primary technique (WLSM) was accepted by the decision makers to calculate the weight and change the linguistic terms into crisp numbers.The subsequent technique was TOPSIS.It was used to compute closeness coefficients to find the ranking order of the alternatives.
Fuzzy ELECTRE-I and fuzzy TOPSIS were used for ranking and selection of the airports (Belbag et al., 2013).The authors considered multiple important criteria such as costs, climatic condition, environmental condition, geographical condition, potential demand, infrastructure, social effects, the extension possibility, and legal regulations and restrictions.Carmona-Benítez et al. (2013) solved the issues of airport location to maximize the total anticipated aircraft passenger requirement as the indispensable aspects by utilizing the wealth index for computing passengers' requirements.Bo et al. (2014) employed the GIS method to rank and choose airport location by collecting the geographic information of the intricate airspace area and using programming of super-map to make the geographic catalog.The airspace construction of multifaceted airspace region was compared.Subsequently, the locations of airport were ranked and selected.Zhao and Sun (2013) compared newfangled airport location selections by two different index methods.The authors measured the evaluating index relative weight and calculated total numerical differences of the schemes.Postorino and Praticò (2012) applied the multi-criteria decision making model in determination of the position of airports contained by a multi-airport organization.Sur and Majumder (2012) applied the entropy weighting method for evaluation and selection of airport location.Construction related cost per individual was considered a criterion in the model.
The gap analysis of the above literature survey clearly explores that though previous researchers have applied some existing MCDM approaches, still there exists absolute necessity of introducing new MCDM model for solving and making appropriate decision regarding airport ranking and selection under new criteria and specific environment.
The objective of the current paper is to develop and introduce a novel MCDM model under fuzzy environment for making appropriate decision in industrial application and demonstrated by illustration suitable example on airport ranking and selection.
The paper is presented by dividing it into some sections for better illustration.Section 1 presents the short introduction and literature survey.Section 2 presents the proposed mathematical algorithm which is the heart of the paper.Section 3 covers numerical examples along with solution and discussions.Section 4 furnishes some essential concluding remarks and scope for further research.

Proposed Algorithm
Let, there is a decision making problem involving multiple alternatives, multiple subjective criteria with vague information and a group of homogeneous decision making experts.For solving such a decision making problem under FMCDM the following algorithm is constructed and proposed.
Step 1: Formation of decision making committee comprising of experts from different important sections of the organizations.The member of the decision making committee can be expressed as follows.
Here, i D denotes the i th decision maker or expert.Whereas p is the number of decision makers.
Step 2: Make a list of the available feasible alternatives.The set of listed alternatives are under consideration for performance assessment.The alternatives can be represented in the form of a row matrix as shown in the Eq. ( 2).
Here, i A denotes the i th alternative.m is the number of alternatives.
Step 3: Identify the significant criteria for decision making regarding evaluation and selection of the alternatives.The set of decision criteria can be represented in the form of the transpose of a row matrix as shown in the Eq.(3).Step 4: Decision matrix: Formation of decision matrix involves alternatives, criteria, decision makers and performance ratings.Each alternative is assessed with respect to each criterion as per the preference of the decision makers in terms of linguistic variables.If all criteria are subjective, then only linguistic variables are used by the decision makers for estimating the performance rating of the alternatives with respect to criteria.The decision matrix is formed by the decision maker applying their knowledge, preference and perceptions.
Here, () ji k x denotes the linguistic performance rating of i th alternative with respect j th criterion, assessed by k th decision maker.
Step 5: Decision matrix in interval: Linguistic terms of decision matrix are transformed into intervals.An interval is expressed by two values viz.lower and upper.It is required for quantification of the assessment of the alternatives with respect to criteria.The decision matrix in interval can be represented in following matrix form.
denotes the fuzzy interval expressing the performance rating of i th alternative under j th criterion by k th decision maker.
Step 6 Determine the geometric mean of performance rating using the following Eq.( 6).
Step 7: Determine the mean crisp performance rating using the Eq.(7a) The mean crisp performance rating s of the alternative with respect to criteria are accommodated in a matrix as provided below.Step 8: Construct the weight matrix in linguistic variables.Importance weights of the different criteria may vary from criteria to criteria, decision maker to decision maker and problem to problem.In the current problem each decision maker estimates impotence weight for each criterion based on own experience, knowledge and perception.Varying degrees of linguistic variables are used for the purpose of measuring the importance weights of the criteria which are accommodated in the following matrix.Here, yjk is the linguistic weight of j th criterion provided by the k th decision maker.Here, m is the number of decision makers and n is the number of criteria.
Step 9: Conversion of linguistic weights into corresponding intervals.This conversion is absolutely necessary for quantification of assessment.The importance weights of the criteria in terms of interval can be represented in the following matrix.
denotes the importance weight of the jth criterion assigned by k th decision maker.
Step 10: Calculate the average criteria weight in interval by calculating the arithmetic mean of the lower and upper values separately by using Eq. ( 10).
( ) Step 11: Compute the crisp weight for each criterion using the Eq. ( 11).Step 12: Measure the normalized crisp weight using the following normalization Eq.( 12).Step 13: Determine individual contribution.This investigation suggests implementation of trigonometric functions for measuring the contribution of individual criterion towards the performance evaluation of the alternatives under consideration.Individual contribution of each criterion to each alternative is computed by applying the Eq. ( 13).
( )  is a modifier.If the modifier is less than unity it can be termed as reducer.If the modifier is greater than unity it can be termed as amplifier.The exact value depends upon the data of the associated problem and the decision of the decision makers.
Step 14: Determine total contribution.It is the aggregate of the total individual contribution of all the criteria under consideration applying the Eq. ( 14). ( ) Arrange the alternative according to decreasing order of the total contribution of.Select the best alternative with the highest total contribution.

Illustrative Example
The proposed algorithm has been illustrated by a suitable decision-making problem on airport selection.The problem is discussed by subdividing it into two subsections viz.problem definition, calculation and discussion as described below.

Problem Definition
The proposed algorithm is demonstrated by illustrating a suitable example on airport selection considering subjective criteria though homogeneous group decision making.This example is partially cited from Wang and Lee (2007).
In this example, a decision-making committee is formed with four rational decision makers having necessary knowledge in the domain.The decision makers are denoted by D1, D2, D3 and D4.The members of the decision-making committee unanimously decided to consider a set of 15 subjective criteria viz.C1: Return to capital (operation profit), C2: Cleanness and comfort at terminal, C3: Trolley move toward travelers, C4: Direction and signal, C5: Aerodrome controlling system, C6: Security, C7: Check-in and check-out system and time, C8: Take-off and loading time, C9: Traffic connecting city, C10: Crew courtesy, C11: Airport scale, C12: Parking lots, C13: Noise pollution system, C14: Navigation controlling system, and C15: Aircraft safety control.
Three alternative airports are initially chosen for further evaluation.The airports are designated A1, A2 and A3.The proposed multi-criteria decision-making algorithm is applied for evaluation, ranking and selection of the airport under consideration.The solution procedure of the airport selection problem is illustrated through the demonstration of the developed and proposed paradigm in the following subsection.

Calculation and Discussions
In the current decision-making problem, there are three alternative airports, fifteen criteria and four decision makers.All criteria are subjective with imprecision, vagueness and ambiguity.Hence linguistic variables are used by the decision makers for estimating the associated performance rating of the alternative airports.Seven degrees of linguistic variables viz.very poor, poor, medium poor, fair, medium good, good and very good are used for estimation of performance ratings.For quantification of each linguistic variable specific fuzzy interval is used.The linguistic variables, abbreviations and corresponding fuzzy intervals for measuring performance rating are represented in Table 1.
In ranking and selection of alternative airports, various decision criteria are given varying importance weight by the experts based on their significance as per the decision makers' experience, knowledge, and perceptions.For extracting this importance weights decision makers generally prefer linguistic terms.The present investigation advocates five degrees of linguistic terms viz.very low, low, medium, high and very high.The linguistic terms, abbreviation and associate fuzzy intervals are accommodated in Table 2. Three alternative airports are assessed by four decision makers using the prescribed seven degrees of linguistic variables which are regarded as the performance ratings of the alternatives.It is seen that decision makers D1, D2, D3 and D4 estimate alternative A1 with MG, G, G, VG with respect to criterion C1.Alternative A2 is assessed with VG, G, MG, MG.Here, VG implies very good, G means good, MG implies medium good.All the other abbreviations bear similar meanings as described earlier.The decision matrix containing performance rating in terms of linguistic variable is presented in Table 3.
The linguistic terms expressing performance ratings are converted into fuzzy interval as per prescribed conversion scale.Each fuzzy interval has two values viz.lower value and upper value.Application of fuzzy interval value is recommended for simplicity in calculation and having capability of conveying information.The geometric mean of the performance rating is determined using the Eq.( 6).The fuzzy intervals of the alternative A1 with respect to criterion C1 assessed by the four decision makers D1, D2, D3 and D4 are [0.6,0.8], [0.7, 0.9], [0.7, 0.9] and [0.8, 1] respectively.Therefore, the geometric mean of the performance rating in fuzzy interval is calculated as 1/ 4 (0.6 0.7 0.70.8)(0.4141, 0.5045)   = .The other mean performance ratings in fuzzy intervals are similarly calculated and accommodated in Table 4. Normalized crisp weights and normalized fuzzy performance ratings are integrated together to compute contribution by individual criterion using Eq. ( 13) and the calculated weighted individual contribution are depicted in Table 9. Aggregate Performance Score (APS) of the airports are determined from the algebraic summing up of the individual contributions for each alternative airport.APS for each alternative is presented in Table 10.Comparison of ranking orders obtained by the proposed method with TOPSIS and SAW are shown in Table 13.Aggregate performance score of the airports are graphically represented in Figure 1 for better visibility and demonstration.

Conclusion
This research work aims to develop and implement a new framework for evaluating, ranking and selecting the best airports considering multiple conflicting criteria incorporating group homogeneous decision makers' experience, opinion, knowledge and perception.The proposed method has been demonstrated through the illustration of a airport selection problem containing three feasible airports, fifteen subjective criteria and four rational decision makers.The result clearly indicates the best airport ensuring the better applicability of the method.The same problem with airport selection is also solved and the result is compared with that of the proposed approach.It is found that the result obtained by the proposed method completely matches with those of the existing approaches.The proposed Interval based multi-criteria homogeneous Group Decision making technique (IMGD) can also be applied for solving similar decision making problems under FMCDM.The approach may be useful FMCDM tool for individual as well as managerial decision makers.Heterogeneous group decision making by considering interdependent multiple conflicting criteria may be the direction of future research.

Figure 1 .
Figure 1.Aggregate performance score of the airportsThe ranking orders of the airports is depicted in Figure2.It is observed that airport A1is ranked 1, airport A2 is ranked 3 and airport A3 is ranked 3. Therefore the preferene order is A1>A2>A3.

Table 2 :
Linguistic variables, abbreviations, and intervals for criteria weight

Table 8 .
Mean weights in interval and in crisp numbers.

Table 13 .
Comparison of ranking order