Multi-Attribute Decision Making in a Dual Hesitant Fuzzy Set using TOPSIS

Transcription

1 eissn: MultiAttribute Decision Making in a Dual Hesitant Fuzzy Set using TOPSIS 1. Pathinathan. T 1., Johnson Savarimuthu. S 2 P.G and Research Department of Mathematics, Loyola College, Chennai34, 2 Department of Mathematics, St. Joseph s College of Arts and Science, Cuddalore1 1 nathanpathi@hotmail.com, 2 johnson22970@gmail.com Abstract: MultiAttribute Decision Making (MADM) which that addresses the problem of making a suitable choice from a set of alternatives which are characterized in terms of their attributes, is a normal human activity. When people make a decision, usually they hesitate to express some of their views about certain given attributes and very often they avoid those situations by not answering them. Many may have extensive knowledge about all the attributes and few others may not. In this paper, we have analyzed the problems faced by the farmers in Villupuram district with the help of Dual Hesitant Fuzzy Sets and we introduce a new methodology to handle the problem of uncertainty in the decision making situations with the application of TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) method in a Dual Hesitant Fuzzy Set (DHFS). Keywords: Multiattribute decision making, Hesitant Fuzzy Set, Dual Hesitant Fuzzy Set, Technique for Order Preference by Similarity to Ideal Solution, TOPSIS 1. INTRODUCTION All those researchers who are keen on contributing to socially relevant research either through pure or applied streams are looking forward to solve real life problems. Fuzzy mathematicians are better equipped to handle issues which are imprecise and vague in nature. Yet it remains a challenge to researchers and they continue to try out different methodologies and theories. Fuzzy sets [19] have provided a wide range of tools to deal with such vagueness. After the introduction of Fuzzy set theory [19] several extensions have been made such as Type2 fuzzy set [3], Typen Fuzzy Set, Intuitionistic Fuzzy Set, Fuzzy MultiSet and Hesitant Fuzzy Set. The first four types of Fuzzy Set had many real life applications in the past and the last one received more attention in recent years. Hesitant Fuzzy Set (HFS) [11,12] discusses the possible membership values among the Decision Makers (DMs) by considering all such alternatives along with their attributes. Decision making by using the above fuzzy extensions [1,3,7,9,14,15,16] are collectively referred as MultiAttribute Decision Making methods (MADM). In classical MultiAttribute Decision Making (MADM) methods, the assessments of alternatives are precisely known [21]. However, because of the inherent vagueness of human preference as well as the objects being Fuzzy and uncertain, the attributes involved in decision making problems are difficult to measure (quantify). In 1970 Bellman and Zadeh [1] first proposed the basic model of fuzzy decision making based on the theory of fuzzy mathematics. Since then, many fuzzy decision making models had been studied and applied in arriving at the suitable conclusions for various kinds social issues. Earlier MADM models had many limitations such as assigning membership values for the attribute, lack of information about the problem concerned, lack of expert s knowledge over the attribute and expert s hesitation in defining the membership values which may lead to an insufficient and sometimes a false conclusion. For instance, Analytic Hierarchy Process (AHP) uses scale importance values [9,10], which are real values written as reciprocal of the integers form 1 to 9. But the attributes involved in decision making problems are not always expressed in real numbers, and some are better suited to be denoted by fuzzy values, such as interval values, fuzzy numbers [6,7], linguistic variables, and intuitionist fuzzy numbers. These deficiencies can be seen in MADM models like Analytic Hierarchy Process (AHP) [9,10], Technique for Order Preference by Similarity to Ideal Solution (TOPSIS), VlseKriterijumska Optimizacija I KompromisnoResenje method (VIKOR), Preference Ranking Organization Pathinathan. T. and Johnson Savarimuthu. S ijesird, Vol. II (I) July 2015/44

2 eissn: Method for Enrichment Evaluation (PROMETHEE), Elimination EtChoixTraduisant la Realite (ELECTRE). Despite all the above developments, Torra [11] introduced a new extension called Hesitant Fuzzy Set (HFSs) to motivate the experts and decision makers in establishing membership degree of an element without any hesitation by choosing the right one. HFSs have attracted the attention of many researchers in a short period of time because hesitant situations are very common and this new approach facilitates the management of uncertainty caused by hesitation. Hesitant Fuzzy Sets does a repair work on limitations and produce a better solution. Also, the above MADM methods are adjusted to accomodate the hesitancy provided by the Decision Makers [21]. Umamaheswari et al [14] extended TOPSIS method for Triangular Hesitant Fuzzy Set. Through this paper, we developed an extension by fitting the Dual Hesitant Fuzzy Set (DHFS) approach with TOPSIS and also we approach the problems faced by farmers in Villupuram District, Tamil Nadu through our newly designed Dual Hesitant Fuzzy MultiAttribute Decision Making (DHF MADM) method. This paper is organized as follows, Section 2 introduces the concept of HFS, DHFS and some of its basic properties. Section 3 presents the working algorithm for TOPSIS. Section 4 gives an application which uses the given hesitant information. Section 5 deals with the adaptation and description of the problem and finally the paper is concluded in Section BASIC DEFINITIONS 2.1 Fuzzy Set Let E be the universal set, let x be an element of E. then the fuzzy subset A of E, ( A of E) is a set of ordered pairs. (2.1) A {( x ( x))}, for all x E A where ( x ) is the grade (or) degree of membership Pathinathan. T. and Johnson Savarimuthu. S ijesird, Vol. II (I) July 2015/45 A of x in A. A( x ) takes the value from the membership set M = [0,1] and A( x ) is the membership function or characteristic function. 2.2 Hesitant Fuzzy Set Let X be a fixed set, a hesitant fuzzy set (HFS) on X is in terms of a function that when applied to X returns a subset of [0,1]. Mathematical representation of Hesitant fuzzy set [18]: A = h A (x) x X } (2.2) where h A (x) is a set of some values in [0,1], denoting the possible membership degrees of the element x X to the set A. 2.3 Hesitant Fuzzy Element (HFE) Every h = h A (x) is defined as hesitant fuzzy element (HFE). Example 2.1 Let X = {x 1,x 2,x 3 } be a fixed set, h A (x 1 ) = {0.2,0.4,0.5}, h A (x 2 ) ={0.3,0.4} and h A (x 3 ) = {0.3,0.2,0.5,0.6} be the HFEs of x i (I =1,2,3) to the set A respectively. Then A can be considered as a HFS: A = { 1, {0.2,0.4,0.5}, 2, {0.3,0.4}, 3, {0.3,0.2,0.5,0.6} } 2.4 Special properties of HFS (i) Empty Hesitant Fuzzy Set: h = {0} (ii) Full Hesitant Fuzzy Set: h = {1}, denoted as I. (iii) Complete Ignorance: h = [0,1] U (All are possible) (iv) Nonsense Set: h = (h returns no value)

3 eissn: Score of an HFS For a Hesitant Fuzzy Element (HFE) h, sh ( ) (2.3) h is called the score of h, where l h is the number of the elements in h. Case (i): For two HFEs h 1 and h 2, If s(h 1 ) s(h 2 ), then h 1 superior to h 2, denoted by h 1 h 2 Case (ii): s (h 1 ) = s (h 2 ) then h 1 is indifferent to h 2, denoted h 1 h Dual Hesitant Fuzzy Set (DHFS) Let X be a fixed set, then a dual hesitant fuzzy set (DHFS) D on X is described as: D ={ l h } (2.4) In which h(x) and g(x) are two sets of some values in [0,1] denoting the possible membership degrees and nonmembership degrees of the element x X to the set D, respectively with the condition, where, h( x); g( x) h ( x) max{ } h( x) g ( x) max{ }, x X h( x) 0, 1, 0, Dual Hesitant Fuzzy Element (DHFE) The pair d(x) = (h(x), g(x)) is called a dual hesitant fuzzy element (DHFE) denoted by d = (h,g) with the conditions, h; g h ( x) max{ } h g ( x) max{ }, x X h 0, 1, 0, Special properties in DHFE (i) Complete uncertainty : d={{0},{1}} (ii) Complete certainty : d={{1},{0}} (iii) Complete illknown (all is possible): d=[0,1]; (iv) Nonsensical element : d ( h, g ),h returnsnovalue. 3. DUAL HESITANT FUZZY MADM ALGORITHM BASED ON TOPSIS This section describes a framework for determining the ranking orders for all the alternatives under hesitant fuzzy environment. The approach involves the following steps: Step 1: For a MADM problem, we construct the dual hesitant decision matrix H = (h ij ) n x m, where all the arguments h ij (i = 1,2, n; j = 1,2, m) are DHFEs, given by the decision makers (DMs), for the alternatives A i A with respect to the attribute x j X. H = Fig 3.1: Dual Hesitant Fuzzy Decision Matrix Step 2: Choose weight vector for each attribute according to their importance over the problem. Step 3: Determine the corresponding dual hesitant fuzzy positive ideal solution and dual hesitant fuzzy negative ideal solution by the following expression; Pathinathan. T. and Johnson Savarimuthu. S ijesird, Vol. II (I) July 2015/46

4 eissn: (3.1) A = {<x j, j = 1,2, m} A = {<x j, j = 1,2, m} (3.2) Step 4: Calculate the separation measures d i and d i of each alternative A i from the dual hesitant fuzzy positive ideal A and negative ideal A respectively by the following expression; d i = { )w j } (3.3) d i = { )w j } (3.4) )w j, )w j, Step 5: Calculate the relative closeness coefficient c(a i ) of each alternative A i to the dual hesitant fuzzy solution by using the formula; c(a i ) = (3.5) Step 6: Rank the alternatives according to the relative closeness to the hesitant positive ideal and negative ideal solution. 4. DUAL HESITANT FUZZY SET APPLICATION The study has been initiated in the backdrop of problems faced by the farmers in Villupuram district over the last few years. Villupuram district is one of the largest districts in Tamil Nadu which is predominantly farmers. Out of the total land area of 7.22 lakh hectares, an extent of about 3.60 lakh hectares is utilized for cultivation. More than 86% of the main workers in the district are engaged in agriculture. Paddy, sugarcane, groundnut, cucumber, banana, cereals, pulses and oilseeds are the important crops produced in the district. The major cultivated area in Villupuram district includes Vanniyapalayam, Arulavadi, Kandambakkam, Arkadu, Payur, Thiruvennainallur, Arasur. The major water source Pathinathan. T. and Johnson Savarimuthu. S ijesird, Vol. II (I) July 2015/47

5 eissn: of the district is from Pennaiyar and Varahanathi river basin and it is of the length 70 and 78 km respectively. Then other water storage sources include Vidur, Sathanur and Komuki dam. The major water sources had lost their identity and now they are all seem to be dry lands. Acute water scarcity in the district hindered the cultivators and now their only hope is ground water. But nowadays the level of ground water has been reduced and in certain places the traces of ground water are hardly seen. [23] The access for ground water gets interrupted due to the lack of electricity.[24] To pump out the water from the ground, the cultivators need electric motors and in the absence of electricity, they seek help from diesel motors which are expensive. To overcome them, the farmers borrow money from the bank or private money lenders with high rates of interest which obviously make them suffer.[25] Apart from all the hindrances, the demand for fertilizers and pesticides are also increased.[26] With all these struggles, the farmers continue with their farming which ultimately ruins their lives. Especially for the last two years, the district had seen severe problems in agriculture which include water scarcity and electricity shortages. We conducted interview from five different experts about the struggle experienced by the farmers in Villupuram district. We made a hierarchical structure with five major cultivating crops as alternatives and the problems into six attributes and we recorded the values for the same. 4.1 Experts (Villupuram District) We collected information the following five experts who are considered as decision makers (DMs). DM 1 Mr. P. Subramanian, farmer and land owner, Vaniampalayam DM 2 Mr. Jayavel, farmer who is doing share cropping (Kuthagai), Arulavadi DM 3 Mr. Muthukrishnan, Private Agricultural Officer, Kandambakkam DM 4 Mr. Raja, Cashier, Agriculture Cooperative Bank, Arkadu DM 5 Mr. Murugan, Teacher as well as farmer, Arkadu 4.2 Alternatives Among the crops, we have considered the following major five crops as our alternatives. A 1 Paddy A 2 Sugarcane A 3 Groundnuts A 4 Cucumber A 5 Banana 4.3 Attributes Through personal interview, we have recorded some of the major problems that hindered farmers lives and consolidated them into six major attributes. These attributes are evaluated into two types viz., (i) Benefit Type (Qualitative in nature) (ii) Cost Type (Quantitative in nature) The set of those six attributes of benefit and a cost type are defined by {X 1, X 2, X 3, X 4, X 5, X 6 }, where X 1 Crop failure (Benefit Type): Reduction in crop yield; drying crop; inability to save the standing crop, decline in the soil strength, irregular and deficient irrigation, inappropriateness in choosing the land for a particular crop cultivation, disproportionate mixture of pesticides and fertilizers, etc., lead to the failure of crop. X 2 Crop debt (Cost Type): The money borrowed for interest to meet out the expenses by a farmer from various sources like agriculture debt from the both government and private banks, availing loan from private fund agencies, land mortgage, jewel mortgage, personal loans from known people on mortgage basis are some of the major crop debts. X 3 Lack of water (Benefit Type) Lack of water sources like no rain, water scarcity in Vidur Dam, Varahanathi Dam, failure of monsoons, poor maintenance of small ponds and lakes which are considered as the main water sources, reduction of water in reservoirs are the few causes which disturb the crop lands resulting in soil eruption. Pathinathan. T. and Johnson Savarimuthu. S ijesird, Vol. II (I) July 2015/48

6 eissn: X 4 Heavy rain and Cyclone (Nilam) (Benefit Type) causes soil erosion, loss of soil fertility and so on. Especially Nilam cyclone in the year 2010 destroyed many lives and farming lands in the recent years is also considered to be one of the natural calamities faced by everyone across the Villupuram District. X 5 Lack of Electricity (Cost Type) electricity problem is considered to be the major reason for the farmers which increases the usage of diesel engines and the cost of the diesel is also raised and once again it harms the farmers. X 6 Demand for Fertilizers and Pesticides (Cost Type) Scarce of fertilizers and pesticides led to increase in their cost and burdened the farmers. 4.4 Hierarchical Structure for DHFS The hierarchical structure of this decision making problem is shown from the below diagram; High Preference Crop Failure Crop Debt Lack of Water Heavy Rain Lack of Electricity Demand for Fertilizers Paddy Sugarcane Groundnut Cucumber Banana Fig 5.1: Preference order hierarchical structure 5. ADAPTATION AND DESCRIPTION OF THE PROBLEM Dual Hesitant fuzzy decision matrix is obtained by considering each and every expert s opinion with their possible membership values and they are recorded as follows: Table 5.1: Hesitant Fuzzy Decision Matrix (by utilizing Step 1 in Algorithm) A 1 {{0.9,0.5,0.7,0.5,0.6}, {0.1,0.5,0.3,0.5,0.4}} A 2 {{0.4,0.6,,,0.8}, {0.6,0.4,0.,,0.2}} A 3 {{,,0.5,,0.6}, {,,0.5,,0.4}} A 4 {{0.2,,0.9,0.5}, {0.8,,0.1,,0.5}} A 5 {{0.5,0.3,0.9,0.6,0.3}, {0.5,0.7,0.1,0.4,0.3}} X 1 X 2 X 3 X 4 X 5 X 6 {{0.9,0.6,0.7,0.1,0.3}, {{0.9,0.8,0.7,0.5,0.2}, {{.5,0.9,0.7,0.5,0.2}, {{0.9,0.8,0.5,,0.1}, {{0.9,0.9,0.9,,0.9}, {0.1,0.4,0.3,0.9,0.7}} {0.1,0.2,0.3,0.5,0.8}} {0.5,0.1,0.3,0.5,0.8}} {0.1,0.2,0.5,,0.9}} {0.1,0.1,0.1,,0.1}} {{0.8,0.9,0.5,0.2,0.1}, {{0.9,0.8,0.6,0.6,0.2}, {{0.5,,0.7,0.5,0.5}, {{0.8,0.6,0.5,,0.1}, {{0.9,0.9,0.9,,0.9}, {0.2,0.1,0.5,0.8,0.9}} {0.1,0.2,0.4,0.4,0.8}} {0.5,,0.3,0.5,0.5}} {0.3,0.6,0.5,,0.9}} {0.1,0.1,0.1,,0.1}} {{0.9,0.8,0.6,0.1,0.2}, {{,0.7,0.6,0.4,0.2}, {{0.8,,0.7,0.5,0.1}, {{0.7,0.3,0.5,,0.1}, {{0.9,0.9,0.9,,0.9}, {0.1,0.2,0.4,0.9,0.8}} {,0.3,0.4,0.6,0.8}} {0.2,,0.3,0.5,0.9}} {0.3,0.7,0.5,0.9}} {0.1,0.1,0.1,,0.1}} {{0.9,0.8,0.5,0.2,0.3}, {{,,0.5,0.4,0.2}, {{,,0.7,0.5,0.2}, {{0.5,0.8,0.5,,0.1}, {{0.9,0.9,0.9,,0.9}, {0.1,0.2,0.5,0.8,0.7}} {,,0.5,0.6,0.8}} {,,0.3,0.5,0.8}} {0.5,0.2,0.5,,0.9}} {0.1,0.1,0.1,,0.1}} {{0.8,0.7,0.4,0.1,0.2}, {{0.9,,0.6,0.4,0.2}, {{0.2,0.9,0.7,0.5,0.2}, {{0.5,0.3,0.5,,0.1}, {{0.9,0.9,0.9,,0.9}, {0.2,0.3,0.6,0.9,0.8}} {0.1,,0.4,0.6,0.8}} {0.8,0.1,0.3,0.5,0.8}} {0.5,0.7,0.5,,0.9}} {0.1,0.1,0.1,,0.1}} Pathinathan. T. and Johnson Savarimuthu. S ijesird, Vol. II (I) July 2015/49

7 eissn: where, A i, X i, i =1,2,3,4,5 (Five Alternatives) i =1,2,3,4,5,6 (Six Attributes) And A 1 (X i ), i =1,2,3,4,5,6 denotes Alternative 1 (Paddy) comparing with all six attributes. And all the five experts are asked to give their opinions and the opinions are tabulated. A 1 (X i ) = (DM 1, DM 2, DM 3, DM 4, DM 5 ) For instance, A 1 (X 1 ) =(0.9, 0.5, 0.7, 0.5, 0.6), denotes on discussing Alternative 1 (Paddy) with attribute 1 Table 5.2: Hesitant Fuzzy Decision Matrix (crop failure), Decision Maker 1 (DM 1 ) provide 0.9 as the membership value, indicates crop failure considered to be the biggest burden for farmer who involve in paddy cultivation. Similarly, DM 2 provide 0.5 as the membership value and so on Suppose, if A 2 (X 1 ) = (0.4, 0.6,,, 0.8), denotes on discussing Alternative 2 (Sugarcane) with the attribute 1 (crop failure), DM 3 and DM 4 failed to record their values due to the lack of knowledge about the respective alternative over the attribute. X 1 X 2 X 3 X 4 X 5 X 6 A 1 {0.9,0.5,0.7,0.5,0.6} {0.9,0.6,0.7,0.1,0.3} {0.9,0.8,0.7,0.5,0.2} {0.5,0.9,0.7,0.5,0.2} {0.9,0.8,0.5,0.1,0.1} {0.9,0.9,0.9,0.9,0.9} A 2 {0.4,0.6,0.4,0.4,0.8} {0.8,0.9,0.5,0.2,0.1} {0.9,0.8,0.6,0.6,0.2} {0.5,0.5,0.7,0.5,0.5} {0.8,0.6,0.5,0.1,0.1} {0.9,0.9,0.9,0.9,0.9} A 3 {0.5,0.5,0.5,0.5,0.6} {0.9,0.8,0.6,0.1,0.2} {0.2,0.7,0.6,0.4,0.2} {0.8,0.1,0.7,0.5,0.1} {0.7,0.3,0.5,0.1,0.1} {0.9,0.9,0.9,0.9,0.9} A 4 {0.2,0.2,0.9,0.2,0.5} {0.9,0.8,0.5,0.2,0.3} {0.2,0.2,0.5,0.4,0.2} {0.2,0.2,0.7,0.5,0.2} {0.5,0.8,0.5,0.1,0.1} {0.9,0.9,0.9,0.9,0.9} A 5 {0.5,0.3,0.9,0.6,0.3} {0.8,0.7,0.4,0.1,0.2} {0.9,0.2,0.6,0.4,0.2} {0.2,0.9,0.7,0.5,0.2} {0.5,0.3,0.5,0.1,0.1} {0.9,0.9,0.9,0.9,0.9} Table 5.3: Dual Hesitant Fuzzy Decision Matrix (by utilizing Step 1 in Algorithm) X 1 X 2 X 3 X 4 X 5 X 6 A 1 {0.1,0.5,0.3,0.5,0.4} {0.1,0.4,0.3,0.9,0.7} {0.1,0.2,0.3,0.5,0.8} {0.5,0.1,0.3,0.5,0.8} {0.1,0.2,0.5,,0.9} {0.1,0.1,0.1,,0.1} A 2 {0.6,0.4,0.,,0.2} {0.2,0.1,0.5,0.8,0.9} {0.1,0.2,0.4,0.4,0.8} {0.5,,0.3,0.5,0.5} {0.3,0.6,0.5,,0.9} {0.1,0.1,0.1,,0.1} A 3 {,,0.5,,0.4} {0.1,0.2,0.4,0.9,0.8} {,0.3,0.4,0.6,0.8} {0.2,,0.3,0.5,0.9} {0.3,0.7,0.5,0.9} {0.1,0.1,0.1,,0.1} A 4 {0.8,,0.1,,0.5} {0.1,0.2,0.5,0.8,0.7} {,,0.5,0.6,0.8} {,,0.3,0.5,0.8} {0.5,0.2,0.5,,0.9} {0.1,0.1,0.1,,0.1} A 5 {0.5,0.7,0.1,0.4,0.3} {0.2,0.3,0.6,0.9,0.8} {0.1,,0.4,0.6,0.8} {0.8,0.1,0.3,0.5,0.8} {0.5,0.7,0.5,,0.9} {0.1,0.1,0.1,,0.1} Table 5.4: Dual Hesitant Fuzzy Decision Matrix X 1 X 2 X 3 X 4 X 5 X 6 A 1 {0.1,0.5,0.3,0.5,0.4} {0.1,0.4,0.3,0.9,0.7} {0.1,0.2,0.3,0.5,0.8} {0.5,0.1,0.3,0.5,0.8} {0.1,0.2,0.5,0.1,0.9} {0.1,0.1,0.1,0.1,0.1} A 2 {0.6,0.4,0.2,0.2,0.2} {0.2,0.1,0.5,0.8,0.9} {0.1,0.2,0.4,0.4,0.8} {0.5,0.3,0.3,0.5,0.5} {0.2,0.6,0.5,0.2,0.9} {0.1,0.1,0.1,0.1,0.1} A 3 {0.4,0.4,0.5,0.4,0.4} {0.1,0.2,0.4,0.9,0.8} {0.3,0.3,0.4,0.6,0.8} {0.2,0.2,0.3,0.5,0.9} {0.3,0.7,0.5,0.3,0.9} {0.1,0.1,0.1,0.1,0.1} A 4 {0.8,0.1,0.1,0.1,0.5} {0.1,0.2,0.5,0.8,0.7} {0.5,0.5,0.5,0.6,0.8} {0.3,0.3,0.3,0.5,0.8} {0.5,0.2,0.5,0.2,0.9} {0.1,0.1,0.1,0.1,0.1} A 5 {0.5,0.7,0.1,0.4,0.3} {0.2,0.3,0.6,0.9,0.8} {0.1,0.1,0.4,0.6,0.8} {0.8,0.1,0.3,0.5,0.8} {0.5,0.7,0.5,0.5,0.9} {0.1,0.1,0.1,0.1,0.1} By adapting the algorithm in the section 3 and by using the equation (3.1 and 3.2), we have the following positive ideal and negative ideal solution for each alternative over the attribute. Table 5.5: Positive Ideal Solution (by Eqn: 3.1) A 1 A 2 A 3 A 4 A 5 A 6 Positive Ideal Solution { :0.9,0.6,0.9,0.6,0.8 } A 1 { 2:0.9,0.9,0.7,0.2,0.3 } A 2 { :0.9,0.8,0.7,0.6,0.2 } A 3 { :0.8,0.9,0.7,0.5,0.5 } A 4 { :0.9,0.8,0.5,0.1,0.1 } A 5 { :0.9,0.9,0.9,0.9,0.9 } A 6 Dual Positive Ideal Solution { :0.8,0.7,0.5,0.5,0.5 } { :0.2,0.4,0.6,0.9,0.9 } { :0.5,0.5,0.5,0.6,0.8 } { :0.8,0.3,0.3,0.5,0.9 } { :0.5,0.7,0.5,0.5,0.9 } { :0.1,0.1,0.1,0.1,0.1 } Pathinathan. T. and Johnson Savarimuthu. S ijesird, Vol. II (I) July 2015/50

8 eissn: Table 5.6: Negative Ideal Solution (by Eqn: 3.2) A 1 A 2 A 3 A 4 A 5 A 6 Negative Ideal Solution { :0.2,0.2,0.4,0.2,0.3 } A 1 { :0.8,0.6,0.4,0.1,0.1 } A 2 { :0.2,0.2,0.5,0.4,0.2 } A 3 { :0.2,0.1,0.7,0.5,0.1 } A 4 { :0.5,0.3,0.5,0.1,0.1 } A 5 { :0.9,0.9,0.9,0.9,0.9 } A 6 Dual Negative Ideal Solution { :0.1,0.1,0.1,0.1,0.2 } { :0.1,0.1,0.3,0.8,0.7 } { :0.1,0.1,0.3,0.4,0.8 } { :0.2,0.1,0.3,0.5,0.5 } { :0.1,0.2,0.5,0.1,0.9 } { :0.1,0.1,0.1,0.1,0.1 } Then, by utilizing the equation (3.3 and 3.4), we have the following distance values; Hamming Distance Calculations, Pathinathan. T. and Johnson Savarimuthu. S ijesird, Vol. II (I) July 2015/51

9 eissn: Table 5.7: Hamming Distance for HFS(by Eqn: 3.3) d i d 1 d 2 d 3 d 4 d 5 PDistance d i d d d d d 5 NDistance d i d 1 d 2 d 3 d 4 d 5 Table 5.8: Hamming Distance for DHFS (by Eqn: 3.4) Dual PDistance d i d d d d d 5 Dual NDistance By using equation 3.5, we recorded the closeness values among the alternatives and they are tabulated as follows; Table 5.9: Closeness for HFS (by Eqn: 3.5) d i d i d i c(a i ) c(a 1 ) c(a 2 ) c(a 3 ) c(a 4 ) c(a 5 ) From the above table we rank the alternatives A i (i=1,2,3,4,5) as: A 5 A 1 A 2 A 3 A 4 Pathinathan. T. and Johnson Savarimuthu. S ijesird, Vol. II (I) July 2015/52

10 eissn: Table 5.10: Closeness for DHFS (by Eqn: 3.5) d i d i d i Dual c(a i ) c(a 1 ) c(a 2 ) c(a 3 ) c(a 4 ) c(a 5 ) From the above table we rank the alternatives A i (i=1,2,3,4,5) as: A 1 A 2 A 3 A 5 A 4 Table 5.11: Relative closeness and order preference ranking c(a i ) Dual c(a i ) c(a 1 ) c(a 2 ) c(a 3 ) c(a 4 ) c(a 5 ) RESULTS AND INTERPRETATION By aggregating the opinion collected from the five Decision Makers as given in table 5.10, we have the preference ranking order relation as A 1 A 2 A 3 A 5 A 4, (i.e.,) this preference ranking order relation reveals the order in which the farmers lives get drastically affect by the failure of crops. As our calculation express, the alternative A 1 (paddy) is dominated by all the other alternatives. That means, when there is a failure of paddy yield due to natural calamities like cyclone or acute dry weather or even attack of insects and disease, the loss is very high for the farmers to bear. Low level yield of Sugarcane (A 2 ) and Groundnuts (A 3 ) almost share same ordering position when compared with the other three alternatives. It shows that the paddy (A 1 ) cultivation and Sugarcane (A 2 ) cultivation are the two major crops failure of which puts the farmer in very bad financial situation. As in fact good harvest of these crops majority of farmers cultivate paddy and sugarcane rescues farmers from their woes. By adapting the Dual Hesitant Fuzzy set concept, we have the preference ranking order relation as A 1 A 2 A 3 A 5 A 4 showing paddy cultivation is dominated by all the other alternatives and it shows the positive response over the attributes. REFERENCES [1] Bellmann, R.E., and Zadeh, L.A., Decision making in a Fuzzy Environment, Management Science, U.S.A, Vol. 17, No. 4, pp , (1970). [2] Chen, N., Xu, Z.S., and Xia, M.M., Correlation coefficients of hesitant fuzzy sets and their applications to clustering analysis, Applied Mathematical Modeling, Vol. 37, pp , (2013). [3] Dubois, D., and Prade, H., Fuzzy Sets and Systems: Theory and Applications, Publisher, Academic press, New York, London, Sydney San Francisco, pp. 14, (1980). [4] Kauffmann, A., Introduction to the theory of fuzzy sets, Academic Press, New York, Vol. 1, (1975). [5] Klir, G., and Yuan, B., Fuzzy sets and fuzzy logic: Theory and applications, Prentice Hall, Upper Saddle River, (1995). [6] Pathinathan, T., and Ponnivalavan, K., Pexxntagonal Fuzzy Numbers, International Journal of Computing Algorithm, Vol. 3, pp , (2014). [7] Pathinathan, T., and Rajkumar., Sieving out the Poor Using Fuzzy Tools, International Journal of computing Algorithm, Vol.03, pp , (2014). [8] Rodriguez, R.M., Martinez, L., Torra, V., Xu, Z.S., and Herrera, F., Hesitant Fuzzy Sets: State of the art and future directions, International Journal of Information Sciences, Elsevier Science Publications, (2014). Pathinathan. T. and Johnson Savarimuthu. S ijesird, Vol. II (I) July 2015/53

11 eissn: [9] Saaty, T.L., How to make a decision: The Analytic Hierarchy Process, European Journal of Operation Research, North Holland, Vol. 48, pp. 926, (1990). [10] Saaty, T.L., Fundamentals of decisionmaking and priority theory with the analytic hierarchy process, RWS Publications, (1994). [11] Torra, V., Hesitant fuzzy sets, International Journal of Intelligent Systems, Vol. 25, pp , (2010). [12] Torra, V., and Narukawa, Y., Modeling decisions: Information fusion and aggregation operators, Springer, (2007). [13] Torra, V., and Narukawa, Y., On hesitant fuzzy sets and decision, 18 th IEEE International Conference on Fuzzy Systems, Jeju Island, Korea, pp , (2009). [14] Umamaheswari, A., and Kumari, P., Fuzzy TOPSIS and Fuzzy VIKOR methods using the Triangular Fuzzy Hesitant Sets, International Journal of Computer Science Engineering and Information Technology Research, Vol. 4, pp. 1524, (2014). [15] Yager, R.R., On the theory of bags, International Journal of General systems, Vol. 13, pp. 2337, (1986). [16] Yager, R.R., Fuzzy Sets and Applications: Selected Papers by L. A. Zadeh, publisher, John Wiley and Sons Inc, Canada, pp. 29, (1987). [17] Xia, M.M., and Xu, Z.S., Studies on the aggregation of intutionistic fuzzy and hesitant fuzzy information, Technical Report, (2011). [18] Xia, M.M., and Xu, Z.S., Hesitant fuzzy information aggregation in decision making, International Journal of Approximate Reasoning, Vol. 52, pp , (2011). [19] Zadeh, L.A., Fuzzy Sets. Information and Control, Vol. 8, pp , (1965). [20] Zadeh, L.A., Probability measures of fuzzy events, Journal of Mathematical Analysis and Applications, Vol. 23, pp , (1968). [21] Zeshui, Xu., Hesitant Fuzzy Sets and Theory, Studies in Fuzziness and Soft Computing, SpringerVerlag Publications, Vol. 314, (2014). [22] Zhu, B., Xu, Z., and Xia, M., Dual Hesitant Fuzzy Sets, Journal of Applied Mathematics, Hindawi Publishing Corporation, Article ID , (2012). [23] f [24] [25] _apr_15_eng.pdf [26] Pathinathan. T. and Johnson Savarimuthu. S ijesird, Vol. II (I) July 2015/54