Fuzzy system theory originates from fuzzy sets, which were proposed by Professor L.A.

Transcription

1 6 Fuzzy-MCDM for Decision Making 6.1 INTRODUCTION Fuzzy system theory originates from fuzzy sets, which were proposed by Professor L.A. Zadeh (University of California) in 1965, and after that, with the cooperation of many researchers, theories of fuzzy logic and fuzzy measure were constructed. The goal of this fuzzy systems theory is to supply one model for when we humans carry out intelligent information processing such as decision making, problem solving and control for large scale, complex systems aand problems. If in this case, the problems to be handled are very complicated or large-scale, it is impossible to integrate the large amount of information and data from them because of the limits of our intellectual capacity and time; therefore we limit it to an amount that we can process and focus our goals there, selecting, as much as possible, what we think is important and retrieving it. Branches of information and data that are not very important are cut out. Fuzzy methods are centered on two things, entering human language into computers and computations within computers. Fuzzy logic is not at all fuzzy as its name implies. Rather than being that, it is based on fuzzy set theory which is an extension of multi-value set theory and Boolean logic. Zadeh contended that computer cannot solve problems as well as human experts unless it is able to think in the characteristic manner of a human being. Conventional Boolean logic is limited in a way that it can be used to describe attributes that is either completely true or completely false. Although it is useful for modeling many situations, conventional Boolean logic does not provide adequate means to model the

2 many imprecise concepts used within human reasoning, Imprecise concepts are attributes of people generally have a cognitive perception, yet are impossible to define precisely. For example, let s consider the human characteristics of tallness. Webster s dictionary defines tall as, having more than average height. For the purpose of this let s take an example; we will assume that the average height of people is 5 8. Thus, using conventional Boolean logic, tall must be defined as Tall(x) = { 0 if height(x) <5 8 { 1 if height(x) >5 8 Now the question arises: Why are someone who is exactly 5 8 considered tall, yet someone that is even slightly less than 5 8 considered not tall? Clearly this is a structural problem, for if we move the lower bound of the range from 5 8 to an arbitrary point the same question cane be posed. It is the strict separation between tall and not tall required by conventional Boolean logic that renders it inadequate for describing imprecise concepts like tallness. Fuzzy logic is a superset of conventional Boolean logic that has been extended to handle the concept of partial truth. Rather than limiting degrees of membership for a particular attribute to the discrete values 0 or 1 as in conventional Boolean logic, fuzzy logic allows degrees of membership to have values within the continuous range 0 or 1. In this way, it is possible to assign a degree of membership for a particular attribute that is neither completely true nor completely false, but somewhere in between. Many of the same logical concepts still apply (such as AND, OR and NOT), but carry slightly different meanings.

3 Fuzzy logic can be used to mathematically formulate imprecise concepts (like tall, warm and cool) so they can be understood and processed by computers. By incorporating fuzzy logic into their code, computer programmers can attempt to apply amore human-like way of thinking into their programs. Fuzzy logic has already been successfully incorporated into such applications as control systems for subways and complex industrial processes, entertainment and household electronics, artificial intelligence and other expert systems. The employment of Fuzzy control is commendable For very complex processes, when there is no simple mathematical model For high nonlinear processes If the processing of (linguistically formulated) expert knowledge is to be performed. The employment of Fuzzy control is no good idea if Conventional control theory yields a satisfying result. An easily solvable and adequate mathematical model already exists. The problem is not solvable The fuzzy logic is a sui generis technology for non linear compensation. Presently it is used in all the domains viz. meteorology, corporate planning and decision making, medicine, trading, engineering, agriculture etc. There is a list of general observations about fuzzy logic. They are:

4 Fuzzy logic is a better way of dealing incomplete information, ill-defined systems, imprecise knowledge linear and nonlinear systems Fuzzy logic is conceptually easy to understand Fuzzy logic can model nonlinear functions of arbitrary complexity The working models and systems have already demonstrated that fuzzy logic can provide more robust and less expensive system as compared to conventional approaches. Fuzzy logic can be built on the top of experience of experts Fuzzy logic is based on easy spoken language Looking at the features of fuzzy management science as in the above, we can see that fuzzy methods are useful and can make a flexible response to programming and management of large-scale complex management organizations. 6.2 GENERAL STRUCTURE OF FUZZY LOGIC The principal design blocks for FLC are the following: Fuzzification module Knowledge base or data base Fuzzy set rules Fuzzy inference mechanism Defuzzification

5 The listed components are necessary for the construction of any FLC. Block diagram of FLC is shown in figure Knowledge base Fuzzifier Fuzzy signal Inference Engine Fuzzy signal Defuzzifier Figure-6.1 Basic architecture of a Fuzzy Logic

6 Figure 6.2: Fuzzy Architecture Fuzzification implies the process of transforming the crisp values of the inputs to a controller, to fuzzy domain. The fuzzified values now fire the rule base and generate an output called fuzzified output. At the final step, the fuzzified output is deduzified to yield the crisp controller output Fuzzification Module Fuzzification means adding uncertainty by design to crisp sets or to sets that are already fuzzy. This may be conceptually new to the fuzzy logic because adding uncertainty seems like going backwards in reference to the classical school of thought in science. On the contrary, Fuzzification is a usual concept and is not an entirely new idea. In practical terms, Fuzzification is spreading the information provided by a crisp number or symbol to its vicinity so that the close neighborhood of the crisp number can be recognized by the computational tools. In the world of numbers, this is analogous to the interpolation

7 method between two data points. Fuzzification corresponds to defining interpolation surfaces between two sets or two numbers by a controlled (designed) uncertainty distribution function. Fuzzification, which is in fact very useful and unique when considering crisply defined logic, formulas, equations, relationships, and variables. The injection of designed uncertainty enables us to carry on computations in those gray areas explicitly defined by crisp entities. The simplest form of Fuzzification can be done using a fuzzifier function F that determines the degree of fuzziness in a set. A fuzzifier function is identical to a membership function in principle except for the objective behind designing it. The objective is to be able to apply a controlled uncertainty distribution to the elements (e.g., inputs, outputs) of a fuzzy system in a consistent manner. It simply modifies the inputs so that they can be interpreted and compared to the rules in the rule-base. Fuzzification is related to the vagueness and imprecision in a natural language. It is a subjective valuation which transforms a measurement into a valuation of an objective value and hence, it could be defined as a mapping form an observed input space to fuzzy sets in certain input universes of discourse. The Fuzzification module performs the following function. Measures the values of input variables

8 Performs a scale transformation which maps the physical values of the current process of discourse. It also maps the normalized value of the control output, variable onto its physical domain. Performs the so called Fuzzification which converts a point wise (crisp), current value of a process state variable into a fuzzy of a process state variables into a fuzzy set, in order to make it compatible with the fuzzy set representation of the process state variables in the rule-antecedent Knowledge base It holds the knowledge, in form of a set of rules, of how best to control the system. The knowledge base of FLC consists of a data base and a rule base. In mathematical models for management systems, coefficient and constrained conditions that are intuitively determined by those responsible for programming can be expressed with ease and flexibility by membership functions, and solutions for these can be found mathematically. The various types of knowledge and experience for management systems can be obtained orally from experts in natural language, and using fuzzy reasoning, computer models and programs can easily be created. In such cases, natural language often uses generalized adjectives and adverbs like very, a little, somewhat, some and about, but these can easily be expressed by membership functions, and entered into a computer. In addition, since inference programs are created using generalized

9 expressions, the amount of software for objectives for which this expression is sufficient can be reduced and this is economical. Instead o squeezing the solutions to a problem down to one, multiple solutions with degree of possibility can be indicated, and since the upper and lower bounds of solut ions can be given, it is easy to add the opinions of experts, managers and administrators later, making for the ability to derive solutions for problems from wide range of vision. In conventional management science methods, extremely approximate models are created, and cases of one solution for these are frequent. Managers and administrators most often make decisions separately from these solutions Data-base The basic function of a data-base is to provide necessary information for the proper functioning of the FM, the rule base and the defuzzification module. This information includes: Fuzzy sets (Membership functions) representing the meaning of the linguistic values of the process and the control output variables. This is Fuzzification of a conventional database, and the relational model for a standard database is extended to a fuzzy relational model. By this means, the fuzzy data represented by fuzzy sets is actively and efficiently used. In terms of uses, it can be thought of as something that will be useful from here on out for expert systems for all fields as a subsystem for various levels of decision making support systems for management and administration.

10 Rule-base The basic function of the rule-base is to represent in a structured way the control policy of an experienced process operator and/or control engineer in the form of a set of production rules such as: If <process state <then <control output > The if part of such a rule is called the antecedents which is the description of a process state. The then part of the rule is called the rule consequent and is again the description of a logical combination of fuzzy propositions. These propositions state the linguistic values which the control outputs take when the current process sate matches the process state description in the rule antecedent. In this technology, a fuzzy control rule is a fuzzy conditional statement in which the antecedent is a condition in its application domain and the consequent is a control action for the system under control. Basically, fuzzy control rules provide a convenient way for expressing control policy and domain knowledge. The source and deviation of fuzzy control rules are: Expert experience and control engineering knowledge. Control operator s control actions Fuzzy model of a process Fuzzy Inference Engine An information processing system that draws conclusions based on given conditions or evidences. A fuzzy inference engine is an inference engine using fuzzy variables. Fuzzy inference refers to a fuzzy IF-THEN structure.

11 It is an algorithm that solves the problems expressed in the basic IF-THEN rule format and formulated as the generalized modus ponens inference. Fuzzy logic applications that do not fit this scheme are often formulated based on a simpler relationship than IF A THEN B, and correspond to one of the main building blocks of the basic fuzzy inference algorithm. However, the application of fuzzy set theory is not confined to this structure. The algorithm, in the form consists of events (computations or other forms of information processing) arranged in the most logical order. The overall layout is invariant, but the order and type of computations can be altered in a number of ways. The basic fuzzy inference algorithm is mostly employed in management applications. However, the inference mechanism is the same for all other types of applications in which approximate reasoning plays a major role. The primary requirement is the availability of a solution articulated in the IF-THEN form. Once this requirement is met, then the basic fuzzy inference algorithm can be designed for any problem regardless of the problem type like classification, forecasting, diagnostics, modeling etc. A fuzzy inference engine has a simple input-output relationship. Input data collected from the external world is processed by the fuzzy inference engine to produce output data to be used back in the external world. In most of the existing management applications, input data received from the external world is analyzed for its validity (in syntax, format, and range) before it is propagated into a fuzzy inference engine. Most of the time, an input data processing step is included within the peripheral computations, and it is not considered as part of the fuzzy inference

12 engine owing to its trivial function. However, this important step cannot be taken without design knowledge. A fuzzy inference can process mixed data. Mixed data in this context refers to the mixture of numerical and symbolic (linguistic) data. The capability of processing mixed data is based on the membership function concept by which all input data are eventually transformed into the same unit (possibility) before the inference computations. Accordingly, the goal of input data processing is to ensure that input data is in an appropriate form for transformation. A fuzzy inference engine normally includes several antecedent fuzzy variables. If the number of antecedent variable is k then there will be k information collected from the external world. This is referred to as input data set. Note that it is a crisp set containing crisp and fuzzy elements. Furthermore, each element in the set may contain more than a single data point, namely a set of distribution point or a fuzzy set. Input data Fuzzy Inference engine Output data Fig 6.3 Fuzzy Inference Engine

13 There are two types of approaches employed in the design of inference engine of FLC. They are: Composition-based inference Individual rule-based-inference In the composition-based inference, the fuzzy max-min composition operator is employed. The basic function of second type of inference is to compute the overall value of the control output variable based on the individual contribution of each rule in the rule base. Each such individual contribution represents the value of the control output variables as computed by a single rule. The output of the Fuzzification module, representing the current crisp values of the process state variables, is matched to each rule antecedent and a degree of a match for each rule is established. Based on the degree of match, the value of the control output variable in the rule-antecedent is modified i.e the clipped fuzzy set representing fuzzy value of the control output variable is determined. The set of all clipped fuzzy set represents the overall fuzzy output. Membership function (MF) are used as a method for computer inputs, and through these words (the approximate quantities included in words: about 5m, about 10 m, etc) are converted to numerical values (from 0 to 1). Fuzzy logic, which is mainly composed of max and min operations is used in the computations we can use the logical sum, logical product and complement for the basic operations and the extension principle for most arithmetic operations with numbers Defuzzification Module

14 It converts the conclusions reached by the inference mechanism into actual inputs for the process. Aggregated results obtained from a fuzzy inference engine are the final results within the scope of fuzzy lofic theory. A need for defuzzification arises from the fact that such results are not practically useful in real life applications. The defuzzification step is an approximation itself based on the assumption that a scalar will represent a fuzzy set in an appropriate manner. When th aggregation is viewed as the contribution of individual decisions, then defuzzification can be viewed as acquiring a popular vote or consensus. The defuzzification technique established in the literature is developed with this view in mind. The functions of a defuzzification module are as follows: Performs the so called defuzzification which controls the set of modified control output values into single point wise values. Performs the so called defuzzification which maps the point-wise value of the central output onto its physical domain. This step is not needed if no normalized fuzzy sets are used. Designing a defuzzification process is again a selection among a few viable options established in the literature as well as in industrial applications. The most widely used defuzzification design options are Center of gravity (centroid), center of area Maximum possibility Center of maximum possibilities, mean of maximum possibilities

15 Center of mass of higher intersected region Others The most frequently used defuzzification method is the center of gravity (Centroid) technique, which is analogous to finding the balance point by calculating the weighted mean of the fuzzy output The membership function µ0 (x) represents the fuzzy set of the final output (either aggregated or single) of one fuzzy variable, and x is the location of each singleton on the universe of discourse. Because of two-dimensional output fuzzy sets, this method is also known as the center-of-area method. The center of gravity C is of each area A and Y is the balancing point. W is the rule importance weight. Note that the index j in these equations corresponds to one fuzzy rule in a rule block Another common method finds the maximum possibility point on the universe of discourse as the answer to defuzzification. If the maximum point is nonsingular (plateau), then the defuzzified out-put is the average of maximums or the center of maximums. Also called the mean of maximums, the defuzzification technique is normally employed after union aggregation just like the centroid method. The center of mass method finds the region that has the highest density of intersecting fuzzy sets. Thus, it is employed with intersection aggregation. The defuzzification computation using this method is performed in parallel with the aggregation process because the location of each individual solution (fuzzy set) needs to be known. The final region, which has the highest density of intersection, is computed by counting the

16 frequency of inclusion, or simply by taking the highest possibility points from intersected fuzzy sets. 6.3 SOFTWARE USED MATLAB is used for the implementation of the above formulated problem. The name MATLAB stands for matrix laboratory. MATLAB is a high-performance language for technical computing. It integrates computation, visualization, and programming in an easy-to-use environment. MATLAB is an interactive system whose basic data element is an array that does not require dimensioning. This allows the user to solve many technical problems especially those with matrix and vector formulations. Fuzzy tool box provided by MATLAB is used by using GUI tool provided by it. This is the MATLAB graphics system. It included high-level commands for two-dimensional and three-dimensional data visualization, Image processing, animation, and presentation graphics. It also includes low-level commands that allow to fully customizing the appearance of graphics as well as to build complete graphical user interface for MATLAB applications. There are five primary GUI tools for building, editing, and observing fuzzy inference systems in the Fuzzy Logic Toolbox: The Fuzzy Inference System or FIS editor The Membership Function Editor The Rule Editor

17 The rule Viewer; and The Surface Viewer

18 Figure 6.4 Components Of Fuzzy Logic GUI Toolbox These GUIs are dynamically linked, in that changes made to the FIS using one of them, can affect any of the other open GUIs. Any one or all of them can be opened for any given system. The primary GUIs can all interact and exchange information The FIS Editor The FIS structure is the MATLAB object that contains all the fuzzy inference system information. This structure is stored inside each GUI tool. Access functions such as getfis and setfis make it easy to examine this structure. All the information for a given fuzzy inference system is contained in the FIS structure, including variable names, membership function definitions, and so on. This structure can itself be brought of as a hierarchy of structures, as shown in the Figure 2.3 Figure 6.5 FIS Structure

19 The point of fuzzy logic is to map an input space to an output space, and the primary mechanism for doing this is a list of if-then statements called rules. All rules are evaluated in parallel, and the order of the rules is unimportant. The rules themselves are useful because they refer to variables and the adjectives that describe those variables The Membership function Editor It is used to define the shapes of all the membership functions associated with each variable The Rule Editor It is for editing the list of rules that defines the behaviour of the system The Rule Viewer and the Surface Viewer These are used for looking at, as opposed to editing, the FIS. They are strictly read-only tools. The rule Viewer is a MATLAB based display of the fuzzy inference diagram; it can show which rules are active, or how individual membership function shapes are influencing the results. The Surface Viewer is used to display the dependency of one of the outputs on any one or two of the inputs- that is, it generates and plots an output surface amp for the system Fuzzy Logic Prerequisites

20 Figure 6.6 Fuzzy Logic connections with physical world Define the control objectives and criteria: what is to be controlled? What should be done to control it? Determine the input and output relationships and choose a minimum number of variables for input to FL engine. Create FL membership functions that define the meaning of input/output terms used in the rules. Using rule based structure of FL, break the control problem into a series of IF x AND Y THEN Z rules that define the desired system output response for given system input conditions. Test the system, evaluate results, tune the rules and membership functions and retest until satisfactory results are obtained.

21 6.2 Fuzzy MCDM Model Fuzzy theory has unique quality of incorporating uncertainties and processing linguistic information. Further Fuzzy models do away with the need of development of complicated and costly mathematical models used in the traditional methods. The information obtained through questionnaire is linguistic in nature and the candidate interviewed is both approximate and abstract in nature. The evaluation of a candidate performance varies from one interview to another. Therefore, to incorporate these uncertainties and to remove the biases, Fuzzy MCDM model has been proposed to select the best candidate based upon test performance ratings. Three Fuzzy input variables CS (Conceptual skills), PS (Programming Skills) and Fundamental Knowledge (FK) are taken based on the advice rendered by HR professionals in the two companies TCS and HCL. Each Fuzzy variable is described by one linguistic term excellence that gives the performance measure of the candidate for a Fuzzy variable. The membership diagrams of the Fuzzy variable CS, PS and FK are shown in Figure 1, Figure 2 and Figure 3 respectively. Figure 1 (Membership function for Conceptual Skills)

22 Figure 2 (Membership function for Programming Skills) Figure 3 (Membership function for Fundamental Knowledge) The IF Then rulebase is constructed using these membership functions. The number of IF Then rules in the rulebase, n, is given by n= n1*n2*n3 (1) Where n1=number of membership functions for its first variable CS n2= Number of membership functions for its first variable PS n3= Number of membership functions for its first variable FK

23 As n1=n2=n3=1 in our case, the number of rules in the rulebase is equal to one only and is given as under: IF CS is excellent AND PS is excellent AND FK is excellent then PR is excellent Where PR is the output variable and denotes performance of the candidate. Further, this variable is numeric and not linguistic. The above model has been intentionally kept simple so that it can easily be comprehended. Therefore each input variable is described by one membership function only. In practical situations, each input variable may be described by 5 or 7 linguistic terms each having its own membership functions. In that case, the number of rules in the rulebase will increase to 125 or 343 respectively. This indicates that recruitment and selection procedure is not very simple and involves complicated analysis but can be simplified using Fuzzy MCDM model. The proposed Fuzzy MCDM is taking into consideration all the three criteria to select the most suitable candidate by matching their preferences in order with CS being given the highest weight, PS being given weight lesser than CS and the PS given weight lesser than FK. It is also assumed that to qualify the minimum marks required in every test is 80. To test the proposed Fuzzy MCDM model a group of 236 students were evaluated with the help of a written test comprising 45 questions (Three sections having 15 questions each and with total mark 100). The questions were framed in such a manner so as to evaluate their Conceptual Skills, Programming Skills and Fundamental Knowledge about information technology. 228 students were rejected out rightly as they failed to score minimum 80 marks in each of three sections. It is to be noted that the membership

24 function for Conceptual Skills, Programming Skills and Fundamental Knowledge which is evident from membership functions shown in figure 1, figure 2 and figure 3 also indicate that performance is equal to zero for score less than 80. Further, from the figures it may be seen that performance measure of each variable is not identical. The remaining eight candidates were left to compete. For example score of 90 in each section means different performances for Conceptual skills it is equal to Ecs 1, same score in Programming skills it is equal to EPS 0.67 and while the same score in Fundamental knowledge is equal to EFK=0.5 only. For example score of 86% in each section means different performances for Conceptual skills it is equal to Ecs 0.6, same score in Programming skills it is equal to EPS 0.4 and while the same score in Fundamental knowledge is equal to EFK= 0.3 only. The scores of these candidates are presented in the table 1 below: Table 1: Candidates scores in CS, PS and FK Conceptual (CS) skills Programming (PS) skills Fundamental knowledge (FK) Devriti (x1) Nalin (x2) Yuvraj (x3) Arjun (x4) Onam (x5) Tanvi (x6) Manisha (x7) Pulkit (x8)

25 The set of alternative is A = {x1, x2, x3. x4, x5, x6, x7, x8} The membership functions for Fuzzy variable CS as given in figure 1 can be described by the following equation: 0 for 0 x 80 x 80 E CS= µ ECS (x) = 10 for 80 x 90 (2) 1 for 90 x 100 The membership functions for Fuzzy variable PS as given in figure 2 can be described by the following equation: 0 for 0 x 80 x 80 E PS = µ EPS(x) = 15 for 80 x 95 (3) 1 for 95 x 100 The membership functions for Fuzzy variable FK as given in figure 3 can be described by the following equation: 0 for 0 x 80 x 80 E FK = µ EFK(x) = 20 for 80 x 100 (4) 1 for x = 100

26 The score of successful eight candidates was fed to the Fuzzy MCDM to find the best performer. Substituting the candidate s scores in CS, PS and FK gives the degree of excellence corresponding to their scores for each of three criteria. The degree of excellence of all the 8 candidate obtained from the respective membership function are tabulated in table 2: Table 2: Candidates degree of excellence in CS, PS and FK Conceptual (CS) skills Programming skills (PS) Fundamental knowledge (FK) Devriti (x1) Nalin (x2) Yuvraj (x3) Arjun (x4) Onam (x5) Tanvi (x6) Manisha (x7) Pulkit (x8) The performance of each candidate is evaluated by applying these Fuzzy degrees of excellence in CS, PS and FK to the rules in the Fuzzy IF-Then rulebase. Our model for the sake of simplicity has only one rule as described in section 5 The Fuzzy AND operation is implemented by applying intersection operator on the Fuzzy values

27 Let G1 be the Fuzzy set describing Excellence in Conceptual skills (CS) for each candidate as given below: = { ( x1, 1 ), (x2, 1), (x3,0.7 ), (x4,1 ), (x5, 0.9 ), (x6,1 ) (x7, 0.3 ), (x8, 0.6)}, Let G2 be the Fuzzy set describing Excellence in Programming skills (PS) for each candidate as given below: = { ( x1,0.93 ), (x2,0.53), (x3,0.67 ),(x4,0.87 ),(x5, 0.80 ),(x6,0.2 ),(x7,1.0 ),(x8, 0.4 )}, Let G3 be the Fuzzy set describing Excellence in Fundamental knowledge (FK) for each candidate as given below: = { ( x1, 0.55 ), (x2,0.5 ), (x3, 0.3 ), (x4, 0.85), (x5,0.6 ),(x6, 0.1 ),(x7, 0.2 ),(x8, 0.85)}, Applying the Fuzzy rule on the corresponding values of each set, we obtain the Fuzzy set for the output variable performance. The Fuzzy set PR describing the performance given by each candidate is given as under: PR= G1 G2 G3 (4) = { ( x1, 0.55 ), (x2, 0.5), (x3,0.3 ), (x4,0.85 ), (x5, 0.6 ), (x6,0.1 ) (x7, 0.2 ), (x8, 0.4)}, Now the best performance in the set PR is given by the maximum degree of performance obtained in the Fuzzy set PR. Hence the conclusion is that x4, i.e. Arjun with the degree of performance 0.85 is the candidate with the best performance to be selected.

28 7 Results and Discussion The simplified Fuzzy MCDM model presented in section 5 is illustrated with a numeric example in section 6. It is seen that out of 236 candidates the best performer was selected with the least amount of difficulty and computational expense. In practical situations it is observed that the three variables CS, PS and FK are not described by one linguistic term each. They are bound to have different opinions about excellence in these fields by different assessors in the interview panel. The Fuzzy MCDM model can be easily extended to include these diversities by employing more than one linguistic term to describe any/all the Fuzzy variable. However it must be kept in mind that the number of rules in Fuzzy MCDM will increase accordingly as described in section 5 also. For example if the 3 variables have 3 linguistic terms each the number of rules will increase up to 3*3*3 i.e. 27. In case the number of linguistic terms is 5 each the number of rules will increase up to 5*5*5 i.e. 125 like wise for seven the linguistic terms the number of rules will increase up to 343. Increased number of rules makes the Fuzzy MCDM more complicated. The Fuzzy MCDM model can easily incorporate these variables. The method and the model remain static. The variations required by the company are incorporated by changing the IF-Then rules in Fuzzy MCDM rules. More the number of rules greater the flexibility introduced in the model. Therefore, it is concluded that proposed Fuzzy MCDM model does away with the confusion of different assessors especially when the candidates are highly competitive and it is difficult to judge who is better than the others.

29 The model is an attempt to minimize subjective judgment in the process of distinguishing between an appropriate employee and an inappropriate employee for a particular job vacancy. Through this method the selector can also short list job applicants compared to the traditional way of selecting an appropriate short-listed applicant based upon selector s few criteria. Compared to the traditional way of selecting an appropriate short-listed job applicant this model minimizes individual judgment at both short-listing and at hiring decision levels. HR professional is having a huge responsibility to hire the best fit person from the available talent pool. At the same time, one needs to be cost conscious.. Whereas generalized assumptions made about ability or ambition, based on applicant s sex, caste, age, religious belief, sexual orientation or any disability is a bad practice. One need to use the proposed kind of technologies through this paper to get the best results from any recruitment process. The use of proposed technology can very well simplify and control the hiring process, improving HR effectiveness and efficiency, while improving speed to hire, quality of hire and reducing cost per hire. Most organizations are constantly seeking better performance which is a difficult goal to achieve without effective staffing practices. When progressive staffing is handled with great care, it can be a powerful source for creating long term competitive advantage for any IT company.