International Journal of SCIENCE AND HUMANITIES
Volume 1, Number 2 : July - December 2015
EDITORS-IN-CHIEF K. Prem Nazeer U. Rizwan A. Noor Mohamed
ISLAMIAH COLLEGE PUBLICATIONS
ISLAMIAH COLLEGE (AUTONOMOUS), VANIYAMBADI 2INTERNATIONAL JOURNAL OF SCIENCE AND HUMANITIES EDITORS-IN-CHIEF Dr. K. PREM NAZEER Dr. U. RIZWAN Dr. A. NOOR MOHAMED Principal Department of Mathematics Department of Commerce Islamiah College Islamiah College Islamiah College (Autonomous) (Autonomous) (Autonomous) Vaniyambadi 635 752 Vaniyambadi 635 752 Vaniyambadi 635 752
EDITORIAL BOARD Dr. G. Shakeel Mohammed Dr. M. A. Jayaraj Islamiah College, Vaniyambadi Joint Director of Collegiate Education, Vellore Dr. Mansur Ahmed Dr. Major. I.B. Syed Shahabuddeen Islamiah College, Vaniyambadi Islamiah Women’s College, Vaniyambadi Dr. A. Liyakath Ali Dr. R. Saravanan Islamiah College, Vaniyambadi VIT University, Vellore Dr. H. S. Muthu Mohamed Dr. P. Nisar Ahmed Islamiah College, Vaniyambadi University of Madras, Chennai Dr. T. Mohamed Ilyas Dr. R. T. Rajendira Kumar Islamiah College, Vaniyambadi Bharathiar University, Coimbatore Dr. C. A. Basheer Ahamed Khan Dr. A. S. Mohamed Rafee Islamiah College, Vaniyambadi Mazharul Uloom College, Ambur Dr. M. Abdul Kadar Dr. T. K. Thiruvengada Mani Islamiah College, Vaniyambadi University of Madras, Chennai Dr. P. Sivaraji Dr. A. Subramanian Islamiah College, Vaniyambadi University of Madras, Chennai Dr. Shaik Abdul Wahab Islamiah College, Vaniyambadi ANNUAL SUBSCRIPTION RATES Within India : Rs. 1000 Outside India : 100 US $ All remittances must be paid in the name of The Principal, Islamiah College, Vaniyambadi. For further details regarding payment of subscription, please contact [email protected] Disclaimer : All views expressed in the articles are that of the authors. The Publishers are not responsible for the contents of the individual authors.
ISLAMIAH COLLEGE PUBLICATIONS ISLAMIAH COLLEGE (AUTONOMOUS) VANIYAMBADI 635 752, INDIA Table of Contents Part A: Science
Some Properties of Intuitionistic Fuzzy sets of Cube Root Type U. Rizwan and I. Mohammed Nabeel ...... 707 - 712
Fuzzy Optimization of Single Period Model with Special Reference to Christmas Tree Problem M. Maragatham and A. Tahseen Jahan ...... 713 - 726
On Intiutionistic Fuzzy Volterra Spaces S. Soundararajan, U. Rizwan and Syed Tahir Hussainy ...... 727 - 738
A New Class of Life Distribution based on Characteristic Function Ordering U. Rizwan, Syed Tahir Hussainy and B. Towfeeq Ahmed ...... 739 - 746
Geometric Processes in Two Dimensions: Basic Results and Applications U. Rizwan and M. Mohamad Yunus ...... 747 - 774
Bivariate Optimal Replacement Policies for a Multistate Degenerative System with Varying Cost Structures U. Rizwan and Zahiruddeen ...... 775 - 792
Growth and Characterization of Brushite Crystal using Silica gel Medium F. Liyakath Ali Khan and M. Saravana Kumar ...... 793 - 800
SCLC Conduction Mechanism in Indium Selenide Thin Films and its Dependence on Local Structure C. Viswanathan, K. Prem Nazeer, M. Tamil Selvan, S. Velumani and J. A. Ascencio ...... 801 - 816
Theoretical Investigations on the Molecular Structure, Infra Red Andraman Spectral Analysis of 4-Methylbenzylcyanide F. Liyakath Ali Khan, A. Md. Sabeelullah Roomy and G. Saravanan ...... 817-828 ii
Stabilization of Super Paramagnetic Iron Oxide Nanoparticles by Polysaccharides: Prepartion, Characterization and Magnetic Studies D. Sivakumar, M. Mohamed Rafi, K. Prem Nazeer and A. Ayisha Begam ...... 829 - 838
Dielectric Polarization Studies of H-Bonded Complexes of Alkyl Methacrylate with O-Substituted Benzoic Acids F. Liyakath Ali Khan, J. Udayaseelan and M. Arvinthraj ...... 839 - 850
FT-Raman and FT-IR Spectra, NBO Analysis, ab initio and Density Functional Studies of 4 cyanothioanisole F. Liyakath Ali Khan, Saravanan and A. Md. Sabeelullah Roomy . 851 - 866
Synthesis of New Chitin Derivatives and their Application as Efficient Heavy Metal Adsorbents S. Raja Mohamed Kamil and S. Jafar Sathik ...... 867 - 878
Synthesis and characterization of novel biodegradable aliphatic copolyesterspoly(butylene succinate-co-butylene adipate), poly (butylene succinate -co-ethylene succinate) CT. Ravichandran, S.Rajamohamed Kamil, M. Karunanidhi and S. Jafar Sathik ...... 879-894
Molecular Based Identification of Colonial Ascidians of Mandapam Coast, India through Sequencing mt-DNA H Abdul Jaffar Ali, V Gopi, A Soban Akram, ML Mohammed Kaleem Arshan and T.V. Jaya Rani ...... 895 - 908
Diversity of Ascidians in the Selected Transects along the Thoothukudi Coast ML Mohammed Kaleem Arshan, A Soban Akram, H Abdul Jaffar Ali and T.V. Jaya Rani ...... 909 - 918
Revolution-Based Optimization Structure for Enhancing the Performance and Cost of Workflows in the Cloud A. Viswanathan and A. A. Khadar Maideen ...... 919 - 928
Conditional Probabilities Applied in Content Mining for Filtering Text Data V. Govindaraj and H. Faheem Ahmed ...... 929 - 942 iii
Attribute Based Encryption with Privacy Preserving Access Control in Clouds V. Govindaraj and P. Magizhan ...... 943 - 954
A Study on Cloud Computing and its Applications to Agriculture Sector A. Viswanathan and H. Faheem Ahmed ...... 955 - 966
Part B: Humanities
Traditional Medicinal Plants Referred in Holy Quran Tanveer Ahmed and N. P. M. Mohamed Tariq ...... 967 - 980
Interpreter of Seerapuranam and Seerapurana’s Edition K. Mohamed Farook ...... 981-984
A Study on Karai Iraiyadiyanin Nabi Mozhikkural M. Mujeebur Rahman ...... 985 - 988
Petrifying Memories Rouses Quest for Freedom in Beloved V. Noorul Hasan and R. Radhika ...... 989 - 994
An Overview of Sea Literature S. Abdullah Shah ...... 995-998
Communal Harmony through Inter-Caste Relations in Rural India H. Munavar Jan ...... 999 - 1012
Nawab Muhammad Ali’s Relationship with Yusuf Khan D. Abul Fazal ...... 1013 - 1018
A Scheme for Understanding Rural Nonfarm Sector among the Indian States by using Cluster Analysis S. Liyahath John and M. Suresh ...... 1019 - 1028
A study on Quality of Work Life and Organizational Commitment of Public Sector Bank Employees in Vellore District C. Nithya and A. Noor Mohamed ...... 1029 - 1044 iv
Service Quality at Railway Junction T. Mohamed Ilyas and H. Ashik Ahamed ...... 1045 - 1056
Employee’s Motivation: An Energizer for Philosophical Thinking M. Krishna Murthy and S. Varalakshmi ...... 1057 - 1070
Organizational Climate and Job Satisfaction of Arts and Science College Teachers N. Rojabai and A. Noor Mohamed ...... 1071 - 1082
Social and Family Role Stress and Job Satisfaction V. Mahmudul Hasan and T. Mohamed Ilyas ...... 1083 - 1106
A study on Working Capital Management of Footwear Companies listed in BSE A. Khaleequzzaman and A. Noor Mohamed ...... 1107 - 1120
Role of NPTEL for e-Learning N. Abdul Latheef ...... 1121 - 1126
An Overview of Internet Usage and Awarness of Open Resources of CBSE School Students in Dubai R. Senthil Kumar, B.S. Swaroop Rani, Joyson Soundarajan and V. Newton ...... 1127 - 1134
Relationship Between Selected Psychological Variable and Achievement of High School Students K. Shakila and A. Subramanian ...... 1135 - 1140
International Journal of Science and Humanities ISSN 2394 9236 Volume 1, Number 2 (2015), pp. 707–712 c Islamiah College Publications http://www.islamiahcollege.edu.in
Some Properties of Intuitionistic Fuzzy sets of Cube Root Type U. Rizwan† and I. Mohammed Nabeel ∗1 Department of Mathematics, Islamiah College, Vaniyambadi 635 752, India † e-mail: [email protected] ∗ Department of Mathematics, Rajalakshmi Engineering College, Chennai, India
Abstract
In this paper, we introduce a new class of Intuitioistic Fuzzy Sets , called Intuitionistic Fuzzy Sets of Cube Root Type (IFSCRT). We also present some of the properties of IFSCRT AMS Subject Classification: 60K10
Keywords: Intuitionistic Fuzzy Sets, .
1. Introduction
Fuzzy sets were introduced by Lofti A. Zadeh in 1965 as a generalization of classical (Crisp) sets. Further the Fuzzy sets are generalized by Krassimir T.Atanassov in which he has taken non membership values also into consideration and he introduced Intuitionistic Fuzzy Set (IFS) and its extension Intuitionistic Fuzzy Set of Second Type (IFSST). Palaniappan and Srinivasan have introduced Intuitionistic Fuzzy Sets of Root Type (IFSRT) and studied some of its properties. In this paper, we define Intuitionistic Fuzzy Sets of Cube Root Type (IFSCRT), some operations and relations over IFSCRT and state few of their properties.
2. Preliminaries
In this section, we give some definition of various types of IFS.
1Part Time Ph.D. Research Scholar in Mathematics at Bharathiar University, Coimbatore 708 Some Properties of Intuitionistic Fuzzy sets of Cube Root Type
Definition 2.1 Let X be a non empty set. An IFS A in X is defined as an object of the form A = {hx, µA(x), γA(x)i : x ∈ X} where the functions µA(x): X → [0, 1] and γA(x): X → [0, 1] denote the membership and non-membership function of A respectively, and
0 ≤ µA(x) + γA(x) ≤ 1 for each x ∈ X
Remark. An ordinary fuzzy set can also be generalized as
A = {hx, µA(x), 1 − µA(x)i : x ∈ X}
Definition 2.2 Let X be a non empty set. An Intuitionistic Fuzzy Set of Second Type (IFSST) A in X is defined as an object of the form
A = {hx, µA(x), γA(x)i : x ∈ X} where the functions µA(x): X → [0, 1] and γA(x): X → [0, 1] denote the membership and non-membership function of A respectively, and
2 2 0 ≤ [µA(x)] + [γA(x)] ≤ 1 for each x ∈ X
Remark. It is obvious that for all real numbers a, b ∈ [0, 1]if 0 ≤ a + b ≤ 1 then 0 ≤ a2 + b2 ≤ 1
Definition 2.3 Let X be a non empty set. An Intuitionistic Fuzzy Set of Root Type (IFSRT) A in X is defined as an object of the form
A = {hx, µA(x), γA(x)i : x ∈ X} where the functions µA(x): X → [0, 1] and γA(x): X → [0, 1] denote the membership and non-membership function of A respectively, and
pµ (x) pγ (x) 0 ≤ A + A ≤ 1 for each x ∈ X 2 2 Definition 2.4 Let X be a non empty set. An Intuitionistic Fuzzy Set of Cube Root Type (IFSCRT) A in X is defined as an object of the form
A = {hx, µA(x), γA(x)i : x ∈ X} where the functions µA(x): X → [0, 1] and γA(x): X → [0, 1] denote the membership and non-membership function of A respectively, and
p3 µ (x) + p3 γ (x) 0 ≤ A √ A ≤ 1 for each x ∈ X 3 U. Rizwan and I. Mohammed Nabeel 709
Remark. It is obvious that for all real numbers a, b [0, 1], if 0 ≤ a + b ≤ 1 then √ √ 3 a + 3 b 0 ≤ √ ≤ 1 3 Definition 2.5 Let X be a non empty set.Let A and B be two IFSCRTs such that
A = {hx, µA(x), γA(x)i : x ∈ X}
B = {hx, µB(x), γB(x)i : x ∈ X} We define the following relations and operations
(i) A ⊂ B if and only if µA(x) ≤ µB(x) and γA(x) ≥ γB(x) ∀ x ∈ X
(ii) A ⊃ B if and only if µA(x) ≥ µB(x) and γA(x) ≤ γB(x) ∀ x ∈ X
(iii) A = B if and only if µA(x) = µB(x) and γA(x) = γB(x) ∀ x ∈ X
(iv) A ∩ B = {hx, min(µA(x), µB(x)), max(γA(x), γB(x))i : x ∈ X}
(v) A ∪ B = {hx, max(µA(x), µB(x)), min(γA(x), γB(x))i : x ∈ X}
(vi) The complement of A is defined by A = {hx, γA(x), µA(x)i : x ∈ X}
Definition 2.6 The degree of non determinacy (uncertainty) of an element x∈X to the IFSCRT A is defined by
p3 p3 3 πA(x) = (1 − µA(x) − γA(x)) Definition 2.7 For every IFSCRT A, we define the following operators. The Necessity measure on A
p3 3 A = hx, µA(x), (1 − µA(x)) i : x ∈ X The Possibility measure on A
p3 3 ♦A = hx, (1 − γA(x)) , γA(x)i : x ∈ X Definition 2.8 For every two IFSCRTs A and B we define the following relations:
p3 µ (x) + p3 µ (x) p3 γ (x) + p3 γ (x) (i) A@B = {hx, A B , A B i : x ∈ X } 2 2 q q p3 p3 p3 p3 (ii) A$B = {hx, µA(x) µB(x), γA(x) γB(x)i : x ∈ X }
It is easy to verify the correctness of the defined relations. 710 Some Properties of Intuitionistic Fuzzy sets of Cube Root Type
3. Properties of IFSCRT
In this section, we prove some properties of IFSCRT.
Proposition 3.1 For every IFSCRTs A and B,we have
(i) A@A = A
(ii) A$A = A
(iii) A@B = A@B
(iv) A$B = A$B
(v) (A@B) = A@B
Proof. (* + ) p3 µ (x) + p3 µ (x) p3 γ (x) + p3 γ (x) (i) A@A = x, A A , A A : x ∈ X 2 2
nD p3 p3 E o = x, µA(x), γA(x) : x ∈ X =A
q q p3 2 p3 2 (ii) A$A = x, ( µA(x)) , ( γA(x)) : x ∈ X
nD p3 p3 E o = x, µA(x), γA(x) : x ∈ X =A
(* + ) p3 γ (x) + p3 γ (x) p3 µ (x) + p3 µ (x) (iii) A@B = x, A B , A B : x ∈ X 2 2 (* + ) p3 µ (x) + p3 µ (x) p3 γ (x) + p3 γ (x) A@B = x, A B , A B : x ∈ X 2 2 = A@B q q p3 p3 p3 p3 (iv) A$B = x, γA(x) γB(x), µA(x) µB(x) : x ∈ X
q q p3 p3 p3 p3 A$B = x, µA(x) µB(x), γA(x) γB(x) : x ∈ X
= A$B U. Rizwan and I. Mohammed Nabeel 711
(* + ) p3 µ (x) + p3 µ (x) p3 γ (x) + p3 γ (x) (v) A@B = x, A B , A B : x ∈ X 2 2
(A@B) = (* + ) p3 µ (x) + p3 µ (x) (1 − p3 µ (x)) + (1 − p3 µ (x)) x, A B , A B : x ∈ X 2 2
and
p3 3 A = {hx, µA(x), (1 − µA(x)) i : x ∈ X}
p3 3 B = {hx, µB(x), (1 − µB(x)) i : x ∈ X},
so that
A@B =
(* + ) p3 µ (x) + p3 µ (x) (1 − p3 µ (x))3 + (1 − p3 µ (x))3 x, A B , A A : x ∈ X 2 2
This completes the proof.
Proposition 3.2 For every IFSCRT A, B and C we have
(i) (A ∪ B)@C = (A@C) ∩ (B@C)
(ii) (A ∩ B)@C = (A@C) ∪ (B@C)
(iii) (A + B)@C ⊂ (A@C) − (B@C)
(iv) (A ∩ B)$(A ∪ B) = A$B
Proof. The proof is simple and hence omitted.
4. Conclusion
We have made an attempt to establish some operations and relations over IFSCRT. It is still open to check whether there exist an IFSCRT in case of operator already defined on an IFS. 712 Some Properties of Intuitionistic Fuzzy sets of Cube Root Type
References
[1] ATANASSOV K.T., (1999), Intuitionistic Fuzzy sets,thoery and Applications, Springer Verlag, New York.
[2] PARVATHI.R AND PALANIAPPAN. N, (2004), Some operations on Intuitionistic Fuzzy sets of second type, Notes on Intuitionistic Fuzzy sets, 10(2), 1-19.
[3] SRINIVASAN.R AND PALANIAPPAN. N, (2006) Some operations on Intuitionistic Fuzzy sets of Root type, Notes on IFS, 12 (3),20-29
[4] SRINIVASAN.R AND PALANIAPPAN. N, (2009) Some properties of Intuitionistic Fuzzy sets of Root type, International journal of Computational and Applied Mathematics, 4(3), pp 383-390.
[5] SRINIVASAN.R AND PALANIAPPAN. N, (2011), Some Topological Operators on Intuitionistic Fuzzy sets of Root type, Research Methods in Mathematical Sciences, Ed: U. Rizwan, PBS Book Publishers, India, pp 23-28. International Journal of Science and Humanities ISSN 2394 9236 Volume 1, Number 2 (2015), pp. 713–726 c Islamiah College Publications http://www.islamiahcollege.edu.in
Fuzzy Optimization of Single Period Model with Special Reference to Christmas Tree Problem
M. Maragatham† and A. Tahseen Jahan∗
Department of Mathemathics, Periyar EVR Govt. College, Trichy - 620023, India Department of Mathemathics, Justice Basheer Ahmed Sayeed College for women, Chennai - 600018, India
Abstract In this paper single period inventory model where demands are considered as bifuzzy variables is enumerated by the particular example, of Christmas tree problem where seller has to decide how many Christmas trees should be purchased for the season otherwise at the end of the period, the items become obsolete. A method for the ordering of fuzzy numbers with respect to their total integral values is adopted to find the optimal quantity of maximizing the expected profit function. If the profit gained from selling one units is less than the loss incurred for each item left unsold, then the inventory policy should be conservative to reduce the leftover. On the contrary, if the profit gained is greater than the loss incurred, then the inventory policy should be aggressive to satisfy the possible demand. When the profit gained is equal to the loss incurred, the order quantity should be equal to the most likely demand and correspond to the quantity with membership grade 1.
Keywords: Fuzzy variable, bifuzzy model, single period inventory.
1. Introduction
Single period inventory problems where only a single procurement is made, has a varied significance in day-to-day life .But the major setback by the decision maker is to forecast the demand, as uncertainty plays a great role as to how many items, customers will buy during single period. Applications of fuzzy sets within the field of inventory management for the most part considered as fuzzification of the crisp set of theories of inventory control .Under conditions of risk and uncertainly probabilistic models cannot deal with vagueness inherent in imprecise determination of preference constraints and objective functions. 714 Fuzzy Optimization of Single Period Model with Special Reference . . .
Fuzzy set theory has been well developed and applied in wide variety of real problems since its introduction in 1965 by Zadeh [1]. The term fuzzy variable was first introduced by Kaufmann [6], then it appeared in Zadeh [5, 10] and Nahmias [11]. Possibility theory was proposed by Zadeh [5] and developed by many researchers such as Dubois and Prade [12, 13]. Some extensions of fuzzy set have been made in the literature, for example, Type 2 fuzzy set was introduced by Zadeh [10] as a fuzzy set whose membership grade are also fuzzy sets. The intuitionistic fuzzy set was proposed by Atanassov [9] as a pair of membership functions whose sum takes values between 0 and 1; Two fold fuzzy set was derived by Dubois and Prade [14]. Bifuzzy variable was initialized by Liu [15] as a function from a possibility space to the set of fuzzy variables.
In this paper, we discuss a type of two-fold uncertainty: bifuzzy variable, Roughly speaking, a bifuzzy variable is a bifuzzy variable defined on the universal set of fuzzy variables or a fuzzy variable taking “Fuzzy Variable” values, i.e a bifuzzy variable is a function from a possibility space to collection of fuzzy variables. Here, fuzzy means the degree of uncertainty. Based on the concept of bifuzzy variables, we develop a bifuzzy model applicable to the single period inventory problem with special reference to Christmas tree problem.
2. Preliminaries
Definition 2.1 If X is the universe of discourse and let µA : X → [0, 1], denote the membership of the elements ‘x’ of X, then the fuzzy set A˜ is defined as ˜ A = {< x, µA(x) >: x ∈ X}. Each fuzzy set is completely and uniquely defined by one particular membership function. The significance of fuzzy variables is that they facilitate gradual transitions between states and consequently possess natural capability to express and deal with observation and measurements and uncertainties.
Definition 2.2 Fuzzy sets defined by membership function in terms of closed interval of real number identified by lower and upper bounds are called interval-valued fuzzy sets. They are of the form A : X → ε([0, 1]) where ε([0, 1]) denotes the family of all closed intervals of real number in [0, 1]. Interval valued fuzzy sets can further be generalized by allowing their intervals to be fuzzy. Each interval now becomes an ordinary fuzzy set defined within the universal set [0, 1]. Since membership grades assigned to elements of the universal set by these generalized fuzzy sets are ordinary fuzzy sets, these sets are referred to as fuzzy sets of type 2. The membership functions have the form
A : X → F([0, 1]) M. Maragatham and A. Tahseen Jahan 715 where F([0, 1]) denotes the set of all ordinary fuzzy set that can be defined within universal set [0, 1].
Bifuzzy variable
Let X denote a universal set. Then a fuzzy subset A˜ of X is defined by its membership function
µA˜ : X → [0, 1], which assigns to each element x ∈ X a real numbers µA˜(x) in the interval [0, 1] where ˜ the value of A˜(x) at x represents the grade of membership of x in A. For example, if F is fuzzy number (set) with degree of membership F˜(x) of an element x in F, then F˜(x) represents the degree of possibility that a parameter has a value x. Thus, the nearer the ˜ value of A˜(x) to unity, the higher the grade of membership of x in A.
Definition 2.3 The set of elements that belong to the fuzzy set A˜ at least to the degree of membership α is called the α−level set, denoted by
˜ Aα = {x ∈ X : A˜(x) ≥ α}. (2.1)
Definition 2.4 A Fuzzy variable is a function from a possibility space (θ, P(θ), Pos) to the real line R.
Definition 2.5 A bifuzzy variable is a function from a possible space (θ, P(θ), Pos) to a collection of fuzzy variable.
n Definition 2.6 Let f : R → R be a function and ξi bifuzzy ( or fuzzy) variables defined on (θ, P(θ), Pos), i = 1, 2, . . . , n, respectively. Then ξ = f(ξ1, ξ2, . . . , ξn) is a bifuzzy (or fuzzy) variable defined on the product possibility space (θ, P(θ), Pos) as
ξ(θ1, θ2, . . . , θn) = f(ξ1(θ1), ξ2(θ2), . . . , ξn(θn))
for any (θ1, θ2, . . . , θn) ∈ θ. 716 Fuzzy Optimization of Single Period Model with Special Reference . . .
The ordering of fuzzy numbers
Definition 2.7 A fuzzy number A˜ is a function on the set of the real line R whose membership function µA˜(x) has the following characteristics with −∞ ≤ aL ≤ a ≤ a ≤ aR ≤ +∞ : µl(x), aL ≤ x ≤ a 1, a ≤ x ≤ a µA˜(x) = µr(x), a ≤ x ≤ aR 0, Otherwise where µL :[aL, a] → [0, 1] is continuous and strictly increasing; µR :[a, aR] → [0, 1] is continuous and strictly decreasing.
Definition 2.8 Let A˜ be a fuzzy number whose membership function is given via (3) and λ ∈ [0, 1] a predetermined parameter. The total λ−integral value of A˜ is defined as
˜ ˜ ˜ Iλ(A) = λIR(A) + (1 − λ)IL(A),
˜ ˜ ˜ where IL(A) and IR(A) are the left and right integral values of A defined as the following, respectively: Z 1 ˜ −1 IL(A) = µL (α) dα 0 and Z 1 ˜ −1 IR(A) = µR (α) dα, 0 −1 −1 ˜ ˜ where µL (α) and µR (α) are the inverse function of IL(A) and IR(A), respectively. The total λ−Integral value of fuzzy number can be used to rank fuzzy numbers.
Definition 2.9 Let λ ∈ [0, 1] be predetermined. For any two fuzzy numbers A˜ and B,˜ ˜ ˜ ˜ ˜ we define A <, = or> B if and only if Iλ(A) <, = or> Iλ(B).
3. Problem Formulation
We are considering single order inventory problem for the entire demand period where demand is uncertain and occurs only once. The seller has to procure certain number of items at the beginning of period and at the end of certain period. It is either of no use or to be sold at a lower price.
Let M. Maragatham and A. Tahseen Jahan 717
p− Unit price of an item at which it is purchased(or produced); s− Unit selling price of the item(s > p); h− holding cost per each item remaining at the end of the period, or it can be considered as unit selling price of the item after the end of the period; Note that h is negative (h < s); r− Unit shortage cost if there is a shortage or lost revenue due to the inability to supply the demand(r > p);
Figure 1: The membership function a˜
Let us assume that there is no initial inventory on hand; Denote by D0 the order quantity at the beginning of the period and the actual demand Di as a bifuzzy variable with the given set of variables {x˜1, x˜2,..., x˜n}. If the demand Di is smaller than the order quantity D0, then an amount of (D0 − Di) will be left at the end of the period, and the total cost is pD0 + h(D0 − Di). On the contrary, if the demand Di is greater than D0, then the inventory is insufficient by an amount of (Di − D0), and the total cost becomes pD0 +r(Di −D0). To summarize, for a prespecified order quantity D0. Hence, the revenue R˜(D) received and the total cost C˜(D) incurred are given as follows: ( s Di,Di ≤ D0 R˜(D) = s D0,Di > D0 ( pD0 + h(D0 − Di),Di ≤ D0 C˜(D) = pD0 + r(Di − D0),Di > D0 and the total profit P˜(D) = R˜(D) − C˜(D) achieved is given by 718 Fuzzy Optimization of Single Period Model with Special Reference . . .
( (s + h)Di − (p + h)D0,Di ≤ D0 P˜(D) = (s − p + r)D0 − rDi,Di ≥ D0 Let P (α) denote the α−cut set of P.˜ There are three cases to be considered for the value ˜ of D0 in discussing the membership function of P (D): aL ≤ D0 ≤ a, a ≤ D0 ≤ a and a ≤ D0 ≤ aR, See Fig. 1.
Case 1. aL ≤ D0 ≤ a. For D0 lying in the range of aL and a. µp˜ is the same as L(·), viz., the left shape function of µ . It is clear from Fig.1 that for α ≤ L(D), the D˜i −1 lower bound of the α− cut P (α) is (s + h)L (α) − (p + h)D0 because the supply −1 is greater than the demand by an amount of (D0 − L (α)), and the upper bound of −1 p(α) is (s − p + r)D0 − rR (α) because the supply is insufficient by an amount of −1 (R (α) − D0). When α ≥ L(D), the supply is always insufficient for any value of the demand defined in the α− cut. Thus we have
−1 −1 (s + h)L (α) − (p + h)D0 (s − p + r)D0 − rR (α) , 0 ≤ α ≤ L(D0) P (α) = −1 −1 (s − p + r)D0 − rL (α)(s − p + r)D0 − rR (α) ,L(D0) ≤ α ≤ 1
Case 2. a ≤ D0 ≤ a. For D0 lying in the range of a and a the lower bound and upper ˜ −1 −1 bound of the α− cut of P are (s+h)L (α)−(p+h)D0 and (s−p+r)D0 −rR (α), respectively, for any value of α. Therefore, the α− cut of P˜ is
−1 −1 P (α) = (s + h)L (α) − (p + h)D0, (s − p + r)D0 − rR (α) , 0 ≤ α ≤ 1.
Case 3. a ≤ D0 ≤ aR. For D0 lying in the range of a and aR, µp˜ is the same as R(·), viz., the right-shape function of µ . Similar to the discussion in Case 1, the α−cut of D˜i P˜ is
−1 −1 (s + h)L (α) − (p + h)D0 (s − p + r)D0 − rR (α) , 0 ≤ α ≤ R(D0) P (α) = −1 −1 (s + h)L (α) − (p + h)D0 (s + h)R (α) − (p + h)D0 ,R(D0) ≤ α ≤ 1.
4. Bifuzzy Model Transformation
Let ξ = (aL, ρ,˜ aR), where ρ˜ = (ρL, ρ0, ρR) is a triangular fuzzy variable with membership function µp˜(x); Then ξ is a bifuzzy variable. Here,ρ˜ = (ρL, ρ0, ρR) is a M. Maragatham and A. Tahseen Jahan 719
fuzzy variable with membership function µp˜(x). See Fig.2.By the concept of α− cuts, denote the α1− cuts (or α1− level sets) of ρ˜ as follows L R ρ˜α1 = [ρα1 , ρα1 ] = {x ∈ X|µp˜(x) ≥ α1}
L R Here αα1 = ρL +α1(ρ0 −ρL) and αα1 = ρR +α1(ρR −ρ0).The parameter α1 ∈ [0, 1] here reflects decision-maker’s degree of optimism. These intervals indicate where the group arrival rate and service rate lie at possibility level α Note that ρ˜α1 are crisp sets than fuzzy sets.
Let X = {xi ∈ X|µ(xi) ≥ α1, i = 1, 2 . . . , n}, so the bifuzzy variable x =
(aLρ˜α1 , aR) can be denoted as ξ = (aL, X, aR) or denoted as follows: ξ˜ (a ,X , a ) 1 L 1 R ˜ ξ2(aL,X1, aR) ξ = . . ˜ ξn(aL,Xn, aR)
L R ˜ where x1 ≤ x2 ≤ ... ≤ xn,X = [x1, xn] = [ρα1 , ρα1 ], i = 1, 2 . . . , n and ξi is fuzzy L R L variable. It is easy to reach that x1 = ρα1 , xn = ρα1 . In other words, ρα1 is the minimum R value that ρ˜ achieves with possibility α1; ρα1 is the maximum value that ρ˜ achieves with ˜ ˜ ˜ possibility α1. The variable ξ can be expressed in another form as ξ = ξ1Uξ2U...Uξn. ˜ Here ξi is fuzzy variable so the bifuzzy variable ξ is transformed into a fuzzy variable with membership function µξ˜(D). On the basis of the concept of α−cuts (or α level sets),denote the α2− cuts (or α2−level sets)of ξ as follows: ˜ h˜L ˜R i ξα2 (D) = ξα2 , ξα2 = {D ∈ X|µξ˜(D) ≥ α2}, 720 Fuzzy Optimization of Single Period Model with Special Reference . . . or ξ˜ (a , ρL , a ) ≥ α , 1 L α1 R 2 ˜ ξ2(aL,X2, aR) ≥ α2, ˜ . ξα2 (D) = . ξ˜ (a ,X , a ) ≥ α , n−1 L n−1 R 2 ˜ R ξn(aL, ρα1 , aR) ≥ α2 By the convexity of a fuzzy number, the bounds of these intervals are function of ! a − a α 0 ≤ α ≤ R L 2 2 a − a + ρR − ρL R L alpha1 alpha1 and can be obtained as
˜ h˜L ˜R i ξα2 = ξα2 , ξα2 .
L ˜L ˜R ˜R ξα2 = max ξiα2 , ξiα2 = min ξiα2 ; and ˜L −1 ˜R −1 ξiα = min µ (α2)ξiα = max µ (α2), i = 1, 2, . . . , n, 2 ξ˜i 2 ξ˜i respectively. Obviously,
L −1 −1 ξα = max min µ (α2) = min µ (α2), 2 ξ˜i ξ˜n and R −1 −1 ξα = max min µ (α2) = min µ (α2). 2 ξ˜i ξ˜i
Consequently, we can use its α2− cuts to construct the corresponding membership L L ˜ function. ξα2 = a, ξα2 = a, viz.ξ = (aL, a, a, aR) is a similar trapezoidal fuzzy number with the membership function µξ(D) at D ∈ [a, a] is considered subjectively to be 1 approximately.
5. Solution methodology
We consider the above single -period inventory problem where the demand is considered as a bifuzzy variable. Di = (aL, ρ,˜ aR) is a bifuzzy variable, so its P (D) is also a bifuzzy variable. Here ρ˜ = (ρL, ρ0, ρR) is fuzzy variable with membership ˜ function µ˜(p)(x) denote the α1 level set of ρ.˜ It follows from (9) and (10) that (D)i = L (aL, a, a, aR) where a = Pα2 is a fuzzy variable. For each α1 and α2 is given by M. Maragatham and A. Tahseen Jahan 721
decision-maker. Depending on the demand Di. Obviously, the demand Di and the profit ˜ ˜ P (D) have the same membership grade. In the other words, the demand Di and P (D) have the same shape of the membership function. In the rest of this section, we will exploit the exact expression of the total integral value of the fuzzy profit P˜(D) for each given order quantity D0 when the imprecise demand Di is characterized by a trapezoidal fuzzy number Di = (aL, a, a, aR).
From the definition 9, we can know that the parameter λ ∈ [0, 1] reflects decision- maker’s degree of optimism. A large λ indicates a higher degree of optimism. More specifically, ˜ ˜ I0(A) = IL(A)(λ = 0) and ˜ ˜ I1((A)) = IR((A))(λ = 1) represent pessimistic and optimistic decision view points, respectively, while ˜ 1 ˜ ˜ 1 I 1 (A) = (IL((A)) + IR((A)) λ = 2 2 2 provides the moderately optimistic decision -maker a comparison criterion for fuzzy numbers. Moreover, when
1 ˜ ˜ ˜ λ = I 1 (A) = I1((A)) = IR((A))(λ = 1) 2 2 is well qualified for a crisp representative(Defuzzification)of the fuzzy number A˜ 1 Though we do this for the case when the parameter λ = , it can be done for all 2 λ ∈ (0, 1) if so desired. The Yager’s ranking method is based on an area measurement index defined as following:
I (P˜) + I (P˜) I(P˜) = L R 2 ˜ ˜ Where IL(P ) represents the area bounded by the left-shape function of P (D), the ˜ x-axis ,the y-axis, and the horizontal line µp˜ = 1.IR(P ) represents the area bounded by ˜ the right -shape function of P (D), the x-axis, y- axis, and the horizontal line µp˜ = 1. The optimal order quantity D∗ can be discussed from the three cases categorized in the preceding sections.
Case 1. aL ≤ D0 ≤ a.For D0 lying in the range of aL and a, the corresponding area measurement index can be calculated as follows based on equations (6) and (11): 722 Fuzzy Optimization of Single Period Model with Special Reference . . .
Z L(D0) ˜ −1 I(P ) = (s − b + r)D0 − 0.5(s + h + r)D0L(D0) + 0.5 (s + h)L (α)dα 0 Z 1 Z 0 −0.5 γL−1(α)dα, 0.5 γR−1(α)dα, L(D0) 1 ˜ The first and second derivatives of I(P ) with respect to D0 are ∂I(P˜) = (s − p + r) − 0.5(s + h + r)L(D0), ∂D0
2 ˜ ∂ I(P ) 0 2 = −0.5(s + h + r)L (D0). ∂D0 2(s − p + r) The first derivative vanishes when L(D ) = . Since s + h + r > 0 and 0 (s + h + r) 2 ˜ 0 ∂ I(P ) L(·) is an increasing function with L (D0) > 0. Therefore 2 is negative, implying ∂D0 that P˜(D) attains the maximum at the 2(s − p + r) D∗ = L−1 . (s + h + r) (s − p + r) However, 2 must lie in the range of 0 and 1 so that Dast will lie between a (s + h + r) L (s − p + r) and a. Since a > b, 2 is always positive. The requirement of (s + h + r) (s − p + r) ≤ 1 (s + h + r) implies s + r ≤ 2p + h. Hence, we have derived that
(2(s − p + r)) D∗ = L−1 , for s + r ≤ 2p + h. ((s + h + s))
Case 2. a ≤ D0 ≤ a. For D0 lying in the range of a and a, the corresponding area measurement index, based on equation (7) and (11),is
Z 1 Z 1 ˜ s −1 −1 I(P ) = 0.5( )D0 + 0.5(s + h) ?L (α)dα − 0.5r R (α)dα r − 2p − h 0 0 ˜ Which is a linear function in D0. If 0.5(s+r −2p−h) is positive, then P (D) attains its maximum when D0 is set to its upper bound a. On the contrary. If 0.5(s+r −2p−h) M. Maragatham and A. Tahseen Jahan 723
˜ is negative, then P (D) is maximized by setting D0 to its lower bound a. When ˜ 0.5(s + r − 2p − h) = 0, any D0 ∈ [a, a] has the same maximum cost P (D0) Now 0.5(s + r − 2p − h). > 0 implies s + r > 2p + h, 0.5(s + r − 2P − h) < 0 implies s + r < 2p + h and 0.5(s + r − 2p − h) = 0 implies s + r = 2p + h, and we have concluded that
a, s + r ≥ 2p + h D∗ = a, a, s + r = 2p + h a, s + r ≤ 2p + h
Case 3. a ≤ D0 ≤ aR When D0 lies in the range of a and aR, the area measurement index can be calculated from equations (8) and (11) as follows:
Z R(D0) ˜ −1 I(P ) = 0.5(s + r + h)D0R(D0) − (p + h)D0 − 0.5 (s + r + h)R (α)dα 0 Z 1 Z 1 +0.5 (s + h)L−1(α)dα + 0.5 (s + r + h)R−1(α)dα, 0 0 The first and second derivatives of I(P˜) are ∂I(P˜) = 0.5(s + h + r)R(D0) − (p + h), ∂D0
2 ˜ ∂ I(P ) 0 2 = 0.5(s + h + r)R (D0). ∂D0 ∂I(P ) By setting to 0, one derives ∂D0 2(p + h) R(D ) = . 0 (s + r + h)
2 ˜ 0 ∂ I(P ) Since s + r + h and R(·) is decreasing function with R (D0) < 0, therefore 2 is ∂D0 negative, which implies that P˜(D) is maximized for 2(p + h) D∗ = R − 1 , (s + r + h) 2(p + h) 2(p + h) provided ≤ 1. It is obvious that > 0. The upper bound (s + h + r) (s + h + r) constraint of 2(p + h) ≤ 1 (s + h + r) 724 Fuzzy Optimization of Single Period Model with Special Reference . . . implies s + r ≥ 2p + h. Thus, we have derived that 2(p + h) D∗ = R−1 , for s + r ≥ 2p + h. s + h + r Combining the three cases, i.e., equations (12), (13) and (14), the optimal order quantity is calculated as −1 2(s − p + r) L , s + r ≤ 2p + h (s + h + r) D∗ = a, a, s + r = 2p + h 2(s − p + r) R−1 , s + r ≥ 2p + h (s + h + r) In an imprecise and uncertain environment, we sometimes do not know the explicit form of membership function, so following (15) we can’t derive the optimal order quantity. We know that the α− cut of this fuzzy number is
˜ −1 −1 A(α) = [L (α),R (α)] = [aL + α(a − aL), aR − α(aR − a)], and 1 R ρα1 = ρL + α(ρ0 − ρL), ρα1 = ρR − α1(ρR − ρ0). When (αR − αL) 0 ≤ (αR − αL) + (ρRα1 − ρLα1) based on (9) and (10) we reach that
a = aL + α2[ρR − α1(ρR − ρ0) − aL] and a = aR + α2[aR − ρL − α1(ρ0 − ρL)] respectively. Then, based on equation (15), the optimal quantity D∗ to order is
2(s − p + r) a + α [ρ − ρ (ρ − ρ ) − a ] , s + r ≤ 2p + h L 2 R 1 R 0 L (s + h + r) D∗ = {aL + α2 [ρR − α1(ρR − ρ0) − aL] , aR − α2 [aR − ρL − α1(ρ0 − ρL)]}, s + r = 2p + h 2(p + h) a − α [a − ρ − α ](ρ − ρ ) , s + r ≥ 2p + h R 2 R L 1 0 L (s + h + r) which is very simple to calculate.
Now, we consider the above single-period inventory problem in two-fold imprecise and uncertain environment where the demand is considered as a bifuzzy variable M. Maragatham and A. Tahseen Jahan 725
Following (16), we can derive the optimal quantity. From the definition of s, p, h and r, s − p represents the profit gained from selling one unit of the item; p + h represents the loss incurred for each item not sold. If the profit gained from selling one unit is less than the loss incurred for each item left unsold, then the inventory policy should be conservative to reduce the leftover. On the contrary, if the profit gained is greater than the loss incurred, then the inventory policy should be aggressive to satisfy the possible demand. When the profit gained is equal to the loss incurred, the order quantity should be equal to the most likely demand, i.e., the quantity with membership grade I.
6. Numerical Example
We consider a single - period inventory problem with a bifuzzy demand Di = (100, ρ,˜ 200), ρ˜ = (pL, p0, ρR). Denote α1 = 0.5, so ρ0˜.5 = [135, 165[150 − 120]. Thus, we derive a set of triangular fuzzy numbers Di = {100, x, 200|x ∈ [135, 165]}. Based on equations (9) and (10), denote α2 = 0.615(α2 ≤ 0.769), we can derive a similar ˜ trapezoidal fuzzy number Di = (100, 140, 160, 200). Suppose the unit cost, selling price, holding cost and unit shortage cost of the item are, respectively , p = 100, s = 140, h = 80 and r = 120. Because s + r = 260 is less than 2p + h = 280, the first formula of equation (16) is applied to find the optimal order quantity D∗ = 137.647. It is the most superior quantity corresponding to discrete value of α1 and α2 when s + r ≤ 2p + h; If the unit cost dropped to p = 90, then the value of s + r = 260 is equal to that of 2p + h = 260. In that case, any amount defined for the plateau of the membership function. viz.., the amount between 140 and 160, is optimal . If the unit cost is further dropped to p = 80, then value of s + r = 260 is greater than that of 2p + h = 240. Third formula of equation (16) is applied to find the optimal order quantity as D∗ = 162.353, It is the most superior quantity corresponding to discrete value of α1 and α2 when s + r ≥ 2p + h.
7. Conclusion
In this paper, we transform a bifuzzy into a similar trapezoidal fuzzy variable and by applying the Yager’s method for ranking fuzzy numbers, a quantity with largest profit is calculated. Since most single-period inventory problems do not have historical data to construct the probability distribution function for calculating the optimal quantity, or one - fold fuzzy variable can’t intactly describe the uncertain environment, the bifuzzy model constructed in this paper has dealt with this problem well. The future scope of research using bifuzzy modelling is very wide and vast as it is highly helpful for solving real world problems providing more avenues towards simplified analysis and problem solving capacities leading to easier methods of finding apt and appropriate solutions. 726 Fuzzy Optimization of Single Period Model with Special Reference . . .
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On Intuitionistic Fuzzy Volterra Spaces S. Soundararajan† U. Rizwan∗ and Syed Tahir Hussainy
Department of Mathematics, Islamiah College, Vaniyambadi 635 752, India † e-mail: [email protected] ∗ e-mail: [email protected]
Abstract In this paper, the concept of intuitionistic fuzzy Volterra spaces and intitionistic fuzzy weakly Volterra spaces are introduced and characterizations of intuitionistic fuzzy Volterra spaces and intuitionistic fuzzy weakly Volterra spaces are studied. Several examples are given to illustrate the concepts introduced in this paper.
AMS Subject Classification: 54A40 Keywords: Intitionistic Fuzzy nowhere dense set, intuitionistic fuzzy first category, intuitionistic fuzzy second category intuitionistic fuzzy baire, intuitionistic fuzzy Volterra and intuitionistic fuzzy weakly Volterra spaces.
1. Introduction
The fuzzy concept has invaded almost all branches of mathematics ever since the introduction of fuzzy set by L.A. Zadeh [15]. The theory of fuzzy topological spaces was introduced and developed by Chang [6]. The idea of Intitionistic fuzzy set was first published by Atanassov [1]. Later, this concept was generalised to Intitionistic L-fuzzy set by Atanassov and Stoneva. The concept of volterra spaces have been studied extensively in classical topology in [8], [9], [11] and [12]. The concept of fuzzy Volterra space is introduces and studied by Thangaraj and Soundararajan [14]. In this paper, we introduce the concept of Intitionistic fuzzy Volterra spaces and Intitionistic fuzzy weakly Volterra spaces. We discuss several characterizations of those spaces and several examples are given to illustrate the concepts introduced in this paper.
2. Preliminaries
Definition 2.1 [3] Let X be a non-empty set. An Intuitionistic Fuzzy Set (IFS) A in
X is defined as an object of the form A = {hx, µA(x), νA(x)i : x ∈ X}, where 728 On Intiutionistic Fuzzy Volterra Spaces
µA(x): X → [0, 1] and νA(x): X → [0, 1] denote the membership and non- membership functions of A respectively, and 0 ≤ µA(x) + νA(x) ≤ 1, for each x ∈ X.
Definition 2.2 [3] Let A and B be two IFSs of the non-empty set X such that
A = {hx, µA(x), νA(x)i : x ∈ X},
B = {hx, µB(x), νB(x)i : x ∈ X}.
We define the following basic operations on A and B.
(i) A ⊆ B iff µA(x) ≤ µB(x) and νA(x) ≥ νB(x), ∀x ∈ X
(ii) A ⊇ B iff µA(x) ≥ µB(x) and νA(x) ≤ νB(x), ∀x ∈ X
(iii) A = B iff µA(x) = µB(x) and νA(x) = νB(x), ∀x ∈ X
(iv) A ∪ B = {hx, µA(x) ∨ µB(x), νA(x) ∧ νB(x)i : x ∈ X}
(v) A ∩ B = {hx, µA(x) ∧ µB(x), νA(x) ∨ νB(x)i : x ∈ X}
c (vi) A = {hx, νA(x), µA(x)i : x ∈ X}.
Definition 2.3 [7] An Intuitionistic fuzzy topology (IFT) on X is a family T of IFSs in X satisfying the following axioms.
(i) 0, 1 ∈ T
(ii) G1 ∩ G2 ∈ T, for any G1,G2 ∈ T
(iii) ∪Gi ∈ T for any family {Gi/i ∈ J} ⊆ T. In this case, the pair (X,T ) is called an Intuitionistic fuzzy topological space (IFTS) and any IFS in T is known as Intuitionistic fuzzy open set (IFOS) in X. The complement Ac of an IFOS A in an IFTS (X,T ) is called an Intuitionistic fuzzy colsed set (IFCS) in X.
Definition 2.4 [7] Let (X,T ) be an IFTS and A = hX, µA, νAi be an IFS in X. Then the Intuitionistic fuzzy interior and an Intuitionistic fuzzy closure are defined by
int(A) = ∪ {G/G is an IFOS in X and G ⊆ A} , cl(A) = ∩ {K/K is an IFCS in X and A ⊆ K} , S. Soundararajan, U. Rizwan and Syed Tahir Hussainy 729
Proposition 2.5 [7] Let (X,T ) be any Intuitionistic fuzzy topological space. Let A be an IFS in (X,T ). Then
(i) 1 − IF cl(A) = IF int(1 − A), (ii) 1 − IF int(A) = IF cl(1 − A).
Definition 2.6 [10] An Intuitionistic Fuzzy set A in an Intuitionistic fuzzy topological space(X,T ) is called Intuitionistic fuzzy dense if there exists no Intuitionistic fuzzy closed set B in (X,T ) such that A ⊂ B ⊂ 1.
3. Intuitionistic Fuzzy Volterra Spaces
Definition 3.1 An Intuitionistic fuzzy set A in an Intuitionistic fuzzy topological space ∞ (X,T ) is called an Intuitionistic fuzzy Gδ set in (X,T ) if A = ∩i=1Ai where Ai ∈ T, ∀i
Definition 3.2 An Intuitionistic fuzzy set A in an Intuitionistic fuzzy topological spaces ∞ (X,T ) is called an Intuitionistic fuzzy Fσ set in (X,T )) if A = ∪i=1 where 1 −
Ai ∈ T, ∀i.
Lemma 3.3 A is an Intuitionistic fuzzy Gδ set in an Intuitionistic fuzzy topological spaces (X,T ) if and only if 1 − A an Intuitionistic fuzzy Fσ set in (X,T ).
Definition 3.4 Let (X,T ) be an Intuitionistic fuzzy topological space. Then (X,T ) N is called an Intuitionistic fuzzy Volterra space if, IF cl(∩i=1Ai) = 1, where Ai’s are
Intuitionistic fuzzy dense and Intuitionistic fuzzy Gδ sets in (X,T ).
Example 3.1 Let X = {z, b, c}. Define the Intuitionistic fuzzy sets A, B A ∩ B and A ∪ B as follows a b c a b c A = x, , , , , , 0.8 0.6 0.7 0.2 0.4 0.3 a b c a b c B = x, , , , , , 0.6 0.9 0.8 0.4 0.1 0.2 a b c a b c A ∩ B = x, , , , , , 0.6 0.6 0.7 0.4 0.4 0.3 a b c a b c A ∪ B = x, , , , , , 0.8 0.9 0.8 0.2 0.1 0.2 730 On Intiutionistic Fuzzy Volterra Spaces
Clearly, T = {0, 1, A, B, A ∪ B,A ∩ B} is an Intuitionistic fuzzy topology on X. Thus (X,T ) is an Intuitionistic fuzzy topological space.
Let G = {A ∩ B ∩ (A ∪ B)} , H = {A ∩ B ∩ (A ∩ B)} , I = {A ∩ B ∩ (A ∩ B) ∩ (A ∪ B)} , where, G, H and I are Intuitionistic fuzzy Gδ sets in (X,T ). Also, we have IF cl(G) = 1, IF cl(H) = 1, IF cl(I) = 1. Also we have IF cl(G ∩ H ∩ I) = 1. Therefore, (X,T ) is an Intuitionistic fuzzy Volterra space.
Example 3.2 Let X = {a, b, c} let the Intuitionistic fuzzy sets are defined by a b c a b c A = x, , , , , , 0.4 0.5 0.5 0.6 0.5 0.5 a b c a b c B = x, , , , , , 0.6 0.4 0.5 0.4 0.6 0.5 a b c a b c A ∩ B = x, , , , , , 0.4 0.4 0.5 0.6 0.6 0.5 a b c a b c A ∪ B = x, , , , , , 0.6 0.5 0.5 0.4 0.5 0.5 Clearly, T = {A, B, A∪B,A∩B} is an Intuitionistic fuzzy topological on X. Therefore (X,T ) is an Intuitionistic fuzzy topological space. But there is no Intuitionistic fuzzy dense Gδ set in (X,T ) therefore (X,T ) is not an Intuitionistic fuzzy Volterra space.
Proposition 3.5 Let (X,T ) be an Intuitionistic fuzzy topological space. If N IF int ∪i=1A1 = 0,Ai’s are Intuitionistic fuzzy nowhere dense and Intuitionistic fuzzy Fσ sets in (X,T ), then (X,T ) is an Intuitionistic fuzzy Volterra space.