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Mathematical Applications of Queueing Theory in Call Centers

V.S. Selvi and M. Nishanthi

Abstract: Queueing theory has a wide range of applications to real world problems. In this paper, we present the concept and work

culture in Call centers and summarize some results. We also present the performance, characterization and properties. Finally, we

mention some applications.

Index Terms - Queues, Call Center, Poisson Process, Non-homogeneous Poisson process.

—————————— ——————————

1. INTRODUCTION

call center is a centralized office used for the

purpose of receiving and transmitting a large

volume of requests by telephone. A call center is

operated by a company to administer incoming

product support or information inquiries from

consumers. Outgoing calls for telemarketing,

clientele, product services, and debt collection are

also made. In addition to a call center, collective

handling of letters, faxes, live chat, and e-mails at

one location is known as a contact center.

A call center is often operated through an

extensive open workspace for call center agents,

with work stations that include a computer for

each agent, a telephone set/headset connected to a

telecom switch, and one or more supervisor

stations. It can be independently operated or

networked with additional centers, often linked to

a corporate computer network, including

mainframes, microcomputers and LANs.

Increasingly, the voice and data pathways into the

center are linked through a set of new

technologies called computer telephony

integration (CTI).

————————————————

V. S. Selvi is currently serving in the Department of Mathematics, Theivanai Ammal College for Women, Viluppuram,, India

M. Nishanthi is currently pursuing. Phil. degree program in Mathematics, Theivanai Ammal College for Women, Viluppuram, India

2. VARITIES

Some variations of call center models are listed below:

Contact center – Supports interaction with customers over a variety of media, including but not necessarily limited to telephony, e-mail and internet chat.

Inbound call center - Exclusively or predominantly handles inbound calls (calls initiated by the customer).

Outbound call center - One in which call

center agents make outbound calls to customers or sales leads.

Blended call center - Combining automatic call distribution for incoming calls with predictive dialling for outbound calls, it makes more efficient use of agent time as each type of agent (inbound or outbound) can handle the overflow of the other.

Telephone answering service - A more personalized version of the call center, where agents get to know more about their customers and their callers; and therefore look after calls just as if based in their customers office.

3. CRITICISM AND PERFORMANCE

Criticisms of call centers generally follow a

number of common themes, from both callers and

call center staff. From callers, common criticisms

include:

Operators working from a script

A

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Non-expert operators (call screening)

Incompetent or untrained operators

incapable of processing customers'

requests effectively.

Obsequious behavior by operators (e.g.,

relentless use of Sir, Ma'am and I'd be

happy to assist you)

Overseas location, with language and

accent problems

Touch tone menu systems and automated

queuing systems

Excessive waiting times to be connected to

an operator

Complaints that departments of companies

do not engage in communication with one

another

Deceit over location of call center (such as

allocating overseas workers false English

names)

Requiring the caller to repeat the same

information multiple times

Common criticisms from staff include:

Close scrutiny by management (e.g.

frequent random call monitoring)

Low compensation (pay and bonuses)

Restrictive working practices (some

operators are required to follow a pre-

written script)

High stress: a common problem associated

with front-end jobs where employees deal

directly with customers

Repetitive job task

Poor working conditions (e.g. poor

facility, poor maintenance and cleaning,

cramped working conditions, management

interference, lack of privacy and noisy)

Impaired vision and hearing problems

Rude and abusive customers

4. PROISSON PROCESS

In probability theory, a Poisson process is a

stochastic process which counts the number of

events and the time that these events occur in a

given time interval. The time between each pair of

consecutive events has an exponential distribution

with parameter 𝛌 and each of these inter-arrival

times is assumed to be independent of other inter-

arrival times. The process is named after the

French mathematician Siméon-Denis Poisson and

is a good model of radioactive decay, telephone

calls and requests for a particular document on a

web server, among many other phenomena.

The Poisson process is a continuous-time process;

the sum of a Bernoulli process can be thought of

as its discrete-time counterpart. A Poisson process

is a pure-birth process, the simplest example of a

birth-death process. It is also a point process on

the real half-line.

Definition. The basic form of Poisson process,

often referred to as the Poisson process, is a

continuous-time counting process {N(t), t ≥ 0}

that possesses the following properties:

N(0) = 0

Independent increments (the numbers of

occurrences counted in disjoint intervals

are independent from each other)

Stationary increments (the probability

distribution of the number of occurrences

counted in any time interval only depends

on the length of the interval)

No counted occurrences are simultaneous.

Consequences of this definition include:

The probability distribution of N(t) is a

Poisson distribution.

The probability distribution of the waiting

time until the next occurrence is an

exponential distribution.

The occurrences are distributed uniformly

on any interval of time. (Note that N(t), the

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total number of occurrences, has a Poisson

distribution over (0, t ], whereas the

location of an individual occurrence on t ∈

(a, b] is uniform.)

Other types of Poisson process are described

below.

The homogeneous Poisson process is one of the

most well known Lévy processes. This process is

characterized by a rate parameter λ, also known as

intensity, such that the number of

events in time interval (t, t + τ] follows a Poisson

distribution with associated parameter λτ. This

relation is given as

P [ (N(t+τ) - N(t)) = k] = 𝑒−𝜆𝜏 (𝜆𝜏)𝑘

𝑘! k=0,1,…,

where N(t+τ) - N(t) = k is the number of events in

time interval (t, t + τ].

Just as a Poisson random variable is characterized

by its scalar parameter λ, a homogeneous Poisson

process is characterized by its rate parameter λ,

which is the expected number of events or arrivals

that occur per unit time.

N(t) is a sample homogeneous Poisson process,

not to be confused with a density or distribution

function.

In general, the rate parameter may change over

time; such a process is called a non-homogeneous

Poisson process or inhomogeneous Poisson

process. In this case, the generalized rate function

is given as λ(t). Now the expected number of

events between time a and time b is

𝜆𝑎 ,𝑏 = 𝜆(𝑡)𝑏

𝑎𝑑𝑡

Thus, the number of arrivals in the time interval

(a, b], given as N(b) − N(a), follows a Poisson

distribution with associated parameter λa,b

P [ (N(b) - N(a)) = k] = 𝑒−𝜆𝑎 ,𝑏 (𝜆𝑎 ,𝑏 )𝑘

𝑘! k=0,1,…

A homogeneous Poisson process may be viewed

as a special case when λ(t) = λ, a constant rate.

An important variation on the (notionally time-

based) Poisson process is the spatial Poisson

process. In the case of a one-dimension space (a

line) the theory differs from that of a time-based

Poisson process only in the interpretation of the

index variable. For higher dimension spaces,

where the index variable (now x) is in some vector

space V (e.g. R2 or R

3), a spatial Poisson process

can be defined by the requirement that the random

variables defined as the counts of the number of

"events" inside each of a number of non-

overlapping finite sub-regions of V should each

have a Poisson distribution and should be

independent of each other.

A further variation on the Poisson process, the

space-time Poisson process, allows for separately

distinguished space and time variables. Even

though this can theoretically be treated as a pure

spatial process by treating "time" as just another

component of a vector space, it is convenient in

most applications to treat space and time

separately, both for modeling purposes in practical

applications and because of the types of properties

of such processes that it is interesting to study.

In comparison to a time-based inhomogeneous

Poisson process, the extension to a space-time

Poisson process can introduce a spatial

dependence into the rate function, such that it is

defined as 𝛌(x,t), where x ∈ 𝑉for some vector

space V (e.g. R2 or R

3). However, a space-time

Poisson process may have a rate function which is

constant with respect to either x or t or both. For

any set S ⊂ 𝑉 (e.g. a spatial region) with finite

measure µ(S) , the number of events occurring

inside this region can be modeled as a Poisson

process with associated rate function λS(t) such

that

𝜆 𝑆(𝑡) = 𝜆(𝑥, 𝑡)

𝑆𝑑µ(𝑥)

5. SEPARABLE SPACE-TIME

PROCESSES

In the special case that this generalized rate

function is a separable function of time and space,

we have:

𝛌(x,t) = f(x) 𝛌(t)

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for some function f(x) . Without loss of generality,

let

𝑓 𝑥 𝑑𝜇(𝑥)

𝑉

= 1

(If this is not the case, λ(t) can be scaled

appropriately.) Now, f(x) represents the spatial

probability density function of these random

events in the following sense. The act of sampling

this spatial Poisson process is equivalent to

sampling a Poisson process with rate function λ(t),

and associating with each event a random vector

X sampled from the probability density function

f(x) . A similar result can be shown for the

general (non-separable) case.

6. CHARACTERIZATION

In its most general form, the only two conditions

for a counting process to be a Poisson process are

Orderliness: which roughly means lim∆𝑡→0

𝑃 𝑁 𝑡 + ∆𝑡 − 𝑁 𝑡 > 1 𝑁 𝑡 + ∆𝑡

− 𝑁 𝑡 ≥ 1) = 0

which implies that arrivals don't occur

simultaneously (but this is actually a

mathematically stronger statement).

Memorylessness (also called evolution

without after-effects): the number of

arrivals occurring in any bounded interval

of time after time t is independent of the

number of arrivals occurring before time t.

These seemingly unrestrictive conditions actually

impose a great deal of structure in the Poisson

process. In particular, they imply that the time

between consecutive events (called

inter-arrival times) are independent random

variables. For the homogeneous Poisson process,

these inter-arrival times are exponentially

distributed with parameter λ (mean 1/λ).

Proof : Let 𝜏1be the first arrival time of the

Poisson process. Its distribution satisfies

Pr[𝜏1 = 𝑥]

= lim𝑑𝑡→0Pr[𝑁𝑥+𝑑𝑡 >0,𝑁𝑥 =0]

𝑑𝑡

= lim𝑑𝑡→01−Pr[𝑁𝑑𝑡 =0]

𝑑𝑡Pr[𝑁𝑥 = 0]

= lim𝑑𝑡→0

1 − (1 − 𝜆𝑑𝑡 + 𝑂(𝑑𝑡2))

𝑑𝑡exp(−𝜆𝑥)

= 𝛌 exp(-𝛌x)

Also, the memorylessness property entails that the

number of events in any time interval is

independent of the number of events in any other

interval that is disjoint from it. This latter property

is known as the independent increments property

of the Poisson process.

7. PROPERTIES

As defined above, the stochastic process {N(t)} is

a Markov process, or more specifically, a

continuous-time Markov process.

To illustrate the exponentially distributed inter-

arrival times property, consider a homogeneous

Poisson process N(t) with rate parameter λ, and let

Tk be the time of the kth arrival, for k = 1, 2, 3, ... .

Clearly the number of arrivals before some fixed

time t is less than k if and only if the waiting time

until the kth arrival is more than t. In symbols, the

event [N(t) < k] occurs if and only if the event

[Tk > t] occurs. Consequently the probabilities of

these events are the same:

P (𝑇𝑘 > 𝑡) = P (N(t) < 𝑘).

In particular, consider the waiting time until the

first arrival. Clearly that time is more than tif and

only if the number of arrivals before time t is 0.

Combining this latter property with the above

probability distribution for the number of

homogeneous Poisson process events in a fixed

interval gives

P (𝑇1 > 𝑡) = P(N(t) =0) = P[(N(t) – N(0)) = 0]

= 𝑒−𝜆𝑡 (𝜆𝑡 )0

0!= 𝑒−𝜆𝑡 .

Consequently, the waiting time until the first

arrival T1 has an exponential distribution, and is

thus memoryless. One can similarly show that the

other inter-arrival times Tk − Tk−1 share the same

distribution. Hence, they are independent,

identically distributed (i.i.d.) random variables

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with parameter λ > 0; and expected value 1/λ. For

example, if the average rate of arrivals is 5 per

minute, then the average waiting time between

arrivals is 1 in 5 minutes.

8. APPLICATIONS

The classic example of phenomena well modeled

by a Poisson process is deaths due to horse kick in

the Prussian army, as shown by Ladislaus-

Bortkiewicz in 1898. The following examples are

also well-modeled by the Poisson process:

Requests for telephone calls at a

switchboard.

Goals scored in a soccer match.

Requests for individual documents on a

web server.

Particle emissions due to radioactive decay

by an unstable substance. In this case the

Poisson process is non-homogeneous in a

predictable manner - the emission rate

declines as particles are emitted.

In queueing theory, the times of customer/job

arrivals at queues are often assumed to be a

Poisson process.

Sample Path of a Poisson process N(t)

REFERENCES

[1] A. Panico, Queueing Theory, Prentice-

Hall, Inc., Englewood cliffs, N.J.

[2] Churchman, C.West,Russel L.Ackott,

E.Leonard Arnoff, Introduction to

Operation Reseach, .New York; John Wiley

and sons,Inc,1963.

[3] Donald Gross Carl M.Harris, Fundamentals

of Queueing Theory, Third Edition, Wiley

India, 1998.

[4] G.Srinivasan, Operations Research

Principles and Applications, PHI Learning

Private Limited. New Delhi, 2007.

[5] E. Page, Introduction of Queuing Theory in

operation Research, 1972.

[6] B. Cooper, Introduction to Queuing Theory,

1972.

[7] Jean Walrand, An Introduction to Queuing

Networks, 1988.

[8] H.M. Wagner, Principles of Operation

Research, 1972.

[9] G. F. Newell, Applications of Queuing

Theory, Second Edition, 1982.

[10] L. Venkata Subramaniam, Call Centers of

the Future, 2008 .

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On Generalized Preregular Closed Sets In Supra Topological Spaces

Vidhya Menon

Abstract - In this paper, a new class of sets called supra generalized preregular closed sets in supra topological spaces is

introduced and its properties are studied. Further the notion of supra preregular T1/2 space and supra generalized preregular

continuity are introduced.

Index Terms - Supra preclosed set, Supra generalized preclosed set, Supra generalized preregular closed set, Supra

generalized preregular open set, supra generalized preregular continuous function.

—————————— ——————————

1. INTRODUCTION

N 1983, Mashhour et al [6] introduced the

concept of supra topological space and studied S

– continuous maps and S*- continuous maps. The

study on supra topological space was further

extended and in 2008, Devi et al [3] introduced

and studied a class of sets called supra α – open

sets and a class of maps called Sα – continuous

maps between topological spaces. In 2010, Sayed

and Noiri [9] introduced supra b – open sets and

supra b - continuity on topological spaces. In

2011, Ravi et al [8] introduced supra sg - closed

sets and supra gs - closed sets. Arockiarani and

M.Trinita Pricilla [1] introduced supra generalized

b-regular closed sets in 2012. In 1997, Gnanambal

[4] introduced generalized preregular closed sets

in general topological spaces. In this paper we

define and study the properties of supra

generalized preregular closed sets (briefly gprµ -

closed) and their relationship with other classes

of sets in supra topological spaces.

2. PRELIMINARIES

Definition 2.1 [6] A subcollection µ ⊂ P(X) is

called a supra topology on X if X ϵ µ and µ is

closed under arbitrary union. (X,µ) is called a

supra topological space.

———————————

Vidhya Menon is working in the Department of Mathematics, CMS

College of Science and Commerce, Coimbatore, India. E-mail:

vidhyamenon77@gmail.com

The elements of µ are said to be supra open in (X,

µ) and the complement of a supra open set is

called supra closed set. The supra closure of a set

A, denoted by clµ(A), is the intersection of supra

closed sets including A . The supra interior of a

set A, denoted by intµ(A), is the union of supra

open sets included in A. The supra topology µ on

X is associated with the topology τ if τ ⊂ µ.

Throughout this paper (X, µ ), (Y, σ) and (Z, η)

(or simply X, Y and Z) denote supra topological

spaces on which no separation axioms are

assumed unless explicitly stated. (X, µ ) will be

replaced by X if there is no chance of confusion.

Definition 2.2 A subset A of a space ( X, µ ) is

called

i) supra preclosed [8] if clµ(int

µ(A)) ⊆ A .

ii) supra α- closed [8] if clµ(int

µ(cl

µ(A))) ⊆ A.

iii) supra semi – preclosed [8] if

intµ(cl

µ(int

µ(A))) ⊆ A.

iv) supra regular closed [1] if A = clµ(int

µ(A))

The complements of the above mentioned closed

sets are called their respective open sets.

Definition 2.3 Let A be a subset of X.

i) The supra pre-closure of a set A is defined as

pclµ(A) = ∩ (B : B is a supra preclosed

set and A ⊆ B)

ii) The supra pre-interior of a set A is defined as

pintµ (A) = ∪ (B : B is a supra preopen

set and B ⊆ A)

I

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Definition 2.4 A subset A of a space (X,µ) is

called

i) supra generalized closed (briefly gµ - closed)

[1] if clµ(A) ⊆ U whenever A ⊆ U and U is

supra open in X .

ii) supra generalized α-closed (briefly gαµ -

closed) [1] if αclµ(A) ⊆ U whenever A ⊆ U

and U is supra α –open in X.

iii) supra α- generalized closed (briefly αgµ -

closed) [1] if αclµ(A) ⊆ U whenever A ⊆ U

and U is supra open in X.

iv) supra regular generalized closed (briefly rgµ -

closed) [7] if clµ(A) ⊆ U whenever A ⊆ U

and U is supra regular-open.

v) supra generalized preclosed (briefly gpµ -

closed) if pclµ(A) ⊆ U whenever A ⊆ U and

U is supra open in X.

vi) supra generalized semi - preclosed (briefly

gspµ - closed) if spcl

µ(A) ⊆ U whenever A

⊆ U and U is supra open in X.

3. SUPRA GENERALIZED PREREGULAR

CLOSED SETS Definition 3.1 A subset A of (X,µ) is called supra

generalized preregular closed (briefly gprµ -

closed) if pclµ(A) ⊆ U whenever A ⊆ U and U

is supra regular open in (X,µ).

Theorem 3.2

i) Every rgµ - closed set is gpr

µ - closed .

ii) Every gµ - closed set is gpr

µ - closed .

iii) Every gpµ - closed set is gpr

µ – closed.

iv) Every αgµ - closed set is gpr

µ – closed.

Proof . Obvious.

However the converse of the above said theorems

are not true.

Example 3.3 Let X = {a, b, c, d} and

µ = { ɸ, X, {a},{a , c},{b , c},{a, b, c}}.

A = {b} is gprµ - closed but not rg

µ - closed.

Example 3.4

Let X = {a, b, c}.

i) If µ = { ɸ, X, {b , c}}, A = {b , c} is gprµ -

closed but not gµ - closed.

ii) If µ = { ɸ, X, {a},{a , b},{a , c}}, A = {a} is

gprµ - closed but not gp

µ - closed.

iii) If µ = { ɸ, X, {c},{a , c}} , A = {c} is gprµ -

closed but not αgµ - closed.

Remark 3.5 gprµ - closed sets and gsp

µ – closed

sets are independent of each other.

Example 3.6 Let X = {a, b, c} and

µ ={ ɸ, X, {a},{a , b},{a , c}}.

Let A = {a}. A is gprµ - closed but not gsp

µ –

closed.

Example 3.7

Let X = {a, b, c, d, e} and

µ = { ɸ, X, {a , b},{c , d},{a, b, c, d}}.

Let A = {a , b}. A is gspµ – closed but not gpr

µ –

closed.

Theorem 3.8 If A is supra regular open and gprµ

- closed, then A is supra preclosed.

Proof. If A is supra regular open and gprµ - closed

then pclµ

(A) ⊂ A. Also A ⊂ pclµ

(A) for any set

A .Thus A is supra preclosed.

Remark 3.9 Union of two gprµ - closed sets need

not be gprµ - closed.

Example 3.10 Let X = {a, b, c, d} and

µ = { ɸ, X, {a},{a , c},{b, c},{a, b, c}}.

Let A = {b} and B = {c}. A and B are gprµ -

closed sets but A ∪ B is not gprµ - closed.

Remark 3.11 Intersection of two gprµ - closed

sets need not be gprµ - closed.

Example 3.12

Let X = {a, b, c} and

µ = { ɸ, X, {a}, {b},{a , b}}.

Let A = {a , b} and B = {a , c}. A and B are gprµ

- closed in (X, µ ). But A ∩ B is not gprµ - closed

in (X, µ).

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Theorem 3.13 Let A be gprµ - closed in (X, µ).

Then pclµ

(A) – A does not contain any non empty

supra regular closed set.

Proof. Let B be a supra regular closed set such

that B ⊆ pclµ

(A) – A. Thus B ⊆ pclµ(A) and

B⊆Ac. Then B ⊆ X– A implies A ⊆ X – B.

Since A is gprµ - closed and X – B is supra regular

open , pclµ

(A) ⊆ X – B. That is B ⊆ X - pclµ

(A).

Hence B ⊆ pclµ

(A) ∩ (X - pclµ

(A)) = ɸ. This

shows B = ɸ.

The converse of the above theorem is not true .

Example 3.14

Let X = {a, b, c ,d,} and

µ = { ɸ, X, {a},{a, c},{b, c},{a, b, c}}. Let A

= {b, c}, then pclµ

(A) – A = {d}, does not

contain any non empty supra regular closed set

but A is not gprµ - closed. .

Corollary 3.15 Let A be gprµ - closed in (X, µ).

Then A is supra preclosed iff pclµ

(A) – A is

supra regular closed.

Proof. Let A be supra preclosed. Then pclµ

(A) =

A implies pclµ

(A) – A = ɸ which is supra regular

closed.

Conversely suppose pclµ

(A) – A is supra regular

closed. Then pclµ

(A) – A = ɸ and A is gprµ -

closed . This shows pclµ

(A) = A or A is supra

preclosed.

Definition 3.16 Let (X, µ) be a supra topological

space, A ⊂ X and x ϵ X . x is said to be a supra

limit point of A iff every

supra open set containing x contains a point of A

different from x. The supra derived set of A

denoted by 𝐷µ [A] is the set of all supra limit

points of A.

Definition 3.17 Let (X, µ) be a supra topological

space, A ⊂ X and x ϵ X. x is said to be a supra

pre-limit point of A iff every supra preopen set

containing x contains a point of A different from

x. The set of all supra pre- limit points of A is said

to be the supra pre-derived set of A denoted by

𝐷𝑃 µ

[A] .

Theorem 3.18 Let A and B be gprµ - closed

sets in (X, µ ) such that 𝐷µ [A] ⊆ 𝐷𝑃

µ [A] and

𝐷µ [B] ⊆ 𝐷𝑃

µ [B]. Then A ∪ B is gpr

µ - closed

set.

Proof. Let U be supra regular open set such that

(A ∪ B) ⊆ U. Then A ⊆ U and B ⊆ U. Since A

and B are gprµ - closed sets pcl

µ (A) ⊆ U and pcl

µ

(B) ⊆ U. For any set E ⊂ (X, µ ), 𝐷𝑃µ

[E] ⊂

𝐷µ [E]. Hence 𝐷𝑃 µ

[A] = 𝐷µ [A] and 𝐷𝑃

µ [B] =

𝐷µ [B]. That is

clµ

(A) = pclµ

(A) and clµ

(B) = pclµ

(B).

clµ

(A∪ B) = clµ

(A) ∪ clµ

(B) = pclµ

(A) ∪ pclµ

(B) ⊆ U .

But pclµ

(A∪B) ⊆ clµ

(A∪ B). Therefore pclµ

(A∪B) ⊆ U. Thus A∪B is gprµ - closed.

Theorem 3.19 If A is gprµ - closed and A ⊆ B

⊆ pclµ

(A), then B is gprµ

- closed.

Proof. Let B ⊆ U where U is a supra regular open

set. Now A ⊆ B implies A ⊆ U . Since A is

gprµ - closed, pcl

µ(A) ⊆ U. Given B ⊆ pcl

µ(A)

implies pclµ(B) ⊆ pcl

µ(A).Thus pcl

µ(B) ⊆ U.

Therefore B is gprµ - closed.

Definition 3.20 A set A ⊂ X is called gµpr - open

set if and only if its complement is gprµ - closed.

Remark 3.21 pclµ(X – A) = X – pint

µ( A ).

Theorem 3.22 A ⊂ X is gprµ - open set if and

only if F ⊂ pintµ(A) whenever F is supra

regular closed and F ⊂A.

Proof. Let A be gprµ - open. Let F be supra

regular closed and F ⊂ A. This implies X – A ⊂

X – F . So X – F is is supra regular open. Since A

is gprµ - open, X – A is gpr

µ - closed. This

implies pclµ

(X – A) ⊂ X – F . Thus X –

pintµ(A) ⊂ X – F by Remark 3.21.Therefore F

⊂ pintµ(A).

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Conversely suppose F is supra regular closed and

F ⊂ A. This implies F ⊂ pintµ(A). Let X – A ⊂

U where U is supra regular open. Then X – U ⊂ A

where X – U is supra regular closed. By

hypothesis X– U ⊂ pintµ(A). This implies X –

pintµ(A ) ⊂ U. Thus pcl

µ(X – A) ⊂ U by remark

3.21. Therefore X – A is gprµ - closed. Hence A is

gprµ - open.

Theorem 3.23 If pintµ(A) ⊂ B ⊂ A and A is gpr

µ-

open then B is gprµ - open.

Proof. Given pintµ(A) ⊂ B ⊂ A, implies X – A

⊂ X – B ⊂ X – pintµ(A). That is X – A ⊂ X – B

⊂ pclµ(X - A). Since A is gpr

µ - open, X - A is

gprµ - closed. Then by theorem 3.19 X – B is

gprµ - closed. Thus B is gpr

µ - open.

4. gprµ -CONTINUOUS AND

gprµ – IRRESOLUTE FUNCTIONS

Definition 4.1 A function 𝑓 : (X, µ) → (Y,σ) is

called a gprµ - continuous if 𝑓−1(V) is gpr

µ -

closed in (X, µ) for every supra closed set V of

(Y, σ).

Definition 4.2 A function 𝑓: (X, µ) → (Y,σ) is

called a gprµ - irresolute if 𝑓−1(V) is gpr

µ -closed

in (X, µ) for every gprµ - closed set V in (Y, σ).

Example 4.3

Let X = {a, b, c, d, e} and µ = { ɸ, X,{a, b, c}}

and σ = { ɸ, X, {a, b, c, d}}. Define a function

𝑓 : (X, µ) → (X, σ) by 𝑓(a) = e, 𝑓(b) = d, 𝑓(c) =

c, 𝑓(d) = a, 𝑓(e) = b. Since for every supra closed

set V of (X, σ) , 𝑓−1(V) is gprµ - closed in (X,

µ). Therefore 𝑓 is gprµ - continuous. Also the

inverse image of every gprµ- closed set is gpr

µ -

closed under 𝑓. Hence 𝑓 is gprµ - irresolute.

Theorem 4.4 Every gprµ - irresolute function is

gprµ - continuous.

Proof. It is obvious.

Converse is not true.

Example 4.5 Consider X = {a, b, c}, µ = { ɸ, X, {a}, {c},{a ,

c}} and σ = { ɸ, X, {a}}. Define a function 𝑓 :

(X, µ) → (X, σ) by 𝑓(a) = b, 𝑓(b) = c, 𝑓(c) =

a. Then 𝑓 is gprµ - continuous but not gpr

µ -

irresolute.

Theorem 4.6

i) Let 𝑓 : (X, µ) → (Y, σ) be rgµ- continuous.

Then 𝑓 is gprµ - continuous.

ii) Let 𝑓 : (X, µ) → (Y, σ) be αgµ- continuous.

Then 𝑓 is gprµ - continuous

Proof.

i) Let V be supra closed in (Y, σ).Then 𝑓−1(V)

is rgµ - closed in (X, µ) as 𝑓 is rg

µ- continuous.

By theorem 3.2( i) 𝑓−1(V) is gprµ - closed.

Hence 𝑓 is gprµ - continuous.

ii) Let U be supra closed in (Y, σ) .Then 𝑓−1(U)

is αgµ

- closed in (X, µ) as 𝑓 is αgµ -

continuous. Since every αgµ

- closed set is

gprµ – closed, we have 𝑓−1(U) is gpr

µ -

closed. Hence 𝑓 is gprµ -continuous.

Converse of the above theorem does not hold.

Example 4.7

Let X = {a, b, c, d};

µ = { ɸ, X, {a},{a , c},{b , c},{a, b, c}} and

σ = { ɸ, X, {a, b, d}}.

Define a function 𝑓 : (X, µ) → (X, σ) by 𝑓(a) =

a, 𝑓(b) = b, 𝑓(c) = c, 𝑓(d) = d. Here 𝑓 is gprµ -

continuous but is not rgµ-continuous .

Example 4.8 Let X = {a, b, c } ,

µ = { ɸ, X, {a , c}, {b , c}} and

σ = { ɸ, X, {a}}.

Let g : (X, µ) → (X, σ ) be defined by g(a) = a,

g(b) = b, g(c) = c. Then g is gprµ - continuous in

(X, µ). But it is not αgµ - continuous .

Theorem 4.9 Let 𝑓 : (X, µ) → (Y, σ) be supra

regular irresolute and supra preclosed. Then

for every gprµ - closed set A of (X , µ) , 𝑓(A) is

gprµ - closed in (Y, σ).

Proof. Let A be a gprµ - closed set of (X,

µ). Let 𝑓(A) ⊂ U where U is supra regular open

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in Y. Then A ⊂ 𝑓−1(U). Since f is supra regular

irresolute and A is gprµ - closed, pcl

µ (A) ⊂

𝑓−1(U). This implies 𝑓(pclµ

(A)) ⊂ U. Thus pclµ

(𝑓(A)) ⊂ pclµ

(𝑓(pclµ (A))) = 𝑓(pcl

µ (A)) ⊂ U.

Hence 𝑓(A) is gprµ-closed in (Y, σ).

Remark 4.10 The composition of two gprµ -

continuous functions need not be gprµ –

continuous.

Example 4.11 Let X = {a, b, c, d},

µ = { ɸ, X,{a},{a, b},{a, b, d},{b, d}},

σ = { ɸ, X ,{b},{b ,c , d}} and

η = { ɸ, X ,{a, b, c}}.

Define a function

𝑓 : (X, µ) → (X, σ)

by 𝑓(a) = b, 𝑓(b) = c, 𝑓(c) = d, 𝑓(d) = a

Define a function

𝑔 : (X, σ) → (X, η )

by 𝑔(a) = a, 𝑔(b) = d, 𝑔(c) = c, 𝑔(d) = b.

Then 𝑓 and 𝑔 are gprµ - continuous. {d} is supra

closed in (X, η ). (𝑔𝑜𝑓)−1{d} = {a} is not gprµ -

closed in (X, µ). Hence 𝑔o𝑓 is not gprµ -

continuous.

Theorem 4.12 Let 𝑓 : (X, µ) → (Y,σ) and

𝑔 : (Y, σ) → (Z, η )

be any two functions. Then

i) 𝑔𝑜𝑓 is gprµ - continuous , if 𝑔 is supra

continuous and 𝑓 is gprµ - continuous.

ii) 𝑔𝑜𝑓 is gprµ - irresolute , if 𝑔 is gpr

µ -

irresolute and 𝑓 is gprµ - irresolute.

iii) 𝑔𝑜𝑓 is gprµ - continuous , if 𝑔 is gpr

µ -

continuous and 𝑓 is gprµ - irresolute.

Proof.

i) Let V be supra closed in (Z, η ). Then

𝑔−1(V) is supra closed in (Y, σ). Since 𝑔 is

supra continuous , gprµ - continuity of 𝑓

implies 𝑓−1(𝑔−1(V)) is gprµ - closed in (X,

µ) .That is (𝑔𝑜𝑓)−1(V) is gprµ - closed in

(X, µ) . Hence 𝑔𝑜𝑓 is gprµ - continuous.

ii) Let V be gprµ - closed in (Z, η ). Since 𝑔 is

gprµ - irresolute, 𝑔−1(V) is gpr

µ - closed in

(Y, σ). As 𝑓 is gprµ - irresolute 𝑓−1(𝑔−1 (V))

= (𝑔𝑜𝑓)−1(V) is gprµ

- closed in (X, µ ).

Hence 𝑔𝑜𝑓 is gprµ

- irresolute.

iii) Let V be closed in (Z, η ). Since 𝑔 is gprµ -

continuous, 𝑔−1(V) is gprµ - closed in (Y, σ).

As 𝑓 is gprµ - irresolute 𝑓−1(𝑔−1 (V)) =

(𝑔𝑜𝑓)−1 (V) is gprµ - closed in (X, µ). Hence

𝑔𝑜𝑓 is gprµ - continuous.

5. SUPRA PREREGULAR T1/2 SPACES Definition 5.1 A space (X,µ) is called supra

preregular T1/2 space if every gprµ - closed set is

supra preclosed.

Remark 5.2 The notions supra preregular T1/2

and supra T1/2 are independent of each other.

Example 5.3 Let X = {a, b, c, d},

µ = { ɸ, X ,{a,},{b},{c},{a , b},{b , c},{a

, c},{a, b, c}} and σ = { ɸ, X, {a , b}, {c , d}}.

(X, µ) is supra T1/2 but not supra preregular T1/2

whereas (X, σ) is supra preregular T1/2 but not

supra T1/2.

Theorem 5.4 For a supra topological space

(X,µ) the following conditions are equivalent

i) X is a supra preregular T1/2

ii) Every singleton of X is either supra regular

closed or supra preopen.

Proof.

(i) →(ii)

Let x ϵ X and assume that {x} is not supra regular

closed. Then X – {x} is not supra regular open

and X – {x} is trivially gprµ - closed . By (i) it is

supra preclosed and thus {x} is supra preopen.

(ii) → (i)

Let A ⊂ X be gprµ - closed and let x ϵ pcl

µ (A).

We will show that x ϵ A . Consider the following

two cases :

Case 1) The set {x} is supra regular closed. Then,

if x does not belongs to A, there exist a supra

regular closed set in pclµ (A) – A. By theorem

3.13, x ϵ A

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Case 2) The set {x} is supra preopen. Since x ϵ

pclµ (A), then {x} ∩ pcl

µ (A) ≠ ɸ. Thus x ϵ A.

So, in both cases x ϵ A. Thus pclµ (A) ⊂ A or

equivalently A is supra preclosed.

The collection of all supra preopen and supra

generalized preregular open subsets of X is

denoted by POµ(X)

and GPRO

µ (X) .

Theorem 5.5 Let (X,µ) be a supra topological

space. Then

(i) POµ(X)

⊂ GPROµ (X).

(ii) A space (X, µ) is supra preregular T1/2 iff

POµ(X ) = GPRO

µ (X) .

Proof. i) Let A be supra preopen. Then X - A is supra

preclosed and so gprµ - closed. This implies A

is gprµ - open. Hence PO

µ(X) ⊂ GPRO

µ (X).

ii) Necessity

Let (X, µ) be supra preregular T1/2. Let A ϵ

GPROµ (X). Then X- A is gpr

µ -closed. By

hypothesis X – A is supra preclosed and thus

A ϵ POµ(X). Hence GPRO

µ(X) = PO

µ(X).

Sufficiency

Let POµ(X) = GPRO

µ (X). Let A be gpr

µ -

closed. Then X- A is gprµ - open. Hence X

– A ϵ POµ(X). Thus A is supra preclosed

thereby implying (X, µ) is supra preregular

T1/2.

ACKNOWLEDGEMENT

The author is thankful to Dr. Gnanambal Ilango,

Department of Post Graduate and Research

Studies in Mathematics, Govt. Arts College,

Coimbatore, for her kind help in preparing this

paper.

REFERENCES

[1] I. Arockiarani and M. Trinita Pricilla, On

generalized b - regular closed sets in supra

topological spaces, Asian Journal of Current

Engineering and Maths 1, 1(2012) 1- 4.

[2] P.Bhattacharrya and B.K. Lahari, Semi –

generalized closed sets in topology, Indian J.

Math, 29(3)(1987), 357-382.

[3] R.Devi, S.Sampathkumar and M.Caldas, On

supra α - open sets and Sα –continuous

functions, General Mathematics, Vol 16, Nr.

2(2008), 77-84.

[4] Y.Gnanambal, On generalized preregular

closed sets in topological spaces, Indian J.

pure appl.Math . 28(3), (1997), 351 – 360.

[5] N. Levine, Generalized closed sets in

topology, Rend.Circ.Mat.Palermo, (2)

19(1970), 89 – 96.

[6] A.S.Mashhour, A.A.Allam , F.S.Mahamoud

and F.H.Khedr , On supra topological spaces,

Indian J. Pure and Appl.Math . No.4,

14(1983), 502-510.

[7] O.Ravi, G.Ramkumar and M.Kamarajand

M.L Thivagar , Mildly supra normal spaces

and some maps, International Journal of

Advances in Pure and Applied Mathematics,

1(4)(2011).

[8] O.Ravi, G.Ramkumar and M.Kamaraj, On

supra sg – closed sets and supra gs – closed

sets,International Journal of Mathematical

Archive – 2(11), (2011), 2413 – 2419.

[9] O.R . Sayed and Takashi Noiri, On supra b –

open sets and supra b – continuity on

topological spaces, European Journal of Pure

and Applied Mathematics, Vol 3, No.2,

(2010), 295 – 302.

[10] O.R Sayed , Supra pre – open sets and supra

pre – continuity on topological spaces,

Scientific Studies and Research Series

Mathematics and Informatics, 20(2)(2010),

79 – 88.

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Mathematical Applications of Queueing Theory in Traffic Congestion

V. S. Selvi and P. Sathya

Abstract - Queueing theory has a wide range of applications. In this paper, we discuss various problems faced because of traffic. We

also summarize the different procedures to measure traffic in different systems.

Index Terms - Queues, Traffic, Erlang distribution.

—————————— ——————————

1. INTRODUCTION

RAFFIC congestion is a condition on road networks that occurs as use increases, and is characterized by slower speeds, longer trip

times, and increased vehicular queueing. The most common example is the physical use of roads by vehicles. When traffic demand is great, then the interaction between vehicles slows the speed of the traffic stream. This results in some congestion. As demand approaches the capacity of a road (or of the intersections along the road), extreme traffic congestion sets in. When vehicles are fully stopped for periods of time, this is colloquially known as a traffic jam or traffic snarl-up.

Traffic congestion occurs when a volume of traffic or modal split generates demand for space greater than the available road capacity; this point is commonly termed saturation. There are a number of specific circumstances which cause or aggravate congestion; most of them reduce the capacity of a road at a given point or over a certain length, or increase the number of vehicles required for a given volume of people or goods. About half of U.S. traffic congestion is recurring, and is attributed to sheer weight of traffic; most of

————————————————

V. S. Selvi is currently serving in the Department of

Mathematics, Theivanai Ammal College for Women,

Viluppuram,, India

P.Sathya is currently pursuing. Phil. degree program

in Mathematics, Theivanai Ammal College for Women,

Viluppuram, India

the rest is attributed to traffic incidents, road work and weather events.

Traffic research still cannot fully predict under which conditions a "traffic jam" (as opposed to heavy, but smoothly flowing traffic) may suddenly occur. It has been found that individual incidents (such as accidents or even a single car braking heavily in a previously smooth flow) may cause ripple effects (a cascading failure) which then spread out and create a sustained traffic jam when, otherwise, normal flow might have continued for some time longer.

The erlang is a dimensionless unit that is used in telephony as a statistical measure of offered load or carried load on service-providing elements such as telephone circuits or telephone switching equipment. It is named after the Danish telephone engineer A. K. Erlang, the originator of traffic engineering and queueing theory.

2. TRAFFIC MEASUREMENTS IN A

TELEPHONE CIRCUIT

When used to represent carried traffic, a value (which can be a non-integer such as 43.5) followed by ―erlangs‖ represents the average number of concurrent calls carried by the circuits (or other service-providing elements), where that average is calculated over some reasonable period of time. The period over which the average is calculated is often one hour, but shorter periods (e.g., 15 minutes) may be used where it is known that there are short spurts of demand and a traffic measurement is desired that does not mask these spurts. One erlang of carried traffic refers to a single resource being in continuous use, or two channels being in use fifty percent of the time, and so on. For example, if an office has two telephone

T

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operators who are both busy all the time, that would represent two erlangs (2 E) of traffic; or a radio channel that is occupied for one hour continuously is said to have a load of 1 Erlang.

When used to describe offered traffic, a value followed by erlangs represents the average number of concurrent calls that would have been carried if there were an unlimited number of circuits (that is, if the call-attempts that were made when all circuits were in use had not been rejected). The relationship between offered traffic and carried traffic depends on the design of the system and user behavior. Three common models are

a) callers whose call-attempts are rejected go away and never come back,

b) callers whose call-attempts are rejected try again within a fairly short space of time, and

c) the system allows users to wait in queue until a circuit becomes available.

A third measurement of traffic is instantaneous traffic, expressed as a certain number of erlangs, meaning the exact number of calls taking place at a point in time. In this case the number is an integer. Traffic-level-recording devices, such as moving-pen recorders, plot instantaneous traffic.

The concepts and mathematics introduced by Agner Krarup Erlang have broad applicability beyond telephony. They apply wherever users arrive more or less at random to receive exclusive service from any one of a group of service-providing elements without prior reservation, for example, where the service-providing elements are ticket-sales windows, toilets on an airplane, or motel rooms. (Erlang‘s models do not apply where the server-providing elements are shared between several concurrent users or different amounts of service are consumed by different users, for instance, on circuits carrying data traffic.)

Offered traffic (in erlangs) is related to the call arrival rate, λ, and the average call-holding time, h, by

,

provided that h and λ are expressed using the same units of time (seconds and calls per second, or minutes and calls per minute).

The practical measurement of traffic is typically based on continuous observations over several days or weeks, during which the instantaneous traffic is recorded at regular, short intervals (such

as every few seconds). These measurements are then used to calculate a single result, most commonly the busy hour traffic (in erlangs). This is the average number of concurrent calls during a given one-hour period of the day, where that period is selected to give the highest result. (This result is called the time-consistent busy hour traffic). An alternative is to calculate a busy hour traffic value separately for each day (which may correspond to slightly different times each day) and take the average of these values. This generally gives a slightly higher value than the time-consistent busy hour value.

The goal of Erlang‘s traffic theory is to determine exactly how many service-providing elements should be provided in order to satisfy users, without wasteful over-provisioning. To do this, a target is set for the grade of service (GoS) or quality of service (QoS). For example, in a system where there is no queuing, the GoS may be that no more than 1 call in 100 is blocked (i.e., rejected) due to all circuits being in use (a GoS of 0.01), which becomes the target probability of call blocking, Pb, when using the Erlang B formula.

There are several Erlang formulae, including Erlang B, Erlang C and the related Engset formula, based on different models of user behavior and system operation. These are discussed below, and may each be derived by means of a special case of continuous-time Markov processes known as a birth-death process.

The busy-hour carried traffic, Ec, is measured on an already-overloaded system, with a significant level of blocking, it is necessary to take account of the blocked calls in estimating the busy-hour offered traffic Eo (which is the traffic value to be used in the Erlang formula). The offered traffic can be estimated by Eo = Ec/(1 - Pb). For this purpose, where the system includes a means of counting blocked calls and successful calls, Pb can be estimated directly from the proportion of calls that are blocked. Failing that, Pb can be estimated by using Ec in place of Eo in the Erlang formula and the resulting estimate of Pb can then be used in Eo = Ec/(1 - Pb) to estimate Eo. Another method of estimating Eo in an overloaded system is to measure the busy-hour call arrival rate, λ (counting successful calls and blocked calls), and the average call-holding time (for successful calls), h, and then estimate Eo using the formula E = λh.

For a situation where the traffic to be handled is completely new traffic, the only choice is to try to model expected user behavior, estimating active user population, N, expected level of use, U

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(number of calls/transactions per user per day), busy-hour concentration factor, C (proportion of daily activity that will fall in the busy hour), and average holding time/service time, h (expressed in minutes). A projection of busy-hour offered traffic would then be Eo = (NUC/60)h erlangs. (The division by 60 translates the busy-hour call/transaction arrival rate into a per-minute value, to match the units in which h is expressed).

Erlang-B also known as the Erlang loss formula, is a formula for the blocking probability derived from the Erlang distribution to describe the probability of call loss on a group of circuits (in a circuit switched network, or equivalent). It is, for example, used in planning telephone networks. The formula was derived by Agner Krarup Erlang and is not limited to telephone networks, since it describes a probability in a queuing system (albeit a special case with a number of servers but no buffer spaces for incoming calls to wait for a free server). Hence, the formula is also used in certain inventory systems with lost sales.

The formula applies under the condition that an unsuccessful call, because the line is busy, is not queued or retried, but instead really lost forever. It is assumed that call attempts arrive following a Poisson process, so call arrivals are independent. Further it is assumed that message length (holding times) are exponentially distributed (Markovian system) although the formula turns out to apply under general holding time distributions.

Erlangs are a dimensionless quantity calculated as the average arrival rate, λ, multiplied by the average call length, h. The Erlang B formula assumes an infinite population of sources (such as telephone subscribers), which jointly offer traffic to N servers (such as links in a trunk group). The rate of arrival of new calls (birth rate) is equal to λ and is constant, not depending on the number of active sources, because the total number of sources is assumed to be infinite. The rate of call departure (death rate) is equal to the number of calls in progress divided by h, the mean call holding time. The formula calculates blocking probability in a loss system, where if a request is not served immediately when it tries to use a resource, it is aborted. Requests are therefore not queued. Blocking occurs when there is a new request from a source, but all the servers are already busy. The formula assumes that blocked traffic is immediately cleared.

The formula provides the GoS (grade of service) which is the probability Pb that a new call arriving at the circuit group is

where

Pb is the probability of blocking m is the number of resources such as

servers or circuits in a group E=𝜆𝑕 is the total amount of traffic offered

in erlangs

This may be expressed recursively as follows, in a form that is used to simplify the calculation of tables of the Erlang B formula:

B(E,0) = 1

B(E , j) = 𝐸𝐵(𝐸,𝑗−1)

𝐸𝐵 𝐸 ,𝑗−1 +𝑗

for all j = 1,2,…,m. Typically, instead of B(E,m)

the inverse 1/B(E,m) is calculated in numerical

computation in order to ensure numerical stability:

1

𝐵(𝐸,0) = 1

1

𝐵(𝐸 ,𝑗 ) = 1 +

𝑗

𝐸

1

𝐵(𝐸,𝑗−1) ,

for all j = 1,2,…,m. The Erlang B formula applies to loss systems, such as telephone systems on both fixed and mobile networks, which do not provide traffic buffering, and are not intended to do so. It assumes that the call arrivals may be modeled by a Poisson process, but is valid for any statistical distribution of call holding times with finite mean. Erlang B is a trunk sizing tool for voice switch to voice switch traffic. The Erlang B formula is decreasing and convex in m.

Extended Erlang B is an iterative calculation, rather than a formula, that adds an extra parameter, the Recall Factor, which defines the recall attempts.

The steps in the process are as follows:

1. Calculate

𝑃𝑏=B(E,m)

as above for Erlang B.

2. Calculate the probable number of blocked calls

a. 𝐵𝑒=E𝑃𝑏

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3. Calculate the number of recalls, R

assuming a Recall Factor, 𝑅𝑓 :

R = 𝐵𝑒𝑅𝑓

4. Calculate the new offered traffic

𝐸𝑖+1 = 𝐸0+R

where 𝐸0 is the initial (baseline) level of traffic.

5. Return to step 1 and iterate until a stable value of E is obtained.

The Erlang C formula expresses the waiting probability in a queuing system. Just as the Erlang B formula, Erlang C assumes an infinite population of sources, which jointly offer traffic of Aerlangs to N servers. However, if all the servers are busy when a request arrives from a source, the request is queued. An unlimited number of requests may be held in the queue in this way simultaneously. This formula calculates the probability of queuing offered traffic, assuming that blocked calls stay in the system until they can be handled. This formula is used to determine the number of agents or customer service representatives needed to staff a call centre, for a specified desired probability of queuing.

𝑃𝑊= 𝐴𝑁

𝑁 !

𝑁

𝑁−𝐴

𝐴𝑖

𝑖! +

𝐴𝑁

𝑁 !𝑁−1𝑖=0

𝑁

𝑁−𝐴

where

A is the total traffic offered in units of erlangs

N is the number of servers 𝑃𝑊 is the probability that a customer has

to wait for service

It is assumed that the call arrivals can be modeled by a Poisson process and that call holding times are described by a negative exponential distribution. A common use for Erlang C is modeling and dimensioning call center agents in a call center environment.

The Engset calculation is a related formula, named after its developer, T. O. Engset, used to

determine the probability of congestion occurring within a telephonycircuit group. It deals with a finite population of S sources rather than the infinite population of sources that Erlang assumes. The formula requires that the user knows the expected peak traffic, the number of sources (callers) and the number of circuits in the network.

3. CONCLUSION

After explaining the concepts of traffic in some

systems, we have described the different erlang

units used to measure traffic.

REFERENCES

[1] Churchman, C.West,Russel L.Ackott,

E.Leonard Arnoff, Introduction to

Operation Reseach, .New York; John

Wiley and sons, Inc, 1963.

[2] Donald Gross Carl M.Harris,

Fundamentals of Queueing Theory, Third

Edition, Wiley India, 1998.

[3] G.Srinivasan Operation Research

Principles of Applications, PHI Learning

Private Limitted.New Delhi, 2007.

[4] E. Pgge, Introduction of Queuing Theory

in operation Research, 1972.

[5] Robert B. Cooper, Introduction to Queuing

Theory, 1972.

[6] Jean Walrand, An Introduction to Queuing

Networks, 1988.

[7] H. M. Wagner, Principles of Operation

Research, 1972.

[8] G. F. Newell, Applications of Queuing

Theory, 1982.

[9] Guoping Zeng, Two common properties of

the ERLONG-B Function, ERLANG-C

Function and Engset blocking function,

Elseuier science (2003).

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Solution and Stability of a Mixed Type Functional Equation in RN-Spaces

K. Ravi

and P. Narasimman

Abstract - In this paper, we obtain the general solution and investigate the generalized Hyers-Ulam Rassias stability problem for the following new

mixed type additive and quadratic functional equation

(2 ) (2 ) (2 ) 2 ( ) ( ) ( ) ( ) ( ) ( )f x y f y z f z x f x y f y z f x z f x f y f z

in random normed spaces with the sense of sherstnev under arbitrary t-norms.

Index Terms – Random normed space, Additive functional equation, Quadratic functional equation, Generalized Hyers-Ulam

stability.

—————————— ——————————

1.INTRODUCTION AND PRELIMINARIES

N 1940, S.M.Ulam[28] presented a list of unsolved problems. One of the problem is the

stability problem. It stated as follows: Suppose that a group G and a metric group H are given. For any 0 , does there exist a 0 such that if a function :f G H satisfies the inequality

( ( ), ( , ( ))d f xy f x f y

for all ,x y G , then a homomorphism :a G H exist with ( ( ), ( ))d f x a x for all

x in G ?.

In 1941, D.H.Hyers[12] answered Ulam‘s

problem for the case of approximately additive

functions under the assumption that G and H

are Banachspaces. Hyers result was further

generalized by Th.M.Rassias[25]. He proved the

following theorem.

Theorem 1.1: Let :f E E be a mapping

from normed vector space E into a Banachspace

E subject to the inequality

( ) ( ) ( )p p

f x y f x f y x y (1.1)

——————————————

K. Ravi is serving in the Department of Mathematics, Sacred Heart

College, Tirupattur, India. E-mail: shckravi@yahoo.com

P. Narasimman is pursuing Ph.D. degree in Mathematics, Bharathiar

University, Coimbatore India. E-mail: simma_gtec@yahoo.co.in

for all ,x y E where and p are constants with 0 and 1p . Then there exists a unique additive mapping :T E E such that

2( ) ( )

2 2

p

pf x T x x

(1.2)

for all x E . If 0p then the inequality (1.1) holds for all , 0x y and (1.2) for 0x .

Also if the function ( )t f tx from in to E

is continuous for each fixed x E , then T is

linear. In 1991, Z.Gajada[5] answered the

question raised by T.M.Rassias that his theorem

also holds good for 1p . This new concept is

known as Hyers-Ulam-Rassias stability of

functional equation (see [1, 3, 4, 6, 9-11]). In

1982-1998, J.M.Rassias [18-22] generalized the

result of Th.M.Rassias and proved the following

theorem.

Theorem: 1.2. Let X be a real-normed linear

space and let Y be real-complete-normed linear

space. Assume in addition that :f X Y is an

approximately additive mapping for which there

exist constants 0 and ,p q such that

1r p q , and f satisfies the Cauchy-

Gavruta-Rassias inequality

( ) ( ) ( )p p

f x y f x f y x y

for all ,x y X . Then there exists a unique

additive mapping :L X Y satisfying

I

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( ) ( )2 2

r

rf x L x x

for all x X . If in addition :f X Y is a

mapping such that the transformation ( )t f tx

is continuous in t for each fixed x X , then

L is -linear mapping.

The stability concept that was introduced by

Th.M.Rassias and J.M.Rassias provided a lot of

influence to a number of mathematicians to

develop the notion of what is known today the

term Hyers-Ulam-Rassias stability of the linear

mapping. Since then, the stability of several

functional equations has been extensively

investigated by several mathematicians (see [2, 8,

14, 23, 24]). In 2002, J.M.Rassias[17] estabilished

the Ulam stability of the following mixed-type

functional equation

3 3

1 1 1 3

( ) ( )i i i j

i i i j

f x f x f x x

on restricted domain. P.Nakmalachalasint [16]

generalized the J.M.Rassias work to the following

n-dimensional mixed-type functional equation

1 1 1

( 2) ( ) ( )n n

i i i j

i i i j n

f x n f x f x x

when 2n and investigated its generalized

Ulam-Gavruta-Rassias stability. E.Eshaghi Gordji

and etal [7] obtained general solution and

investigated the generalized Hyers-Ulam-Rassias

stability for the following Mixed- type functional

equations

4[ (3 ) (3 )]

12[ ( ) ( )]

12[ (2 ) (2 )] 8 ( )

192 ( ) (2 ) 30 (2 )

f x y f x y

f x y f x y

f x y f x y f y

f x f y f x

2 2 2

( ) ( )]

( ) ( ) 2(1 ) ( )

and f x ky f x ky

k f x y k f x y k f x

0, 1.k In 2005, K.W. Jun and H.M. Kim [13]

obtained the general solution of a generalized

quadratic and additive type functional equation of

the form

( ) ( ) ( ) ( )f x ay af x y f x ay af x y

for any integer a with 1,0,1.a A.Najati and

M.B.Moghimi [15] dealt the functional equation

(2 ) (2 )

( ) ( ) 2 (2 ) 2 ( )

f x y f x y

f x y f x y f x f x

(1.3)

which is derived from quadratic and additive

functions and established the general solution of

equation (1.3) and investigated the Hyers-Ulam-

Rassias stability for equation (1.3).

Before we proceed to the main theorems, we

present the necessary terminologies notations and

definitions which will be useful to do our main

theorems concerning random normed space. In

this study the space of all probability distribution

functions is denoted by

: , 0,1 :

(0) 0 ( ) 1

F R

F and F

where F is left continuous and non-decreasing on

R . Also the subset is the set

: ( ) 1D F l F

where ( )l f x denotes the left limit of the

function f at the point x , ( ) lim ( )t x

l f x f t

.

The space is partially ordered by the usual

point-wise ordering of functions, i.e., F G if

and only if ( ) ( )F t G t for all t R . The

maximal element for in this order is the

distribution function given by:

0

0, 0,( )

1, 0.

if tt

if t

Definition 1.3. ([26]). A mapping

: 0,1 0,1 0,1T

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is a continuous- norm, if T satisfies the following

conditions:

1. T is commutative and associative;

2. T is continuous;

3. T (a, 1) = a for all 0,1a ;

4. ( , ) ( , )T a b T c d whenever a c and

b ≤ d for all , , , 0,1 .a b c d

Typical examples of continuous t norm are

( , )T a b ab , ( , ) min( , )T a b a b and

( , ) max( 1,0)LT a b a b (the Lukasiewicz

t norm ). Now t norm are recursively defined

by 1T T and

1 2 3 1

1

1 2 3 1

( , , ,..., )

( , , ,..., ) ,

n

n

n

n n

T x x x x

T T x x x x x

for all 2n and 0,1ix , for all

1,2,..., 1 .T n The t norm T is Hadzic type if

for given 0,1 there is 0,1 such that

(1 ,...,1 ) 1 ,mT m N .

A typical example of such t- norm is

( , ) min( , )T a b a b .

Recall that if T is a t – norm and nx is a given

sequence of numbers in [0, 1], 1

n

i iT x is defined

recursively by

1

1i i iT x x and 1

1 1( , )n n

i i i i nT x T T x x

for 2n is

defined as 1lim .n

i in

T x

Definition 1.4. ([27]). A random normed space

(briefly, RN space) is a triple (X, μ, T), where X is

a vector space, T is a continuous t- norm and μ is

a mapping from X into D such that the following

conditions hold:

(RN1) 0( ) ( )x t t for all t > 0

if and only if x = 0;

(RN2) ( )x x

tt

for all x X , 0

and 0t ;

(RN4) ( ) ( ( ), ( ))x y x yt s T t s for all

,x y X and , 0t s . Clearly every normed

space ,X defines a RN-space (X, μ, TM),

where ( )x

tt

t x

for all t > 0,

and TM is the minimum. t- norm. This space is

called the induced random normed space.

Definition 1.5. Let (X, μ, T) be an RN-space. A

sequence nx in X is said to be convergent to x

in X if, for every 0t and 0 there exists

positive integer N such that ( ) 1nx x t

whenever .n N

Definition 1.6. Let (X, μ, T) be an RN-space. A

sequence nx in X is called Cauchy sequence if,

for every 0t and 0 there exists positive

integer N such that ( ) 1n mx x t whenever

.n m N

Definition 1.7. A RN-space (X, μ, T) is said to be

complete if and only if every Cauchy sequence in

X is convergent to a point in X.

Theorem 1.8. ([26]). If (X, μ, T) is an RN-space

and nx is a sequence such that

lim ( ) ( )nn x x

nx x then T T

almost everywhere.

In this paper, we are discussing a new mixed type

of additive and quadratic functional equation

(2 ) (2 ) (2 )

2 ( ) ( ) ( )

( ) ( ) ( )

f x y f y z f z x

f x y f y z f x z

f x f y f z

(1.4)

te its general solution and studied its stability in

random normed spaces with the sence of sherstnev

under arbitrary t- norms.

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In Section-2, we investigate the General Solution

of the Functional Equation (1.4) and in Section-3,

we discuss the stability of functional equation

(1.4) in random normed spaces with the sence of

sherstnev under arbitrary t-norm.

2. THE GENERAL SOLUTION OF THE

FUNDAMENTAL EQUATION (1.4)

In this section we establish the general solution of

functional equation (1.4).

Theorem 2.1. Let X and Y be a vector spaces,

and let :f X Y be a function satisfies (1.4).

Then the following assertions hold

a) If f is even function, then f is quadratic

b) If f is odd function, then f is additive

Proof: a) By putting 0x y z in (1.4), we

get (0) 0f . By evenness of f , equation (1.4)

can be written as

(2 ) (2 ) (2 )

2 ( ) ( ) ( )

( ) ( ) ( )

f x y f y z f z x

f x y f y z f x z

f x f y f z

(2.1)

, ,x y z X . Setting ( , , ) ( ,0,0),x y z x ( , ,0)x x

and ( , , )x y y in (2.1), we obtain the following

equations

(2 ) 4 ( )f x f x , (3 ) 9 ( )f x f x and

(2 ) ( 2 )

4 ( ) ( ) ( )

f x y f x y

f x y f x f y

(2.2)

respectively, for all ,x y X . Setting

( , , ) ( , ,0)x y z x y in (2.1), we obtain

(2 ) ( ) 2 ( ) 2 ( )f x y f y f x y f x (2.3)

(2 ) (2 ) (2 )

(2 ) ( 2 )1

(2 ) ( 2 )2

(2 ) ( 2 )

e e ef x y f y z f z x

f x y f x y

f y z f y z

f z x f z x

2 ( ) ( ) ( )

( ) ( ) ( )

e e e

e e e

f x y f y z f x z

f x f y f z

for all , ,x y z X . This means that ef holds in

(1.4). Similarly we can show that ef satisfies

(1.4). By above theorem, ef and ef are quadratic

and additive respectively. Thus there exists a

unique symmetric bi-additive function

:B X X Y such that ( ) ( , )ef x B x x for all

x X . Put ( ) ( )oA x f x for all x X . It

follows that ( ) ( , ) ( )f x B x x A x for all

x X . The proof of the converse is trivial.

3. STABILITY OF THE FUNCTIONAL

EQUATION (1.4)

In the section, the authors present the stability for

the functional equation (1.4) in random normed

space. Throughout this section, assume that X is a

real linear space and ( , , )Y T is a complete RN-

space.

Theorem 3.1. Let :f X Y be a function with

(0) 0f for which there is : X X D

with the property:

(2 ) (2 ) (2 ) , ,2 ( ) ( ) ( )

( ) ( ) ( )

( ) ( )f x y f y z f z x x y zf x y f y z f x z

f x f y f z

t t

(3.1)

for all , ,x y z X and all 0t . If

1 1

1 1

1

1

2

2 ,2 ,0

2

2 ,2 2 ,0

12

2 ,0,0

2

2 2 ,0,0

2

2

2lim 1

2

5

2

i n i n

i n i n

i n

i n

n i

x x

n i

x x

in in

x

n i

x

t

tT

t

t

(3.2)

and 2

2 ,2 ,2lim 2 1n n n

n

x x znt

(3.3)

for all , ,x y z X and all 0t , then there exists a

unique quadratic mapping :Q X Y such that

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1 1 1 1

1 1

(2 ) 2 ( ) ( )

2 ,2 ,0 2 ,2 2 ,0

1

2 ,0,0 2 2 ,0,0

( )

22

2

22

5

i i i i

i i

f x f x Q x

ii

x x x x

ii

i

x x

t

tt

Tt

t

(3.4)

for all ,x y X and all 0t .

Proof. Putting , 0y z in (3.1), we get

5 (2 ) 15 ( ) 5 ( ) ,0,0( )

5f x f x f x x

tt

(3.5)

for all x X . Putting ( , , ) ( , ,0)x y z x x and

( ,2 ,0)x x in (3.1) and adding the resultant

equations, we arrive

4 (2 ) 7 ( ) 5 ( ) 2 (4 ) ( 2 )

, ,0 ,2 ,0

( )

2

f x f x f x f x f x

x x x x

t

tt

(3.6)

for all x X . Adding (3.5) and (3.6), we obtain

9 (2 ) 8 ( ) 2 (4 ) ( 2 )

, ,0 ,2 ,0 ,0,0

( )

2 5

f x f x f x f x

x x x x x

t

t tt

(3.7)

for all x X . Setting ( , , ) (2 ,0,0)x y z x in

(3.1) and adding the resultant equation with (3.7),

we arrive

(4 ) 6 (2 ) 8 ( ) , ,0( )2

f x f x f x x x

tt

,2 ,0 ,0,0 2 ,0,05

x x x x

tt t

(3.8)

for all x X . Let

, , , ,0

,2 ,0 ,0,0 2 ,0,0

( )2

5

x x x x x

x x x x

tt

tt t

(3.9)

for all x X . Then equation (3.8) becomes

(4 ) 6 (2 ) 8 ( ) , ,( ) ( )f x f x f x x x xt t (3.10)

for all x X and all 0t . Let :g X Y be a

mapping defined by

( ) (2 ) 2 ( )g x f x f x .

Then we conclude that

(2 ) 4 ( ) , ,( ) ( )g x g x x x xt t (3.11)

for all x X . Thus we have

2

2

(2 ) , ,( )

2

( ) (2 )g x x x xg x

t t

(3.12)

for all x X and all 0t . Hence

1

2( 1) 2

2( 1)

(2 ) (2 ) 2 ,2 ,2

2 2

( ) (2 )k k k k k

k k

k

g x g x x x xt t

(3.13)

for all x X and all k N . This means that

1

2( 1) 2

1

1(2 ) (2 ) 2 ,2 ,2

2 2

(2 )2

k k k k k

k k

k

kg x g x x x x

tt

(3.14)

for all x X , 0t and all k N . By the

triangle inequality from 2

1 1 11 ...

2 2 2n , it

follows that

2

1

2( 1) 2

1 1 1

(2 )( )

2

1

0 1(2 ) (2 )

2 2

1 1

0 2 ,2 ,2

1 2 ,2 ,2

( )

2

2

2

n

n

k k

k k

k k k

i i i

g xg x

n

k kg x g x

n k

k x x x

n i

i x x x

t

tT

T t

T t

(3.15)

for all x X and all 0t . In order to prove the

convergence of the sequence 2

(2 )

2

n

n

g x , we replace

x with 2m x in (3.15) to obtain that

2( ) 2

(2 ) (2 )

2 2

n m m

n m m

g x g xt

1 1 1

2

1 2 ,2 ,2(2 )i m i m i m

n i m

i x x xT t

(3.16)

Since the right hand side of the inequality (3.16)

tends to 1 as m and n tend to infinity, the

sequence 2

(2 )

2

n

n

g x is a Cauchy sequence. Thus we

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may define 2

(2 )( ) lim

2

n

nn

g xQ x

for all x X .

Now we show that Q is a quadratic mapping.

Replacing ,x y with 2n x and 2n y in (3.1),

respectively, we get

2

(2 ) (2 ) (2 ) 2 ,2 ,22 ( ) ( ) ( )

( ) ( ) ( )

( ) (2 )n n n

n

g x y g y z g z x x y zg x y g y z g x z

g x g y g z

t t

(3.17)

Taking the limit as n , we find that Q

satisfies (1.4) for all ,x y X . By Lemma 2.1 the

mapping :Q X Y is quadratic. Letting the limit

as n in (3.15), we get (3.4) by (3.9). Finally,

to prove the uniqueness of the quadratic mapping

Q subject to (3.4), let us assume that there exists

another quadratic mapping Q which satisfies

(3.4). Since

2 2(2 ) 2 ( ), (2 ) 2 ( )n n n nQ x Q x Q x Q x

for all x X and n N , from (3.4), it follows

that 2 1

( ) ( ) (2 ) (2 )(2 ) (2 )n n

n

Q x Q x Q x Q xt t

1 1

1 1

1

1

1 1

1 1

2

2 ,2 ,0

2

2 ,2 2 ,0

12

2 ,0,0

2

2 2 ,0,0

2

2 ,2 ,0

2

2 ,2 2 ,0

1

2

2

2,

2

5

2

2

2

2

i n i n

i n i n

i n

i n

i n i n

i n i n

n i

x x

n i

x x

in i

x

n i

x

n i

x x

n i

x x

i

t

tT

t

tT

t

tT

1

1

2

2 ,0,0

2

2 2 ,0,0

2

5

2

i n

i n

n i

x

n i

x

t

t

(3.18)

for all x X and all 0t . By letting n in

(3.18), we conclude that Q Q .

Theorem 3.2. Let :f X Y be a function with

(0) 0f for which there is : X X D

with the property:

(2 ) (2 ) (2 ) , ,2 ( ) ( ) ( )

( ) ( ) ( )

( ) ( )f x y f y z f z x x y zf x y f y z f x z

f x f y f z

t t

(3.19)

for all , ,x y z X and all 0t . If

1 1

1 1

1

1

2 ,2 ,0

2 ,2 2 ,0

1

2 ,0,0

2 2 ,0,0

2

2

2lim 1

2

5

2

i n i n

i n i n

i n

i n

n

x x

n

x x

inn

x

n

x

t

tT

t

t

(3.20)

and

2 ,2 ,2

lim 2 1n n n

n

x x znt

(3.21)

for all , ,x y z X and all 0t , then there exists

a unique quadratic mapping :A X Y such that

1 1 1 1

1 1

(2 ) 4 ( ) ( )

2 ,2 ,0 2 ,2 2 ,0

1

2 ,0,0 2 2 ,0,0

( )

2

5

i i i i

i i

f x f x A x

x x x x

i

x x

t

tt

Tt

t

(3.22)

for all ,x y X and all 0t .

Proof. The steps are same as in Theorem 3.1 up

to the equation (3.10). Let :h X Y be a

mapping defined by ( ) (2 ) 4 ( )g x f x f x .

Then we conclude that

(2 ) 2 ( ) , ,( ) ( )h x h x x x xt t (3.23)

for all x X . Thus we have

(2 ) , ,

( )2

( ) (2 )h x x x xh x

t t

(3.24)

for all x X and all 0t . Hence

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1

1

1

(2 ) (2 ) 2 ,2 ,2

2 2

( ) (2 )k k k k k

k k

k

h x h x x x xt t

(3.25)

for all x X and all k N . This means that

1

1

1(2 ) (2 ) 2 ,2 ,2

2 2

( )2

k k k k k

k k

kh x h x x x x

tt

(3.26)

for all x X , 0t and all k N . By the

triangle inequality from 2

1 1 11 ...

2 2 2n , it

follows

1

1

1 1 1

1

0 1(2 ) (2 ) (2 )( )

2 2 2

1

0 2 ,2 ,2

1 2 ,2 ,2

( )2

n k k

n k k

k k k

i i i

n

k kh x h x h xh x

n

k x x x

n

i x x x

tt T

T t

T t

(3.27)

for all x X and all 0t . In order to prove the

convergence of the sequence (2 )

2

n

n

h x , we replace

x with 2m x in (3.27) to obtain that

1 1 1

(2 ) (2 )

2 2

1 2 ,2 ,2(2 )

n m m

n m m

i m i m i m

h x h x

n m

i x x x

t

T t

(3.28)

Since the right hand side of the inequality (3.28)

tends to 1 as m and n tend to infinity, the

sequence (2 )

2

n

n

h x is a Cauchy sequence. Thus we

may define

(2 )( ) lim

2

n

nn

h xA x

for all x X . Now we show that A is a additive

mapping. Replacing ,x y with 2n x and 2n y in

(3.19), respectively, we get

(2 ) (2 ) (2 ) 2 ,2 ,22 ( ) ( ) ( )

( ) ( ) ( )

( ) (2 )n n n

n

h x y h y z h z x x y zh x y h y z h x z

h x h y h z

t t

(3.29)

Taking the limit as n , we find that A

satisfies (1.4) for all ,x y X . By Lemma 2.1 the

mapping :A X Y is additive. Letting the limit

as n in (3.27), we get (3.22) by (3.9).

Finally, to prove the uniqueness of the additive

mapping Asubject to (3.4), let us assume that

there exists another additive mapping A which

satisfies (3.4). Since

(2 ) 2 ( ),n nA x A x (2 ) 2 ( )n nA x A x

for all x X and n N , from (3.4), it follows

that

1

( ) ( ) (2 ) (2 )

(2 ) (2 ) (2 ) (2 )

(2 ) (2 )

(2 ), (2 )

n n

n n n n

n

A x A x A x A x

n n

A x h x h x A x

t t

T t t

1 1

1 1

1

1

1 1

1 1

1

2 ,2 ,0

2 ,2 2 ,0

1

2 ,0,0

2 2 ,0,0

2 ,2 ,0

2 ,2 2 ,0

1

2 ,0,0

2

2

2,

2

5

2

2

2

2

2

5

i n i n

i n i n

i n

i n

i n i n

i n i n

i n

n

x x

n

x x

in

x

n

x

n

x x

n

x x

i n

x

t

tT

t

tT

t

tT

t

12 2 ,0,02i n

n

xt

(3.30)

for all x X and all 0t . By letting n in

(3.30), we conclude that A A .

Theorem 3.3. Let :f X Y be a function with

(0) 0f for which there is : X X D

with the property:

(2 ) (2 ) (2 ) , ,2 ( ) ( ) ( )

( ) ( ) ( )

( ) ( )f x y f y z f z x x y zf x y f y z f x z

f x f y f z

t t

(3.31)

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for all , ,x y z X and all 0t . If

1 1

1 1

1

1

2

2 ,2 ,0

2

2 ,2 2 ,0

12

2 ,0,0

2

2 2 ,0,0

2

2

2lim 1

2

5

2

i n i n

i n i n

i n

i n

n i

x x

n i

x x

in in

x

n i

x

t

tT

t

t

1 1 1 1

1 1

2 ,2 ,0 2 ,2 2 ,0

1

2 ,0,0 2 2 ,0,0

22

2lim

22

5

i n i n i n i n

i n i n

nn

x x x x

i nnn

x x

tt

Tt

t

(3.32)

and

2

2 ,2 ,2lim 2 1n n n

n

x x znt

=

2 ,2 ,2lim 2n n n

n

x x znt

(3.33)

for all , ,x y z X and all 0t , then there exists

a unique quadratic mapping :Q X Y and

unique additive mapping :A X Y such that

1 1 1 1

1 1

1 1 1 1

1 1

( ) ( ) ( )

2 ,2 ,0 2 ,2 2 ,0

1

2 ,0,0 2 2 ,0,0

2 ,2 ,0 2 ,2 2 ,0

1

2 ,0,0 2 2 ,0,0

( )

2 2 2

2 22 2

5

2

22

5

i i i i

i i

i i i i

i i

f x Q x A x

i i

x x x x

ii

i

x x

x x x x

i

x x

t

t t

T tt

t t

T tt

(3.34)

for all ,x y X and all 0t .

Proof. By Theorem 3.1 and Theorem 3.2, there

exist a quadratic mapping :Q X Y and a

additive mapping :A X Y such that

1 1 1 1

1 1

(2 ) 2 ( ) ( )

2 ,2 ,0 2 ,2 2 ,0

1

2 ,0,0 2 2 ,0,0

( )

22

2

22

5

i i i i

i i

f x f x Q x

ii

x x x x

ii

i

x x

t

tt

Tt

t

And

1 1 1 1

1 1

(2 ) 4 ( ) ( )

2 ,2 ,0 2 ,2 2 ,0

1

2 ,0,0 2 2 ,0,0

( )

2

5

i i i i

i i

f x f x A x

x x x x

i

x x

t

tt

Tt

t

for all x X and all 0t . So it follows from the

last inequalities that

1 1 1 1

1 1

( ) ( )( )

2 2

2 ,2 ,0 2 ,2 2 ,0

1

2 ,0,0 2 2 ,0,0

( )

2 2 2

2 22 2

5

i i i i

i i

Q x A xf x

i i

x x x x

ii

i

x x

t

t t

T tt

1 1 1 1

1 1

2 ,2 ,0 2 ,2 2 ,0

1

2 ,0,0 2 2 ,0,0

2

22

5

i i i i

i i

x x x x

i

x x

t t

T tt

for all x X and all 0t . Hence we obtain

(3.34) by letting ( )

( )2

Q xQ x

and

( )( )

2

A xA x

for all x X . The uniqueness

property of Q and A , are trivial.

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Approximate homomorphisms,

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27 (1941) 222-224.

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[23] K.Ravi R. Kodandan, P.Narasimman,

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Analysis of Worm Propagation in Computer

Networks with A Discrete Epidemic Model

M. Reni Sagaya Raj, A. George Maria Selvam and T.Sivagnanam

Abstract - Computer worms are self replicating programs that run independently and self-propagating across computer networks.

Computer worms have devastating effects on the economy. In order to defend against the worms, there is a need to understand the

propagation mechanism of worm spreading. An effective worm propagation model helps us to develop strategies to guard against the

worm attacks. In this paper, a modified epidemiological model is proposed. The model is constructed with difference equations and

certain dynamical behavior is investigated. Numerical simulations are performed with MATLAB.

Index Terms – Computer Network, virus, Epidemic model.

—————————— ——————————

1. INTRODUCTION

worm is a program that copies itself without

human intervention. Worms are malicious

computer program codes. In order to prevent

worms from propagating in networks, users need

to understand and predict the dynamic behaviours

of worm propagation in networks. There are

different types of worms such as email worms,

instant messaging (IM) worms, Internet worms,

Internet Relay Chat (IRC) worms and file sharing

networks worms and so on. Worms have

enormous adverse impact on the Internet. Network

worms have the potential to infect many

vulnerable hosts on the Internet before human

countermeasures take place. The aggressive

scanning traffic generated by the infected hosts

have caused network congestion, equipment

failure, and blocking of physical facilities such as

subway stations, 911 call centres, etc. The

detection count of malicious programs in June

2012 was 25,399. The worm called Bancos steals

IDs and passwords for on line banking. Code Red,

SQL Slammers, and Sasser are some of the most

famous examples of worms that have caused

considerable damage. Jerusalem is one of the

earliest worms discovered in 1987. Viruses cost

——————————————

M. Reni Sagayaraj is serving in the Department of Mathematics, Sacred

HeartCollege, Tirupattur, India. E-mail: reni.sagaya@gmail.com

A. George Maria Selvam is serving in the Department of Mathematics, Sacred HeartCollege, Tirupattur, India.

T.Sivagnanam is serving in the Department of Mathematics, St. Joseph University, Dar Es Salaam, Tanzania.

organizations millions of dollars. The TK worm

inflicted an estimated $ 5.5m of damage across the

Internet. In the following table, we present the

damage caused by some of the famous worm in

the history of Internet.

Name of the

worm

Year Damage in

dollars

Morris worm 1988 10 million

Melissa March 26, 1999 1.1 billion

I Love You May, 3, 2000 8.75

billion

Anna

Kournikova

Virus worm

February 2001 166,827

billion

Code Red July 13, 2001 2.6 billion

Sircam July 19, 2001 1.03

billion

NIMDA September 2001 645

million

Klez October 26,

2001

18.9

billion

SQL

Slammer

January 25,

2003

1.2 billion

Sobig January 2003 36.1

billion

Blaster August 11, 2003 1.3 billion

Mydoom January 26,

2004

38.5

billion

W March 19, 2004 11 million

Sasser April 30, 2004 14.8

billion

As per the latest statistics, there are 2,267,233,742

Internet Users worldwide. Leading global cyber

security firms in May 2012 announced the

A

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detection of a sophisticated new type of malicious

code on hundreds of computers throughout the

Middle East, with particular concentration in Iran.

The malicious code named Flame has been

capturing sensitive user information such as

screen shots, emails, documents and audio files

using a computers microphone. Flame is a data-

mining virus that in May 2012 penetrated the

computers of high-ranking Iranian officials,

sweeping up information from their machines.

2.EPIDEMIC MODELS

Mathematical modelling of infectious diseases can

be traced back to Bernoulli (1790). Bernoulli

developed mathematical models to study the

spread smallpox. In 1906, Hamer formulated a

model to investigate the spread of measles. In

1911, Dr. Ross described the transmission of

malaria between human population and

mosquitoes using a system of differential

equations. Epidemic models with vital dynamics

were constructed by Kermack and Mckendrick in

1927 [6].

A computer is susceptible to a worm if it could

become infected with the worm, provided the

worm is somehow introduced to the computer. In

the SIR model, a machine stays in one of the

following three states: susceptible state, infectious

state and removed state. When an infectious

machine is cleaned of worms, the machine

becomes a removed machine and it is immune to

the same type of worms. A susceptible machine

may become an infectious machine with the

possibility in a unit time. An infectious machine

is cured and becomes a removed machine with the

possibility in a unit time. At time t, ( )S t is the

number of susceptible machines, ( )I t is the

number of infectious machines and ( )R t is the

number of removed machines. The corresponding

differential equations are given by

( )( ) ( )

dS tS t I t

dt

( )

( ) ( ) ( )dI t

S t I t I tdt

( )( )

dR tI t

dt

( ) ( ) ( )S t I t R t N

where N is the total number of machines in the

system. The process of worm propagation on the

Internet is very similar to that of biological viruses

in populations. It can pass from one computer to

others like a biological virus between persons.

Spreading of worms in computer network is

epidemic in nature. Hence epidemic models have

been applied to study the propagation of viruses

and worms. Many authors analysed the process of

worm propagation on the Internet based on the

epidemic models [1, 5, 7, 9, 10].

3. MODEL DESCRIPTION

In this model, the host machines recovering from

the infective are allowed to go into a temporarily

immune state. Let be the rate at which

removals loose the immunization and becomes

susceptible. The assumptions are formulated in to

the following discrete SIRS model.

(1)

The initial conditions are

(0) (0) (0)S I R N

where (0), (0), (0) 0S I R . From the system of

above equations (1), we find

  ( 1) ( 1) ( 1) ( ) ( ) ( )S t I t R t S t I t R t N

.

If we replace ( )R t by ( ) ( )N I t S t in (1), we

obtain the following system of two equations in S

and I ,

( 1) ( ) ( ) ( ) ( )

( 1) ( )(1 ) ( ) ( )

( 1) ( )(1 ) ( )

S t S t S t I t R tN

I t I t S t I tN

R t R t I t

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( 1) ( ) ( ) ( ) [ ( ) ( )]

( 1) ( )(1 ) ( ) ( )

S t S t S t I t N S t I tN

I t I t S t I tN

(2)

4. EQUILIBIRIUM POINTS

There exists two equilibria for the system of equations (2) in

S and I . They can be found by solving the following

equations in S and

*I .

* * * * * *

* * * *

[ ]

(1 )

S S S I N S IN

I I S IN

(3)

Simplifying and solving the equations (3)

simultaneously, we obtain the following two

equilibrium solutions.

* *

* *

1. , 0

( )2. , .

( )

S N I

NS N I

The first equilibrium is the disease-free

equilibrium and the second one is the endemic

equilibrium. The Jacobian matrix has the form

1

( , )

1

I SN N

J S I

I SN N

At the disease-free equilibrium, the Jacobian

matrix is

1( ,0)

0 1J N

The Jacobian matrix is upper triangular. Hence the

eigen values are 1 1 and 2 1 .

The disease free equilibrium is locally

asymptotically stable if1,2 1 . Suitably the

restrictions can be imposed on the parameters

such that 10 1 . The second eigen value

satisfies 20 1 if 1

. The basic

reproduction number is defined as (0)R

. If

(0) 1R , then there exists disease-free

equilibrium and it is locally asymptotically stable.

We shall continue with the analysis of the model

and investigate the stability of the system at the

second equilibrium point for various values of the

parameters. At the second equilibrium point the

Jacobian matrix has the form

* *

( 2 )1 ( )

( , )( )

1

J S I

We shall assume that

( 2 )2 0

.

The equilibrium point is locally asymptotically

stable if the eigen values of the Jacobian matrix J

satisfy 1i if and if only if 1 det 2Tr J J

[3, 8]. Hence

( 2 )2

( 2 )2 ( )

2

yields

0 ( )( ) ( 2 ) (4)

The equilibrium point is locally asymptotically

stable if and only if (4) is satisfied.

5. NUMERICAL SIMULATIONS AND

DISCUSSIONS

In the following discussion we take 100N . Now

we take 0.2, 0.3 and 0.02 so that

(0) 1R . The Jacobian matrix is

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1.02625 0.32

0.00625 1J

For J , we have 2.02625Tr J and

det 1.0243J . The eigen values are

1 2 11.0597, 0.9665, 1 and 2 1 .

Since (0) 1R , the infection dies out. But most of

the machines remain in susceptible state. Hence,

they are vulnerable to another variant of the same

worm.

Choosing the values 0.8, 0.2 and 0.1

and so (0) 1R , the equilibrium point is

(25,25)E . Calculations show that 1.9Tr J and

det 0.96J . Therefore

1 det 2Tr J J

is satisfied. Hence equilibrium point (25,25)E is

locally asymptotically stable. Also the eigen

values are

1 0.95 0.2398i

and 2 0.95 0.2398i

where 0.9798 1i . Due to the fact (0) 1R ,

the worm is not removed from all the machines

and they continue to reside in the network and

they pose a threat to the system. Hence the rate of

removal should be accelerated.

Considering 0.5, 0.2, 0.07 so that

(0) 1R , the equilibrium point is (40,15.6)E . For

the Jacobian matrix J , we have 1.9922Tr J and

det 1.01132J . Also det 1J . The eigen values

are

1,2 0.9961 0.1449i

and 1,2 1.0066 1 .

Hence the equilibrium (40,15.6)E is unstable.

6. CONCLUSION

The paper discusses discrete-time epidemic model

for worm propagation in computer networks with

a modified SIR model allowing fraction of the

hosts in the class R to go back in to susceptible

state. Numerical simulation are presented for the

cases (0) 1R and (0) 1R . This helps the

organization to frame their policy regarding the

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use of Anti Virus software‘s and the frequency of

updating the data base of the Anti Virus software.

REFERENCES

[1] Bimal Kumar Mishra, Dinesh Saini,

Mathematical models on computer viruses,

Applied Mathematics and Computation, 187

(2007)929936.

[2] Essam Al Daoud, Iqbal H. Jebril and Belal

Zaqaibeh, Computer Virus Strategies and

Detection Methods, Int. J. Open Problems

Compt. Math., Vol. 1, No. 2, September

2008.

[3] Leah Edelstein-Keshet, Mathematical

Models in Biology, SIAM, Random House,

New York, 2005.

[4] Fangwei Wang, Yunkai Zhang, Jianfeng Ma,

Defending passive worms in unstructured

P2P networks based on healthy file

dissemination, Computers and Security,

28(2009), 628 - 636.

[5] Jonghyun Kim, Sridhar Radhakrishnan,

Sudarshan K.Dhall, Measurement and

analysis of worm propagation on Internet

network Topology, ICCCN 2004.

[6] J.D.Murray, Mathematical Biology I: An

Introduction, 3-e, Springer International

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[7] Onwubiko.C, Lenagham A.P and Hebbes.L,

An Improved worm mitigation model for

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worms, EUROCON 2005.

[8] L.Perko, Differential Equations and

Dynamical Systems, 3-e, Springer - Verlag,

New York Inc, 2001.

[9] Jose R.C. Piqueira, Betyna Fernandez

Navarro and Luiz Henrique Alvez Monteiro,

Epidemic-logical Models Applied to Viruses

in Computer Networks, Journal of Computer

Science, 1 (1): 31 - 34, 2005.

[10] Vasileios Vlachos, Diomidis Spinellis, and

Stefanos Androutsellis-Theotokis, Biological

Aspects of Computer Virology, 3rd

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23-25 September 2009, Athens, Greece.

[11] Xiang Fan, Yang Xiang, Modelling the

propagation of peer-to-peer worms, Future

Generation Computer Systems, 26(2010),

1433 - 1443.

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Sum Labeling for Arbitrary Supersubdivision of Path, Cycle and Star

J. Gerard Rozario, J. Jon Arockiaraj, P. Lawrence Rozario Raj and U. Rizwan

Abstract - A sum labeling is a mapping 𝜆 from the vertices of G into the positive integers such that, for any two vertices

u, v 𝜖 V (G) with labels 𝜆(u) and 𝜆(v), respectively, (uv) is an edge iff 𝜆(u) + 𝜆(v) is the label of another vertex in V (G).

Any graph supporting such a labeling is called a sum graph. It is necessary to add (as a disjoint union) a component to

sum label a graph. This disconnected component is a set of isolated vertices known as isolates and the labeling scheme

that requires the fewest isolates is termed optimal. The number of isolates required for a graph to support a sum labeling

is known as the sum number of the graph. In this paper, we will give optimal sum labeling scheme for some cycle related

graphs.

Index Terms - Sum Labeling, Sum number, Sum graph, isolates

—————————— ——————————

1. INTRODUCTION

LL the graphs considered here are simple,

finite and undirected. For all terminologies

and notations we follow Harary [2] and graph

labeling as in [1]. Sum labeling of graphs was

introduced by Harary [3] in 1990. Following

definitions are useful for the present study.

Definition 1.1 A Sum Labeling is a mapping 𝜆

from the vertices of G into the positive integers

such that, for any two vertices u, v 𝜖 V (G) with

labels 𝜆(u) and 𝜆(v), respectively, (uv) is an edge

iff 𝜆(u) + 𝜆(v) is the label of another vertex in V

(G). Any graph supporting such a labeling is

called a Sum Graph.

——————————————

J. Gerard Rozario is serving in the Department of Mathematics, St. Joseph’s College of Arts and Science, Cudallore, India. E-mail: rozario.gerard@yahoo.com

J. Jon Arockiaraj is serving in the Department of Mathematics, St. Joseph’s College of Arts and Science, Cudallore, India. E-mail: jonarockiaraj@gmail.com

P. Lawrence Rozario Raj is serving in the Department of Mathematics, St. Joseph College, Trichy, India. E-mail : lawraj2006@yahoo.co.in

U. Rizwan is serving in the Department of Mathematics, Islamiah College, Vaniyambadi, India. E-mail : rizwanmaths@yahoo.com

efinition 1.2 It is necessary to add (as a disjoint

union) a component to sum label a graph. This

disconnected component is a set of isolated

vertices known as Isolates and the labeling

scheme that requires the fewest isolates is termed

Optimal.

Definition 1.3 The number of isolates required for

a graph G to support a sum labeling is known as

the Sum Number of the graph. It is denoted

as 𝜎 𝐺 .

Definition 1.4 Let G be a graph with q edges. A

graph H is called a Super subdivision of G if H is

obtained from G by replacing every edge ei of G

by a complete bipartite graph im2,K for some mi,

1 i q in such a way that the end vertices of

each ei are identified with the two vertices of 2-

vertices part of im2,K after removing the edge ei

from graph G. If mi is varying arbitrarily for each

edge ei then super subdivision is called arbitrary

super subdivision of G.

In this paper, we will prove that graphs obtained

by arbitrary super subdivision of path 𝑃𝑛 , cycle 𝐶𝑛

and star 𝐾1,𝑛 are optimal summable with sum

number 2.

A

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2.OPTIMAL SUM LABELING SCHEME FOR

ARBITRARY SUPER SUBDIVISION OF

PATH, CYCLE AND STAR

Sethuraman et.al [5], introduced a new method of

construction called Supersubdivision of graph and

proved that arbitrary supersubdivision of any path

and cycle Cn are graceful. Kathiresan et.al [4],

proved that arbitrary supersubdivision of any star

is graceful.

In this section, we prove that the arbitrary super

subdivision of path 𝑃𝑛 , cycle 𝐶𝑛 and star 𝐾1,𝑛 are

optimal summable with sum number 2.

Theorem 2.1 Arbitrary supersubdivision of path

𝑃𝑛 are optimal summable with sum number 2.

Proof. Let G be a path 𝑃𝑛 with n vertices. Let vi

(1 i n) be the vertices of G. Let H be the

arbitrary supersubdivision of G which is obtained

by replacing every edge of G with 𝐾2,𝑚 𝑖.

Let 𝑚 = 𝑚𝑖𝑛−11 . Let uj be the vertices which

are used for arbitrary supersubdivision of G

where 1 j m. Let x and y be two isolated

vertices. Therefore, the vertex set of H is

V(H) = { v1, v2,……,vn,u1,u2,……,um}.

Define f : V(G) {1,2,3,…, N}

𝑓 𝑣𝑖 = 𝑖 ; 1 ≤ 𝑖 ≤ 𝑛

𝑓 𝑢1 = 𝑚 + 𝑛

𝑓 𝑢𝑗 = 𝑓 𝑢𝑗−1 − 1 ; 2 ≤ 𝑗 ≤ 𝑚

Then

𝑓 𝑥 = 𝑓 𝑢1 + 1 𝑎𝑛𝑑 𝑓 𝑦 = 𝑓 𝑢1 + 2

Thus, we are able to identify all the edges of path

with just two isolated vertices.

Hence, arbitrary supersubdivision of path 𝑃𝑛 is

optimal summable with sum number 2.

Illustration 2.1 Sum labeling for arbitrary

supersubdivision of path 𝑃5 is given in figure 2.1

Figure 2.1

Theorem: 2.2 Arbitrary super subdivision of

cycle 𝐶𝑛 is optimal summable with sum number 2.

Proof. Let G be a cycle 𝐶𝑛 with n vertices. Let vi

(1 i n) be the vertices of G. Let H be the

arbitrary super subdivision of G which is obtained

by replacing every edge of G with 𝐾2,𝑚 𝑖.

Let 𝑚 = 𝑚𝑖𝑛1 . Let uj be the vertices which are

used for arbitrary supersubdivision of G

where 1 j m. Let x and y be two isolated

vertices. Therefore, the vertex set of H is

V(H) = { v1, v2,……,vn,u1,u2,……,um}.

Define f : V(G) {1,2,3,…, N}

𝑓 𝑣𝑖 = 𝑖 ; 1 ≤ 𝑖 ≤ 𝑛

𝑓 𝑢1 = 𝑚 + 𝑛

𝑓 𝑢𝑗 = 𝑓 𝑢𝑗−1 − 1 ; 2 ≤ 𝑗 ≤ 𝑚

Then 𝑓 𝑥 = 𝑓 𝑢1 + 1 𝑎𝑛𝑑 𝑓 𝑦 = 𝑓 𝑢1 + 2 Thus, we are able to identify all the edges of path

with just two isolated vertices.

Hence, arbitrary supersubdivision of cycle 𝐶𝑛 are

optimal summable with sum number 2.

Illustration 2.2 Sum labeling for arbitrary

supersubdivision of cycle 𝐶5 is given in figure 2.2

Figure 2.2

Theorem: 2.3 Arbitrary supersubdivision of star

𝐾1,𝑛 are optimal summable with sum number 2.

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Proof. Let G be a star 𝐾1,𝑛 with n+1 vertices. Let

vi (1 i n+1) be the vertices of G. Let H be the

arbitrary super subdivision of G which is obtained

by replacing every edge of G with 𝐾2,𝑚 𝑖.

Let 𝑚 = 𝑚𝑖𝑛−11 . Let uj be the vertices which

are used for arbitrary supersubdivision of G

where 1 j m. Let x and y be two isolated

vertices. Therefore, the vertex set of H is

V(H) = { v1, v2,……,vn+1,u1,u2,……,um}.

Define f : V(G) {1,2,3,…, N}

𝑓 𝑣𝑖 = 𝑖 ; 1 ≤ 𝑖 ≤ 𝑛 + 1

𝑓 𝑢1 = 𝑚 + 𝑛 + 1

𝑓 𝑢𝑗 = 𝑓 𝑢𝑗−1 − 1 ; 2 ≤ 𝑗 ≤ 𝑚

Let x and y be two isolated vertices. Then

𝑓 𝑥 = 𝑓 𝑢1 + 1 𝑎𝑛𝑑 𝑓 𝑦 = 𝑓 𝑢1 + 2

Hence, arbitrary supersubdivision of star 𝐾1,𝑛 are

optimal summable with sum number 2.

Figure 2.3

Illustration 2.3 Sum labeling for arbitrary

supersubdivision of star 𝐾1,5 is given in figure 2.3

REFERENCES

[1] Gallian J A, A dynamic survey of graph

labeling, The Electronics Journal of

Combinatorics, 16, (2009) DS6.

[2] Harary F, Graph theory, Addison Wesley,

Reading, Massachusetts, 1972.

[3] Harary F, Sum graphs and Difference

graphs, Congress Numerantium, no.72, 101-

108, 1990.

[4] K.M. Kathiresan, S. Amutha, ―Arbitrary

supersubdivisions of stars are graceful‖,

Indian Journal of pure and applied

Mathematics. 35(1), pp. 81-84, 2004.

[5] G. Sethuraman, P. Selvaraju, ―Gracefulness

of arbitrary supersubdivisions of graphs‖,

Indian Journal of pure and applied

Mathematics, 32(7), pp. 1059-1064, 2001.

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An Extreme Shock Maintenance Model for a Multistate Degenerative System Under a

Bivariate Replacement Policy

M. Mohamad Yunus, P. Govindaraju and U. Rizwan

Abstract - In this paper, we consider a simple repairable multistate degenerative system which is subjected to random shocks from its environment. The long-run average cost for the extreme shock maintenance model of a degenerative multistate system under a bivariate replacement policy (T, N) , where T is the working age of the system and N is the number of failures of the system, is studied. Explicit expressions for the long-run average cost is given.

Index Terms - Geometric Process, Replacement Policy, Renewal Reward Process, Shock models.

—————————— ——————————

1. INTRODUCTION

HE study of a multistate degenerative system

in a maintenance model plays an important

role in reliability. A multistate degenerative

system is subject to damage and the damage

occurs randomly in an operating stage. Most of

the maintenance models just pay attention on the

internal cause of the system failure, but do not on

an external cause of the system failure. A system

failure may be caused by some external cause,

such as a shock. The shock models have been

successfully applied to different fields, such as

physics, communication, electronic engineering

and medicine, etc. A very few authors considered

the deteriorating systems interrupted by random

shocks. Barlow and Proschen (1983) considered

an imperfect repair model, in which a repair is

perfect with probability p and a minimal repair

with probability 1– p. There were many papers

which consider extreme shock models. In their

models, the system will fail if the amount of

shock exceeds a specific threshold. In these

——————————————

U. Rizwan is serving in the Department of Mathematics, Islamiah

College, Vaniyambadi, India. E-mail: rizwanmaths@yaoo.com

P. Govindaraju is serving in the Department of Mathematics, Islamiah

College, Vaniyambadi, India. E-mail : govindrajmaths69@gmail.com

M. Mohamad Yunus is pursuing Ph.D. in Mathematics, Islamiah College,

Vaniyambadi, India.

models, a shock is called a deadly shock or

extreme shock. Thangaraj and Rizwan (2001)

have introduced and studied the shock model with

NONN repair times are discussed an extended

extreme shock maintenance model for a

deteriorating system and so on. Chen and

Li(2008) have introduced and studied the extreme

shock model.

The rest of the paper is organized as follows: In

section 2, we present an extreme shock model for

the maintenance problem of a multistate

repairable system. In section 3, explicit

expressions for the long-run average cost under a

bivariate replacement policy ),( NT is derived.

2. DESCRIPTION OF THE MODEL

In this section, we first give some definitions.

Next, we describe the model of a one-component

multistate system. We also evaluate the

conditional probabilities of the operating times

and failure times given the state of the system.

Definition 1 A random variable X is said to be

stochastically smaller than another random

variable Y , if )>()>( YPXP , for all real

. It is denoted by YX st . A stochastic process

1,2,=,nX n is said to be stochastically

increasing, if 1 nstn XX , for 1,2,=n .

T

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Definition 2 A Markov process 1,2,=,nX n

with state space 0,1,2, is said to be

stochastically monotone, if

1 1 1 2| = | = ,n n st n nX X i X X i

1 20 .for any i i

Clearly, the stochastically monotone concept for a

Markov process is defined and based on the

transition probabilities from one state to another

state, conditioning on the former state. However,

the stochastically monotone concept for a

stochastic process defined here is for a general

process and is based on the conditional

distribution of two successive random variables in

the process.

Definition 3 A stochastic process

1,2,=,nX n is a geometric process, (GP) if

there exist a constant 0>a such that

1,2,=,1 nXa n

n forms a renewal process. The

number a is called the ratio of the geometric

process.

If 1<<0 a , then the GP is stochastically

increasing; if 1>a , the GP is stochastically

decreasing and if 1=a , the GP will reduce to a

renewal process.

Definition 4 An integer valued random variable

N is said to be a stopping time for the sequence

of independent random variables ,, 21 XX , if the

event nN = is independent of ,, 21 nn XX , for

all 1,2,=n .

We shall now describe the system states. Consider

a one-component multistate system with lk

states ( k -working states and l -failure states).

The system state at time t is given by

( = 1,2, , )( ) =

( = 1,2, , )

if the systemis in the i thi

working state at time t

i kS t

if the system is in the j thk j

failure state at time t

j l

The set of working states is },{1,2,=1 k ; the

set of failure states is },2,1,{=2 lkkk

and the state space is 21= . Initially,

assume that a new system in working state 1 is

installed. Whenever the system fails, it will be

repaired. Let nt be the completion time of the n -

th repair, 0,1,=n with 0=0t and let ns be the

time of occurrence of the n -th failure, 1,2,=n

Then

.<<<<<< 110 nn tstst

We next describe the probability structure of the

model.

Assume that the transition probability from

working state kii ,1,2,=, , to failure state

ljjk ,1,2,=, , is

jnn qitSjksSP =)=)(|=)(( 1

with 1.=1= j

l

jq Moreover, the transition

probability from failure state ljjk ,1,2,=, ,

to working state kii ,1,2,=, is given by

inn pjksSitSP =)=)(|=)((

with 1.=1= i

k

ip

Let 1X be the operating time of the system after

installation. In general, let 2,3,=, nX n be the

operating time of the system after 1)( n -st repair

and 1,2,=, nYn be the repair time after n -th

failure. Assume that there exist a life-time

distribution )(tU and kiai ,1,=0,> such that

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)(=)( 1 tUtXP (1)

and

kitaUitStXP i ,1,2,=),(=)=)(|( 12 (2)

where .1 21 kaaa

In general, for

,,1,2, ki j

).(==)(,,=)(|11

1111 taaUitSitStXPn

iinnn

(3)

Similarly, assume that there exist a life-time

distribution )(tV and libi ,1,2,=0,> such that

),(=)=)(|( 11 tbViksStYP i

(4)

where 0>1 21 lbbb and in general, for

,,1,2, li j

1 1

1

( | ( ) = , , ( ) = )

= ( )

n n n

i in

P Y t S s k i S s k i

V b b t

(5)

In particular, if '

k aaaba ===1,== 211 and '

l bbb ===2 , then the )( lk -state system

reduces to a two-state system. In this case,

equations (3) and (5) become

taUtXP n'

n

1)(=

and ,)(= tbVtYP n'

n

respectively.

Thus the sequence 1,2,=,nX n forms a GP

with ratio 1>'a , while the sequence

1,2,=,nYn forms a GP with ratio 1<<0 'b . In

this case, our model reduces to the GP model for

the one component two-state system introduced

by Lam [1988].

For two working states kii 21 <1 , we have

.=)(|=)(| 112212 itSXitSX st

Therefore, the working state 1i is better than the

working state ,2i in the sense that, the system in

state 1i has a stochastically large operating time

than 66

it does in state 2i . Consequently, the k working

states are arranged in decreasing order, such that

state 1 is the best working state and state k is the

worst working state. Similarly, for two failure

states 1ik and 2ik such that

lkikikk 21 <1 , we have

.=)(|=)(| 211111 iksSYiksSY st

Therefore, the failure state 1ik is better than the

failure state 2ik in the sense that the system in

state 1ik has a stochastically smaller repair time

than it does in state 2ik . Thus, the l failure

states are also arranged in decreasing order, such

that the state 1k is the best failure state and the

state lk is the worst failure state.

Consider a monotone process model for a

multistate one-component system described in this

section and make the following package of

assumptions, 101 AA .

A1 At the beginning, a new system is

installed. The system has )( lk

possible states, where the states

k,1,2, denote, respectively, the first-

type working state, the second-type

working state , , k -th-type working

state and the states

)(,2),(1),( lkkk denote,

respectively, the first-type failure state,

the second-type failure state and the

l -th type failure state of the system. The

occurrences of these types of failures

are stochastic and mutually exclusive.

A2 Whenever the system fails in any of the

failure states, it will be repaired. The

system will be replaced by an identical

one some times later.

A3 Once the system is operating, the

shocks from the environment arrive

according to a renewal process. Let

1,2,=, iX ni be the intervals between

the 1)( i -st and the i -th shock, after

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the 1)( n -st repair. Let =)( 11XE .

We assume that ,1,2,=, iX ni are iid

sequences for all n

A4 Let 1,2,=, iYni be the sequence of the

amount of shock damage produced by the

i -th shock, after the 1)( n -st repair. Let

=)( 11YE . Then }1,2,=,{ iYni are iid

sequences, for all n . If the system fails, it

is closed, so that the random shocks have

no effect on the system during the repair

time.

A5 Let 1,2,=, nZn be the repair time after

the n -th repair and 1,2,=,nZn

constitute a non decreasing geometric

process with =)( 1ZE and ratio b , such

that 1<0 b . )(tNn is the counting

process denoting the number of shocks

after the 1)( n -st repair. The distribution

of nZ is denoted by )(nG . It is clear that

.=)(1nn

bZE

A6 Let r be the reward rate per unit time of

the system when it is operating and c be

the repair cost rate per unit time of the

system and the replacement cost is .R

The replacement time is a random variable

Z with =)(ZE .

A7 If the system in working state i is

operating, then let the reward rate be r. If

the system in failure state k + i is under

repair, the repair cost is c. The

replacement cost comprisesof two parts :

one part is the basic replacement R and the

other is proportional to the replacement

time Z at rate cp. In other words, the

replacement cost is given by R + cp Z.

A8 Assume that 1 ≤ a1 ≤ a2 ≤ . . . . ≤ ak and

1≥ b1 ≥ b2 ≥ b3 ≥ . . . . ≥ bl > 0.

A9

Assume that )(tFn be the cumulative

distribution of i

n

in XL 1== and )(tGn be

the cumulative distribution of

.=1= i

n

in YM

A10 The working time nX , the repair time nY

and the replacement time Z , )1,2,=( n

are independent random variables.

3. THE POLICY ),( NT

In this section, we introduce and study a bivariate

replacement policy ),( NT for the multistate

degenerative system, under which system is

replaced at working age T or at the time of N -th

failure, whichever occurs first. The problem is to

choose an optimal replacement policy ),( NT

such that the long-run average cost per unit time is

minimized.

The working age T of the system at time t is the

cumulative life-time given by

1

1 1 1 1

<=

<

n n n n n

n n n n n

t M L M t L MT

L L M t L M

where i

n

in XL 1== and i

n

in YM 1== and

0== 00 ML .

Following Lam and Zhang (2004), the distribution

of the survival time nX in A3 and the distribution

of the repair time nY in A4 are given by

),(!!

1)!(=)( 1

11

1

1

1=

1=

taaUppjj

ntXP k

j

k

jk

j

k

j

k

ni

j

k

i

n

(6)

where Zjjj k,,, 21 and

),(!!

!=)( 1

11

1

1

=

1=

tbbVqqjj

ntYP l

j

l

jl

j

l

j

l

ni

j

l

i

n

(7)

where Zjjj l,,, 21 . If =)( 1XE , then the

mean survival time is

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,=)(

1nna

XE

(8)

for 1,>n where

1

1=

=

i

ik

i a

pa

(9)

and if =)( 1YE , then the mean repair time is

nn

bYE

=)(

(10)

for 1>n , where

.=

1

1=

j

jl

j b

qb

(11)

Further, if

kiisShererR nin ,1,2,=,=)(w= 1

denotes the reward earned after the n -th repair,

then mean reward earned after 1)( n -st repair is

rXRE =)( 11 and for 2n then expected reward

after installation is given by

,=)(1nnn

a

rXRE

(12)

where

.=1= i

iik

i a

prr

(13)

and if liiksSherecC nin ,1,2,=,=)(w=

denotes the repair cost after the n -th failure, then

mean repair cost after n -th failure is

,=)(1nnn

b

cYCE

(14)

where

.=1= i

iil

i b

qcc

(15)

3.1 The Length of a cycle and its Mean

The length of a cycle under the bivariate

replacement policy ),( NT is

,= >

1=

1

1=1=

ZYTYXW TN

Li

i

TN

Li

N

i

i

N

i

where 1,0,1,2,= N is the number of failures

before the working age of the system exceeds T

and

( )

1 ,=

0 .A

if the event A occurs

if the event A does not occur

From Leung (2005), we have

( < ) = ( < )L T L i N

i NE P L T L

)()(= TLPTLP Ni

).()(= TFTF Ni

Lemma 3.1 The mean length of a cycle is

.)(d)(=)(1

1

1=0

TFb

uuFWE ii

N

i

N

T

(16)

Proof. Consider

1

( )

=1 =1

( > )

=1

( ) =

( )

N N

i i L TN

i i

i L TN

i

E W E X Y

E T Y E Z

uLYXEE NTN

Li

N

i

i

N

i

=|= )(

1

1=1=

)()>(

1=

)>( ZEYETE TN

Li

i

TN

L

)(d)()(d=1

1=00

uFYEuFu Ni

N

i

T

N

T

1

1=1

( ) <N

N i Nii

T F T E L T Lb

)(d)(=

0uFuTFT N

T

N

1 1

1 1=1 =1

( ) ( ) ( )N N

i N Ni ii i

F T F T F Tb b

,)(d)(=1

1

1=0

TFb

uuF ii

N

i

N

T

as desired. ■

3.2 Mean Reward and Mean Repair cost

Lemma 3.2 If TLN and 2n , then the

expected Reward earned is

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.)(=)(R02

2=2=

uudFa

rTLXE N

T

n

N

n

Nnn

N

n

(17)

Proof.

=2

=2

R ( )

= R ( ) |

N

n n N

n

N

n n N N

n

E X L T

E E X L T L

)(=|R=2=

0udFuLXE NNnn

N

n

T

,)(=02

2=

uudFa

rN

T

n

N

n

as desired. ■

Lemma 3.3 If TLN > and 2n then the

expected reward earned is

.)()(=)>(R2

2=2=

TFTFa

rTLXE Nnn

N

n

Nnn

N

n

(18)

Proof.

=2

=2

( > )

= ( < < )

n n N

n

N

n n n N

n

E R X L T

E R X L T L

)<<()(=2=

Nnnn

N

n

LTLEXRE

.)()(=2

2=

TFTFa

rNnn

N

n

■

Lemma 3.4 If ,TLN then the expected repair

cost is

.)(=)(C1

1

1=

1

1=

TFb

cTLYE Nn

N

n

Nnn

N

n

(19)

Proof.

1

=1

1

=1

C ( )

= C | = ( )

N

n n N

n

N

n n N N

n

E Y L T

E E Y L u L T

)(=|C=1

1=0

udFuLYE nNnn

N

n

T

)()C(=1

1=0

udFYE Nnn

N

n

T

.)(=1

1

1=

TFb

cNn

N

n

■

Lemma 3.5 If ,> TLN then the expected repair

cost is

.)()(=)>(C1

1

1=

1

1=

TFTFb

cTLYE Nnn

N

n

Nnn

n

(20)

Proof.

1

=1

1

=1

C ( > )

= C ( < < )

n n N

n

N

n n n N

n

E Y L T

E Y L T L

)<<()C(=1

1=

Nnnn

N

n

LTLEYE

.)()(=1

1

1=

TFTFb

cNnn

N

n

■

4. THE LONG-RUN AVERAGE COST

UNDER ),( NT POLICY

Let 1T be the first replacement time and let nT

2)( n be the time between 1)( n -st

replacement and n -th replacement. Then the

sequence ,1,2,=, nTn forms a renewal process.

The interarrival time between two consecutive

replacements is a renewal cycle. By the renewal

reward theorem (Ross (1996) ), the long-run

average cost per unit time under the multistate

bivariate replacement policy ),( NT is

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( , ) =the expected cost incurredin a cycle

C T Nthe expected length of a cycle

.)(

)(

=

)>(

1=1=

)(

1=

1

1=

WE

ZEcRTYCE

RXRYCE

pTN

Ln

n

nn

n

TN

Lnn

N

n

nn

N

n

Using Lemmas 3.2 to 3.5, we obtain

1

1 2 0=1 =2

1

1 1=1

12=2

1

10=1

( ) ( )

( ) ( )

( ) ( )

( , ) = .

( )d ( )

N N T

N Nn nn n

N

p n Nnn

N

n Nnn

NT

N nnn

c rF T udF u

b a

cc R r F T F T

b

rF T F T rT

aC T N

F u u F Tb

Summarizing the above results, we have the

following.

Theorem 3.1 For the model described in Section

2, under the assumptions A1 to A10, the long-run

average cost per unit time under the bivariate

replacement policy ),( NT for a multistate

degenerative system is given by

1

1 2 0=1 =2

1 2=2

1

10=1

( ) ( )

( ) ( ) ( )

( , ) = .

( )d ( )

N N T

n Nn nn n

N

n Nnn

p

NT

N nnn

c rF T udF u

b a

rr T F T F T

a

c RC T N

F u u F Tb

The standard minimization procedure can be

adapted to determine the optimal values.

REFERENCES

[1] Barlow, R.E. and Proschan, F. (1975)

Statistical Theory of Reliability and life

testing, John Wiley, New York.

[2] Chen, J. and Li, Z. (2008) An extended

extreme shock maintenance model for a

deteriorating system, Relia.Engg and Sys

Saf., 93, 11231129.

[3] Govindaraju. P, Rizwan. U and Thangaraj,

V, (2011) An extreme shock maintenance

Model under a Bivariate Replacement

Policy, Research Methods in Mathematical

Sciences, 110.

[4] Lam, Y. (1988) Geometric Processes and

Replacement Problem, Acta. Math. Sinica,

4, 366377.

[5] Lam,Y. (1991) An Optimal Repairable

Replacement Model for Deteriorating

System, J. App. Prob., 28, 843851.

[6] Lam, Y. and Zhang, Y.L. (2004) A shock

model for the maintenance problem of a

repairable system, Computers and

Operations Research, 31, 18071820.

[7] Leung, K.N.G., (2005), A Note on a

Bivariate Optimal Replacement Policy for a

Repairable System, Engineering

Optimization, 38, 621 -625.

[8] Rizwan. U and Mohamad Yunus. M,

(2011), (2011) An extreme shock

maintenance Model for a Multistate

Degenerative System – I, Research

Methods in Mathematical Sciences, 49 – 62.

[9] Ross, S.M. (1996) Stochastic Processes,

(2nd ed), John Wiley and Sons, New York.

[10] Stadje, W. and Zuckerman, D.(1990)

Optimal strategies for some repair

replacement models, Adv. Appl. Prob., 22,

641656.

[11] Thangaraj, V. and Rizwan, U. (2000) Burn-

in with Optimal Replacement Policies for a

system subject to shocks, Proc. of the

National Conference on Optimization

Techniques in Industrial Mathematics, Ed.,

Elumalai, S, University of Madras, 207215.

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Study of Ionic Conductivity in Li I Grafted Solid Biopolymer Electrolyte

A. Ayisha Begam, K. Prem Nazeer and Rugmini Radhakrishnan

Abstract - Solid Chitosan Acetate Electrolytic Films were prepared by Grafting the Lithium Iodide salt (0.1 – 0.5 wt %) with Chitosan in

the presence of acetic acid. These films were subjected to impedance spectroscopy and ionic conductivity studies. It was found that the

ionic conduction behaviour in the electrolyte systems depends on the concentration of the salt used. The highest room temperature

conductivity (1.05×10−4

S/cm) obtained from impedance measurement for LiI grafted chitosan electrolyte along with the dielectric and

relaxation studies supported the segmental motion of the ion. Study on transference number revealed that the highest conducting

samples were ionic conductors.

Index Terms - Chitosan acetate solid electrolyte films – Impedance analysis – Dielectric studies – Transference number

—————————— ——————————

1. INTRODUCTION

HE science of polymer electrolytes has

attracted both in academia and industry, for

the past two decades due to the potentially

promising applications of such electrolytes, not

only in all solid-state rechargeable batteries, but

also in other electrochemical devices such as

supercapacitors, electrochromic windows, and

sensors (Scrosati, 1930), Gray, 1991, 1997). The

study of polymer electrolytes was launched by

Fenton et al., in 1973, but their technological

significance was not appreciated until the research

undertaken by Armand et al., (1979) a few years

later. These authors claimed that the polymer

complexes formed from alkali metal salts were

capable of demonstrating significant ionic

conductivity, and highlighted their possible

application as battery electrolytes. This work

inspired intense research and development on the

synthesis of new polymer electrolytes, physical

studies of their structure and

——————————————

A. Ayisha Begam is serving in the Department of Physics, Avinashilingam University for Women, Coimbatore, India,

K. Prem Nazeer is serving in the Department of Physics, Islamiah

College, Vaniyambadi, India. E-mail : nazeerprem@gmail.com

Rugmini Radhakrishnan served in the Department of Physics, Avinashilingam University for Women, Coimbatore, India

charge transport, theoretical modeling of the charge-

transport processes, the physical and chemical

properties and their relationship etc. (MacCallum and

Vincent (1987, 1989), Song et al.,(1999). This paper

deals with ionic conductivity of alkali metal doped

solid chitosan acetate electrolyte analyzed with the

help of impedance spectroscopy, ac conduction and

transference number studies.

2. EXPERIMENTAL

Chitosan-salt complexes were formed by grinding

1 g of chitosan powder (Fluka - medium

molecular weight) with different concentration (10

– 50 wt %) of LiI salts and by mixing this

compound with 50 ml of acetic acid in a 500 ml

glass beaker. This solution was stirred for about 1

hour continuously using a Teflon pellet, which

was rotated with a help of a magnetic stirrer

maintained at room temperature to form 2 % (w/v)

solution. Most of the chitosan salt complexes

dissolved to give a transparent solution

(electrolyte). Minor insoluble solids were

removed using a syringe filter with a pore size of

1micron and the required Chitosan electrolyte

solutions were collected for the preparation of

solid Chitosan electrolyte films.

Film casting technique is employed in the present

study for the preparation of solid biopolymer

electrolyte. The casting of the films was carried

T

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out by pouring the filtered chitosan solution (5ml)

onto optically plane glass moulds (10×10 cm) and

were allowed to dry at room temperature (25 2

C) in a closed atmosphere for 3 days. The dried

films were carefully removed from the mould and

its edges were clamped onto a well cleaned

optically plane glass plate and finally dried (50C,

24 Hrs) and stored under dry condition. The film

thickness was determined using a universal

length-measuring instrument (TRIMOS,

Switzerland) to an accuracy of 0.110–6

m (the

pressure maintained in the ball contact was 2.47

Pa). Typical thicknesses of ~ 50µm were used for

all the studies. In the present work, an automated

Electrochemical Impedance Analyzer (Princeton

Applied Research potentiostat - (Model 2273) was

used to measure various parameter of the

biopolymer electrolyte. DC polarization

technique was adapted to measure transference

number with the help of home made instruments.

3. RESULTS AND DISCUSSION

3.1 Impedance Spectroscopy of Li I Grafted

Chitosan Acetate (CHA) Electrolyte:

Impedance spectroscopy is a powerful method of

characterizing many of the electrical properties of

electrolyte materials and their interfaces with

electronically conducting electrodes. Impedance

plot (plot between real and imaginary parts of

impedance) for CHA doped with various

concentration (10, 30 and 50 wt %) of LiI

biopolymer electrolytes at room temperature are

shown in Fig.1. In the Nyquist plot (Z’ vs -Z”),

one observes a typical spectrum of the ionic

conductors consisting of high frequency

semicircle and low frequency tail. The observed

semicircle in the high frequency region is due to

the bulk effect of the electrolytes, and the linear

region seen at low frequency range is attributed to

the effect of the blocking electrodes. Since the

complex impedance will be dominated by the

ionic conductance when the phase angle is close

to zero, normally, the bulk resistance is directly

obtained from the intercept of complex impedance

plot with real axis (Z0 axis).

Fig. 1 Impedance plot of CHA electrolyte

grafted with Li I at 303 K

From Fig.1, some complex impedance curves

have not touched the real axis though they are

near to the real axis. For these cases, the complex

impedance plot is extrapolated to its intersection

with the real axis and the conductivity of the

electrolyte is calculated. The point where the

semi-circle intersects the real axis (Z’) gives the

value of bulk resistance (Rb). By knowing the

value of bulk resistance (Rb) along with the

dimensions of the sample, the conductivity of the

sample has been calculated by using the relation σ

= d/RbA, where d (≈50μm) is the thickness of the

polymer electrolyte films and A (0.5cm2) the

surface area of the films. As a general trend, in

many studies for the dependence of salt

concentration on the ionic conductivity in solid

polymer electrolytes at low salt concentrations,

the conductivity increases due to build-up of

charge carriers. And at high salt concentrations,

the conductivity decreases due to build-up of

charge carriers offset by the retarding effect of ion

cloud (Anji Reddy Polu and Ranveer Kumar,

2011).

The impedance spectra of Li grafted CHA showed

the retarding effect beyond 50 wt% may be due to

preferential site for interaction as suggested in IR

studies or due to phase transformation found in

the case of UV and XRD studies (Ayisha Begam

(2012)) . Therefore, further studies on LiI grafted

CHA will certainly through more light on the

nature of the required biopolymer electrolyte.

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3.2 Conductivity Studies on Li I Grafted

Chitosan Acetate (CHA) Electrolyte

There are various factors that influence ionic

conductivity; the number of charge carriers, ionic

mobility as well as the availability of a connecting

polar domain as the conduction pathway (Linford,

1993). In this present work, the increase in

conductivity could be attributed to the increment

of charge carriers, while the decline in

conductivity value could simply be explained by

the retarding behaviour of ions at higher

concentration that arises due to some structural

change as reported in IR, UV-Vis and XRD

studies (Ayisha Begam, 2012). The behavior of

the conductivity–metal salt variation can be

explained in terms of weak electrolyte theory. The

weak electrolyte theory states that σ = ηqμ, where

η is the number density of mobile ions which will

be in motion under the action of an electric field;

q is the electronic charge and μ is the mobility of

ionic species (Idris et al., 2009).

From the graph (Fig.2), it can be observed that

CHA grafted with LiI (50 wt %) has the highest

room temperature conductivity of 1.05 × 10−4

S/cm is more than the basic requirement for

electrolyte used for battery applications (Gray et

al. (1997)). Similar behaviour was also observed

for many polymer electrolytes (Ramya et al.,

2005), in which the high ionic conductivity is

attributed to increased ionic mobility and

increased ionic charge carrier concentration. The

temperature-dependent ionic conductivity

measurements were taken to analyze the

mechanism of ionic conduction in polymer

electrolytes. Fig.3 shows the plot of log σac versus

103/T for selected samples in the system. From

these plots, the activation energy was calculated

using the Arrhenius equation:

𝜎 = 𝜎0 𝑒𝑥𝑝 −𝐸𝑎 𝑘𝑇

where σo is the pre-exponential factor; Ea is the

activation energy; T is the absolute temperature

and k is the Boltzmann‘s constant. From the

conductivity–temperature data which obeys

Arrhenius relationship, it can be deduced that the

nature of cation transport is quite similar to that

occurring in ionic crystals, where ions jump into

Fig. 2 Ionic conductivity and Activation Energy

of CHA grafted with various amount

of LiI at 303K

Fig. 3 Temperature-dependent ionic conductivity

of CHA grafted with LiI electrolyte

neighboring vacant sites and hence increase the

ionic conductivity (Kulkarni et al., 2010). A

linear variation observed from this plot suggests

that no phase transition occurred in the polymer

matrix or domain formed by addition of LiI. The

conductivity values do not show any abrupt jump

with temperature, which indicate that, these

electrolytes exhibit amorphous nature (Samsudin

et al., 2011) as reported in the XRD analysis

(Ayisha Begam 2012). Therefore, no dynamic

conformational change in the polymer matrix and

Li ions might migrate through the conduction path

formed by the lattice structure Selvasekarapandian

et al., 2005) of the chitosan chains. The increase

in conductivity with temperature in solid polymer

electrolyte is attributed to segmental motion

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which facilitates faster ionic movement (Khiar

and Arof, 2011).

The activation energy, Ea

(combination of the

energy of defect formation and the energy for

migration of ion) calculated by linear fit from the

Fig. 3. The Ea

was assumed to be the energy

required to move the ion, presupposing that the

structure remain unchanged, plus the energy

required to deform the structure enough to allow

the ion to pass. It is observed from the Fig. 2 that

the activation energy decreases gradually with the

salt concentration up to 30 wt% and beyond that a

fast fall is observed. The decrease in activation

energy is due to the density of ions in the polymer

electrolyte increase with increasing of LiI

concentration; hence, the energy barrier to the ion

transport decreases, which would lead to a

decrease in the activation energy (Idris et al.,

2009). Rice and Roth hypothesized that energy

gap exists in the ionic conductor, in which the ion

conducting species can be thermally excited from

localized ionic states to free ion-like states. Since

the ion transfer is greatly affected by the polymer

segmental motion (Samsudin et al., 2011), which

is very active beyond the concentration of 30

wt%, lowers the value of Ea imply rapid ionic

conduction and then increase in mobility of ions

also the ionic conductivity.

3.3 Dielectric Relaxation and Transference Number

of LiI Grafted CHA Electrolyte

The strength and frequency of relaxation depend

on characteristic property of dipolar relaxation

(Pradhan et al., 2008). The study of dielectric loss

will throw more information on relaxation

process. Fig.4. depicts the frequency dependence

of loss tangent for selected samples at ambient

temperature. For maximum dielectric loss (tan

δmax) at a particular frequency, the absorption peak

is described by: τω =1. Here τ is the relaxation

time, ω is the angular velocity with ω=2πf, f is the

frequency value corresponding to maximum tan δ

in Hz. Relaxation time occurs as a result of ionic

charge carriers within the sample to obey the

change in the direction of the applied field. It is

observed from Fig. 5, that the peak frequency

shifted towards higher frequency as the salt

concentration increases. The shift of the peak

towards higher frequency suggests quicker

relaxation time.

As discussed earlier, the increase in ionic

conductivity is due to the enhancement of the

number of carrier ions. On addition of LiI, it is

believed that there is an increase in the amorphous

content in the materials, which in turn speed up

the segmental motion by increasing the available

free volume. It is evidenced by the peak shifting

towards higher frequency side, thereby reducing

the relaxation time. Thus, the relatively fast

segmental motion coupled with mobile ions

enhances the transport properties of the sample

which explains the decrease in relaxation time

with concentration as depicted in Fig. 4 (Khiar

and Arof, 2011).

The transference number is defined as the ratio of

the conductivity of a species to the final

conductivity of the sample (Osman et al., 2001).

Since conductivity could be attributed to ions and

electrons, the ion transference number was

analyzed according to the following equation:

𝑡𝑖𝑜𝑛 + 𝑡𝑒 = 1

where tion and te represent the ionic and electronic

transference number. When a voltage V = 2V,

is applied to the cell below the decomposition

potential of the electrolyte, ionic migration will

occur until steady state is achieved. At the steady

state, the cell is polarized and any residual current

flows because of electron migration across the

electrolyte and interfaces. This is because the

ionic currents through an ion-blocking electrode

fall rapidly with time if the electrolyte is

primarily ionic. The plot of polarized current

versus time is shown in Fig. 6. The initial total

current decreases with time due to the depletion of

the ionic species in the electrolyte and becomes

constant in the fully depleted situation.

The ratio of saturation current to initial current

that gives the lithium transference number 0.22.

Thus the ionic transference number 𝑡+ =𝐼𝑐𝑎𝑡𝑖𝑜𝑛

𝐼0

obtained is 0.78 indicating the sample to be an

ionic conductor. Reports on lithium transference

number ranges from 0.1 to 0.5 (Yahya and Arof,

2002, Khiar and Arof, 2011).

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Fig. 4. Dependence of tan δ with frequency

for selected samples

Fig. 5 Variation of relaxation time as a

function of salt concentration

Fig. 6 The transference number polarization curve

for the highest conducting LiI grafted

CHA electrolyte

These values show that the polymers grafted

/doped with lithium salts are ionic conductors and

the main conducting species is the anions. Fig. 6

presents the temporal change in polarization

current, which has been normalized for

convenience. Comparing the reported results, LiI

grafted CHA samples are eligible for ionic battery

applications as a suitable biopolymer electrolyte

complex. Since the transference measurements

were attempted manually, there may be some

error (±5%) in the measurement that results in

little lower value (0.78) than the theoretical value.

4. CONCLUSION

The highest room temperature conductivity (1.05

× 10−4

S/cm) was obtained from impedance

measurement. A linear variation in activation

energy suggests that there was no phase transition

in the polymer matrix and it supported the

thermally activated segmental motion of ions.

Dielectric and relaxation studies further support

the segmental motion coupled with ion enhanced

transport property of the samples. Study on

transference number revealed that the highest

conducting samples were ionic conductors.

Further work on the fabrication of biopolymer

battery will open a new avenue in the field of

green and biodegradable electronics.

ACKNOWLEDGEMENTS

The authors would like to thank the authorities of

Avinashilingam University for Women,

Coimbatore and Islamiah College, Vaniyambadi

for their support to carry out this work.

REFERENCES

[1] Ayisha Begam A (2012), ‗Synthesis,

Characterization and Application of

Chitosan and its Composite Films‘,

(Doctoral Thesis), Avinashilingam

University for Women, India.

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[2] Anji Reddy Polu and Ranveer Kumar

(2011), ‗AC impedance and dielectric

spectroscopic studies of Mg2+ ion

conducting PVA–PEG blended polymer

electrolytes,‘ Bull. Mater. Sci., 34, 5, 1063–

1067.

[3] Armand, M.B; Chabagno, J.M and Duclot,

M (1979), in: Vashista, P; Mundy, J.N and

Shenoy, G.K (Eds), ‗Fast Ion Transport in

Solids‘, Elsevier, Amsterdam.

[4] Gray, F.M (1991), ‗Solid Polymer

Electrolytes—Fundamentals and

Technological Applications‘, VCH, New

York.

[5] Gray, F.M (1997), ‗Polymer Electrolytes‘,

RSC Materials Monographs, The Royal

Society of Chemistry, Cambridge, London.

[6] Idris, N.K; Nik Aziz, N.A; Zambri, M.S.M;

Zakaria, N.A and Isa, M.I.N (2009), ‗Ionic

conductivity studies of chitosan-based

polymer electrolytes doped with adipic acid,

Ionics, 15, 643-646.

[7] MacCallum, J.R and Vincent, C.A (1987),

‗Polymer Electrolyte Reviews-1‘, Elsevier,

London.

[8] MacCallum, J.R and Vincent, C.A (1989),

‗Polymer Electrolyte Reviews-2‘, Elsevier,

London.

[9] Scrosati, B (1930), ‗Applications of

Electroactive Polymers‘, Chapman and Hall,

London.

[10] Song, J.Y; Wang, Y.Y and Wan, C.C

(1999), Review of gel type polymer

electrolyte for lithum ion batteries, J. Power

Sources, 77, 183 -197.

[11] Ramya, C.S; Selvasekarapanidan, S;

Savitha, T and Hirankumar, G (2005),

‗Transport mechanism of Cu-ion conducting

PVA based solid polymer electrolyte‘,

Ionics, 11, 5-6, 436-441.

[12] Linford, R.G (1993), ‗Electrical and

electrochemical properties of ion conducting

polymers in Applications of Electroactive

Polymers‘, B. Scrosati (Ed.), Chapman and

Hall, London.

[13] Khiar, A.S.A and Arof, A.K (2011),

‗Electrical Properties of Starch/Chitosan-

NH4NO3 Polymer Electrolyte‘, World

Academy of Science, Engineering and

Technology, 59, 23-27.

[14] Kulkarni, A.R; Balaji, R and Srinivasa, R.S,

(2010), ‗Structural Investigation of

Polyurethane Based Gel Polymer

Electrolytes Using Small Angle X-ray

Scattering (SAXS)‘ Proc. 3rd Int. Conf.

Physics of Solid State Ionics (ICPSSI-3), J.

Phys. Soc. Jpn, 79, Suppl. A, 154-159.

[15] Pradhan, D.K; Choudhary, R.N.P and

Samantaray, B.K (2008), ‗Studies of

Dielectric Relaxation and AC Conductivity

Behavior of Plasticized Polymer

Nanocomposite Electrolytes‘, Int. J.

Electrochem. Sci., 3, 597-608.

[16] Samsudin, A.S; Kuan, E.C.H and Isa, M.I.N

(2011), ‗Methyl Cellulose – Glycolic Acid

System: Study on the Potential as Proton

Conducting Bio-Polymer Electrolytes‘,

Proc. Conf. Empowering Science,

Technology and Innovation Towards a

Better Tomorrow-UMTAS-2011, 375-379.

[17] Selvasekarapandian, S; Hirankumar, G;

Kuwata, N; Kawamura, J and Hattori, T

(2005), ‗1H Solid State NMR studies on the

Proton conducting polymer electrolytes‘,

Materials Letters, 59, 22, 2741-2745.

[18] Osman, Z; Ibrahim, Z.A and Arof, A.K

(2001), ‗Conductivity enhancement due to

ion dissociation in plasticised chitosan based

polymer electrolytes‘, Carbohydrate

Polymer, 44, 167-173.

[19] Yahya, M.Z.A and Arof, A.K

(2002),‘Studies on lithium acetate doped

chitosan conducting polymer system‘,

European Polymer Journal, 38, 1191–1197.

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Neighborhood Critical Edges of an M -strong Guzzy Graph

S. Ismail Mohideen and A. Mohamed Ismayil

Abstract - A set VS is a neighborhood set of G if = [ ]u V

G N u

and is denoted by setn . The neighborhood

number of G is the minimum scalar cardinality taken over all n-set and is denoted by 0n . 0n -set is a neighborhood set of G

with minimum scalar cardinality. In this paper, we investigate the properties of critical edges. That is whether the end vertices of critical edges are fixed, free and totally free.

Index Terms – Fuzzy Sets, Fuzzy Graph.

—————————— ——————————

1. INTROCUCTION

HE notion of fuzzy graph and several fuzzy

analogs of graph theoretical concepts such as

path, cycle and connectedness are introduced by

Rosenfeld in the year 1975[5]. Mordeson and

Peng introduced the concept of fuzzy line graph

and developed its basic properties in the year

1993[4]. The neighborhood numbers )( 0n of

various known fuzzy graphs are introduced by S.

Ismail Mohideen and A. Mohamed Ismayil in the

year 2010[3]. Neighborhood critical vertex in

crisp graph is introduced by E. Sambathkumar and

Prabha S. Neeralagi in the year 1992[6]. In this

paper, Neighborhood critical edges of an M -

strong fuzzy graph are discussed. Theorems

related to these critcal edges are stated and

proved. In a fuzzy graph G, the neighborhood

number may increase or decrease or remain

unaltered, if a vertex or an edge is removed from

G. Some results based on neighborhood critical

vertices are given in section 3.

———————————

S. Ismail Mohideen is serving in the Department of Mathematics, Jamal

Mohamed College, Tiruchirappalli, India. E-mail:

simohideen@yahoo.co.in

A. Mohamed Ismayil is serving in the Department of Mathematics, Jamal

Mohamed College, Tiruchirappalli, India. E-mail:

amismayil1973@yahoo.co.in

2. PRELIMINARIES

Definition 2.1 Let V be a finite non empty set and

E be the collection of two element subsets of V .

A fuzzy graph ),(= G is a set with two

functions [0,1]: V and [0,1]: E such

that )()(),( vuvu for all ., Vvu

Definition 2.2 Let ),(= G be a fuzzy graph on

V and VS . Then the scalar cardinality of S is

defined by )(uSu

. The order )( p and size

)(q of a fuzzy graph ),(= G are the scalar

cardinality of V and E respectively.

Definition 2.3 A fuzzy graph ),(= 111 G is

called the fuzzy sub graph induced by 1V if

)()(1 uu for all 1Vu and

),()()(),( 111 vuvuvu for all 1, Vvu

and is denoted by 1V . A fuzzy graph

),(= 111 G is called the full fuzzy sub graph

induced by 1V if )(=)(1 uu for all 1Vu and

),()()(=),( 111 vuvuvu for all 1, Vvu

and is denoted by 1V .

Definition 2.4 A vertex u of a fuzzy graph

),(= G is said to be isolated vertex if

)()(<),( vuvu for all uVv \ . An edge

T

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),(= vue of a fuzzy graph is called an effective

edge if )()(=),( vuvu . Here the vertex u

is adjacent to v and the edge e is incident to u

and v . A fuzzy graph ),(= G is said to be M -

strong fuzzy graph [1] if )()(=),( vuvu for

all Evu ),( .

Definition 2.5 Let Vvu , and ),(= vue then

)}()(=),(:{=)( vuvuVvuN is called

open neighborhood of u and uuNuN )(=][ is

called closed neighborhood of u .

Definition 2.6 Let ),(= G be a fuzzy graph on

V and let Vvu , . If )()(=),( vuvu then

u dominates v (or v is dominated by u ) in G .

A subset D of V is called a dominating set in G

if for every DVv then there exist Du such

that u dominates v .The minimum fuzzy

cardinality of a dominating set of G is called the

domination number of G and is denoted by )(G

or .

Definition 2.7 Let ),(= G be an M -strong

fuzzy graph. A set VS is a neighborhood set of

G if ][= uNG Su and is denoted by setn .

The neighborhood number of G is the minimium

scalar cardinality taken over all n-set and is

denoted by 0n . 0n -set is a neighborhood set of G

with minimium scalar cardinality.

In a fuzzy graph G, the beighborhood number may

increase or decrcease or remain unaltered if a

vertex is removed from G.

3. NEIGHBORHOOD CRITICAL VERTICES

Definition 3.1 A vertex v of G is

(i) critical if )()( GvG

(ii) critical if )(>)( GvG

(iii) critical if )(<)( GvG

(iv) fixed if v belongs to every -set

(v) free if v belongs to some -set but

not all

(vi) etotallyfre if v does not belong any

-set.

Here the parameter is used as a common

symbol for neighborhood number 0n and

domination number .

Definition 3.2 The set of all critical

( critical , critical , fixed , free ,

totallyfree ) vertices are called setc (

c ,

c , fx ,

fr , settf ).

Example 3.1 Consider the fuzzy graph given in

figure 3.1.

0.5 0.2 0.4 0.4 0.6 0.5 0.7 0.3

. ——— . ——– . ——— . . ——– . ——– . .

1 2 3 4 5 6 7 8v v v v v v v v

Figure: 3.1

Let the set be 8642 ,,, vvvv .

(i) 86421 ,,,, vvvvv is c -set

(ii) 62 ,vv is

c -set

(iii) 841 ,, vvv is

c -set

(iv) 862 ,, vvv is fx -set

(v) 43,vv is fr -set

(vi) 751 ,, vvv is tf -set.

Observation 3.1

1. If the vertex v is isolate then

)(<)( GvG , that is fxv -set.

2. The union of fx-set, fr

-set and tf -set is

V, that is

psetsetset tffrfx = ,

where p is an order of G .

3. Every vertex of -set is .critical

Coverse is not true, for example, In figure

3.1, 1v is critical but not in -set.

Theorem 3.1 Every critical vertex of G is

fixed .

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Remark 3.1 Converse of the Theorem given in

sec.1.5 need not be true. For example, In figure

3.1, 8v is fixed but not critical .

Theorem 3.2 Let G be a fuzzy graph, if -set is

unique in G , then every vertex of V is either in

fx -set or tf -set. In this case, the union of

fx -

set and tf -set is V. That is

psetset tffx = .

Remark 3.2 If -set is unique, then

1. fr -set is empty.

2. Intersection of fx -set and

tf -set is

empty.

Theorem 3.3 Every vertex of V not in setc is

either in fr -set and

tf -set.

Theorem 3.4

1. A vertex v is critical if and only if

)()( uNuN vDu for some -set D

containing v .

2. v is criticaln

0 if and only if )(vN is a

full induced fuzzy subgraph of

)(uNvDu for some 0n -set D

containing v .

4. NEIGHBORHOOD CRITICAL EDGES

In this section, we intestigate critical edges

and investigate whether the end vertices of critical

edges are fixed, free and totally free edges.

Definition 4.1 An edge e of G is

(i) critical if )()( GeG

(ii) critical if )(>)( GeG

(iii) critical if )(<)( GeG

Observation 4.1

1. If an edge e in G is critical , then

)(>)( GeG . Thus a critical edge is

always critical .

2. An criticaln 0 edge e is either criticaln

0

or criticaln

0. For example, removal of any

one edge of an odd fuzzy cycle of length 5

is criticaln

0, but the edge e in the figure

1.1 is criticaln

0.

Theorem 4.1 An edge e in G is critical if

and only if there is no dominating set of eG

with scalar cardinality )(G .

Proof. If e is ,critical then ( ) > ( ).G e G

Suppose there exists a dominating set of eG

with scalar cardinality )(G . Then

)(=)( GeG , which is a contradiction.

conversely, suppose there is no dominating set of

eG with scalar cardinality )(G . Then

)()( GeG and e is critical .

Theorem 4.2 An edge e in G is criticaln

0(

criticaln

0) if and only if there does not exists

an 0n -set of G ( eG ) with scalar cardinality

)(0 Gn ( )(0 eGn ).

Proof. The proof is on similar lines to that of

theorem 4.1 and is omitted.

Corrolory 4.1 Let ),(= vue be critical . Then

any -set D of G contains exactly one of the

end verex of e .

Proof. Let D be a -set of G and let

Gvue ),(= . If D contains both u and v or

none of u and v , then D is a dominating set of

eG with scalar cardinality )(G and hence e is

not critical which is a contradiction by the

theorem 4.1.

Corrolory 4.2 Let ),(= vue be criticaln

0 .

Then any 0n -set D of G contains exactly one of

the end verex of e .

Proof. Let D be a 0n -set of G and let

Gvue ),(= . If D contains both u and v or

none of u and v , then D is an 0n -set of eG

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with scalar cardinality )(0 Gn and hence e is not

criticaln

0 which is a contradiction by the

theorem 4.2.

Observation 4.2 If an edge ),(= vue is

critical , then either

1. u and v are fixed or

2. u is fixed and v is totallyfree or

3. both u and v are free .

5. RELATIONSHIP BETWEEN 0n CRITICAL

AND 0 CRITICAL EDGES

For an critical0 edge e in a fuzzy graph

without isolated vertices, )()( 00 GeG .

Observation 5.1 If G has no fuzzy triagles, then

)(=)( 00 GGn . If r is the scalar cardinality of

the isolated vertices of )( eG , then

reGeGn )(=)( 00

Theorem 5.1 Let e be an edge in an M -strong

fuzzy graph G without any fuzzy triangle. Then

1. e is criticaln

0 if and only if e is

critical0 and e is not a pendent edge.

2. e is criticaln

0 if and only if e is a pendent

edge and if e does not form a component by

itself, then e is not an 0 .critical

Proof.

1. If e is criticaln

0 and G has no fuzzy

triagles, then

0 0

0

0

0

( ) = ( )

< ( )

= ( )

< ( )

G e r n G e

n G

G

G e

Therefore, if = 0r , then e is not a pendent

edge and if 0r , then e is critical0 .

Conversely, if e is critical0 and e is not

pendent edge, then )(<)( 00 GeG and

= 0r then by obsevation 5.1,

)(=)(<)(=)( 0000 GnGeGeGn .

Hence e is criticaln

0.

2. If e is criticaln

0, then

iGneGn )(=)( 00 (for some i)

iG )(= 0 .

by obsevation 5.1. reGG i )(=)( 00 .

(i) If ir = , then )(=)( 00 GeG (or)

(ii) suppose, If e is a component of G , then

ir > . Hence )(>)( 00 eGG .

Conversely, suppose e is a pendent edge such tha

if e does not form a component by itself, then e is

not criticaln 0 . Then either (i) or (ii) holds. If (i)

holds, it follows from the obsevation given in sec.

3.1 that )(>)( 00 GneGn and if (ii) holds, then

trivially e is criticaln

0.

REFERENCES

[1] K.R. Bhutani and A. Battou, On M-strong

fuzzy graphs, Information Sciences 155,

pp.103-109 (2003).

[2] George J.Klir and Bo Yuan, Fuzzy sets and

Fuzzy logic-Theory and Application, Prentice

Hall of India, 2005

[3] S. Ismail Mohideen and A. Mohamed Ismayil,

The vertex neighborhood number of a fuzzy

graph, Int. Jour. Mathematics Research, vol.2,

Number 3, pp. 71-76(2010).

[4] J.N. Mordeson, C.S. Peng, Operations on

fuzzy graphs, Information Sciences 79, 159-

170 (1994).

[5] A. Rosenfeld, Fuzzy graphs, in: L.A. Zedeh,

K.S. Fu, K. Tanaka, M. Shimura (Eds.), Fuzzy

sets and Their Applications to Cognitive and

Decision Processes, Academic Press,

NewYork, 1975, pp 77-95.

[6] Sampathkumar, E and Prabha S. Neeralagi,

Domination and neighborhood critical, fixed,

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free and totally free points, Indian J. of

Statistics, Special Vol.54,pp. 403-407(1992)

[7] Zimmermann,H.J, Fuzzy Set Theory and its

Application, Springer International Edition,

Fourth Edition(2001).

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On Pricing an Exotic Option in a Jump-Diffusion with a Switching Type Volatility

M. Reni Sagaya Raj, P. Manoharan and A. George Maria Selvam

Abstract - In Jump Diffusion models, the frequency of jumps is finite. They are the prototypes for a large class of more complex

models such as the stochastic volatility with jumps. In this paper, we consider the time horizon T (strike time) of a call option and

K (strike price) of the asset. The call option which is called a cliquet option, have its terminal claim given by

( , ) = max ( ),( ),0C T T KC

C T T S K S

where the date CT is called cliquet date. The results obtained can be applied to other areas like Number Theory, Stochastic

Processes and Probabilistic Models.

Index Terms – Jump diffusion, exotic option.

—————————— ——————————

1. INTRODUCTION

ESIDES the standard vanilla options, exotic

options such as barrier options, look back

options, floating-strike options and cliquet options

have become very popular financial trading

instruments. Unlike the vanilla options, the pay -

off functions of the exotic options are path-

dependent and hence the problem of obtaining

closed form solutions for each options are very

much complicated. Several studies have been

made in the last few decades in obtaining pricing

formulas for the exotic options.

However, not much work has been done for

pricing exotic options in stochastic volatility

models. Accordingly, we present the problem

of pricing a cliquet option when the underlying

asset price satisfies a jump-diffusion equation and

the volatility changes according to the occurrence

of the jumps in the asset price.

——————————————

M. Reni Sagayaraj is serving in the Department of Mathematics, Sacred

HeartCollege, Tirupattur, India. E-mail: reni.sagaya@gmail.com

P. Maoharan is serving in the Department of Mathematics, Sacred

HeartCollege, Tirupattur, India.

A. George Maria Selvam is serving in the Department of Mathematics,

Sacred HeartCollege, Tirupattur, India.

2. FINANCIAL MARKET MODEL

Let T be a positive constant representing the time

horizon and let the market consist of a risk-free

asset (bond) and a risky asset (stock). At time t ,

let the price of the bond be tB and that of the

stock be tS . We assume that 1=0B and AS =0 .

Let tB satisfy the equation

tt rBdB = (1)

where r is a positive constant representing the

risk-free interest rate of the bond price. Then rt

t eB = . We assume that the price tS of the risky

asset satisfies the stochastic jump-diffusion

differential equation

1

2

, 0 ,=

, ,

t t ct

t t ct

dt dW dN t TdS

dt dW dN tS

(2)

where we have assumed

21,, and are positive constants;

tW is a standard Brownian motion on a

probability space ),,( WWW PF ;

tN is a Poisson process on a probability

space ),,( NNN PF ;

cT is random variable independent of tW

and tN; tW

and tN are independent of

each other.

B

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Let the Poisson process tN be of constant

intensity and the probability density function of

cT be )(uf . Let Tt

W tF 0 be the natural

filtration generated by tW and Tt

NtF 0 be that

generated by tN . Let ),,( PF be the product

space formed by ),,( WWW PF and

),,( NNN PF . Let Tt

tF 0 be the filtration

generated by the direct product of Tt

W tF 0 and

Tt

NtF 0 . Then the asset process TttS0

is

defined on ),,( PF . The equation (1) can be

written as

,<,0= TtdNdWdtS

dSttt

t

t (3)

where c

Ttc

Ttt II 201= with AI denoting

the indicator function of the set A defined by

1( ) =

0A

if x AI x

otherwise

3. AN EQUIVALENT MARTINGALE

MEASURE AND THE STICK PRICE

We now proceed to solve the equation (3) for the

asset price tS and obtain an equivalent martingale

measure Q under which the discounted asset price

becomes a martingale. For this, we let

tNM tt = Then the equation (3) becomes

,<,0)(= TtdMdWdtS

dSttt

t

t (4)

Define the discounted price of the asset

t

rt

t

tt Se

B

SS ==~

Then the equation (4) becomes

,<,0)(=~

~

TtdMdWdtrS

Sdttt

t

t (5)

although tW and tM are P martingale in the

above equation (5), the process tS~

is not a P

martingale. We now seek a probability measure

Q such that tS~

is a Q martingale. First, for each

pair of constants and with 0 , we define

a process tL defined by

ttt NWtL log2

1)(1exp= 2

(6)

Then, we have

1 2 1 2(1 )( ) 2 2

(1 ) log

=0

(1 )

=0

(1 ) (1 )

[ ] = [(log ) ]

( )

!

( )

!

1

t tP

t t

nt n t

n

nt t

n

t t

E L e e E N

te e e

n

te e

n

e e

For each tFA , we define a set function Q such

that

)(=)( dPLAQ t

A

Then Q is a probability measure equivalent to P

such that the Radon-Nikodym derivative tLdP

dQ=

We define

.=,= )()( tNMtWW t

Q

tt

Q

t

Then, with respect to the measure Q , we note that

( ) ( )

( )

( )

( ) 2

1 2( ) 2

[ ]

[ ]

[{ } ]

1[( )exp[{ (1 ) }

2

(log ) ]]

[( ) ]

Q Q

t

Q

t

P

t t

P

t

t t

tP W

t

E W

E W t

E W t L

E W t t

W N

E W t e e

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0.=][ )()( Q

t

Q WE (7)

Similarly, we note that ( ) ( )

( )

( )

( ) 2

( ) (1 )

[ ]

[ ]

[( ) ]

1[( )exp[{ (1 ) }

2

(log ) ]]

[( ) ]

Q Q

t

Q

t

P

t t

P

t

t t

NP t tt

E M

E N t

E N t L

E N t t

W N

E N t e

0==][ )()( ttME Q

t

Q (8)

Further the process Q

tW is a standard Brownian

Motion with respect to the measure Q . The

equations (5) becomes

( )

( )

( )

( )

( ) { }=

{ }

( )=

Q

t t

Qt t

Q

t t t

Q

t

r dt d W tdS

S d M t t

r dt dW

dM

(9)

We eliminate the presence of in (9) by

choosing and such that

0= tr (10)

Then from the equation (10), we obtain

t

r

)(=

(11)

Then the equation (9) yields

)()(=~

~Q

t

Q

tt

t

t dMdWS

Sd (12)

The equation (12) clearly establishes the fact that

tS~

is a Q martingale. We solve the equation

(12) by putting tt SX~

log= and noting the fact that

( )

2( )

( )

( )

= ( ),

= ,

= ,

= , = 2,3,...

Q

t

Q

t

Q

t t

nQ

t t

dw o dt

dw dt

dM dN dt

dM dN n

We obtain the equation (12) that

2 3

( ) ( ) 2

3 4

= log 1

1 1= ...

2 3

1

2=

1 1...

3 4

tt

t

t t t

t t t

Q Q

t t t t

t t

dSdX

S

dS dS dS

S S S

dW dM dt

dN dN

t

Q

tttt dNdWdtdX )(1log2

1= )(2

(13)

When cTt 0 , we have from the equation (13),

t

Q

tt NWtXX )(1log2

1= )(

1

2

10

That is, we have

2

0 1

( )

1

1log = log

2

log(1 )

t

Q

t t

S S t

W N

(14)

Similarly, when TtTc , we have from the

equation (13),

2

2

( ) ( )

2

1log = log ( )

2

log(1 ) ( )

t T Cc

Q Q

t TC

t TC

S S t T

W W

N N

(15)

Using (14) and (15), we can solve the problem of

pricing a cliquet option.

REFERENCES

[1] Chesney, M. and Jeanblanc, M. (2004):

Pricing American currency options in an

exponential Levy model, Applied

Mathematical Finance, 11, 207 - 225.

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[2] Cont, R., and Tankov, P.(2004): Financial

modeling with jump processes, Chapman and

Hall, CRC.

[3] Cox, J.C., and Ross, S.A. (1976): Them

valuation of options for alternative stochastic

processes, Journal of Financial Economics, 3,

145 - 166.

[4] Devydov, D. and Linetsky, V.(2001): Pricing

and hedging path - dependent options under

the CEV process, Management Science, 47,

949 - 965.

[5] Duffle, D.(2000): Dynamic Asset Pricing

Theory, 3rd Edition, Princeton University

Press.

[6] Etheridge, A. (2002): A course in Financial

Calculus, Cambridge University Press.

[7] Geman, H. and Yor, M. (1996): Pricing and

Hedging double - barrier options: A

probabilistic approach, Math. Fin., 6, 365 -

378.

[8] Gukhal, C.R. (2001): Analytical valuation of

American options on jump - diffusion

processes, Mathematical Finance, 11, 97 -

115.

[9] Haifeng, Y., Jianqi, Y. and L. Limin (2005):

Pricing cliquet Options in Jump - Diffusion

Models, Stochastic Models, 21, 875 - 884.

[10] Heston, S.L.(1992): A closed form solution

for options with stochastic volatility with

applications to bond and curency options,

Rev. Financial Studies, 6, 333 - 343.

[11] Kallianpur, G. and R.L. Karandikar (2000):

Introduction to Option Pricing Theory

Birkhauser.

[12] Karatzas, I. (1998): On Pricing of American

Options, Appl. Math. Optim., 17, 37 - 60.

[13] Kou, S.G. (2002): A Jump - Diffusion Model

for Option Pricing, Management Science,

48, 1086 - 1101.

[14] Mercurio, F., and Runggaldier, W.J. (1993):

Option pricing for jump - diffusions:

approximations and their interpretation,

Mathematical Finance, 3, 191 - 200.

[15] Myneni, R. (1992): The pricing of American

Option, Ann. Appl. Probob., 2, 1 - 23.

[16] Schoutens, W. (2006): Exotic options under

Levy models: An overview, Journal of

Computational and Applied Mathematics,

189, 526 - 538.

[17] Sharpe, W.F. (1964): Capital Asset Prices:

A theory of market equilibrium under

conditions of risk, J. Financial Studies, 4,

425 - 442.

[18] Smith, C.W. JR. (1976): Option Pricing: A

Review, J. Financial Studies, 3, 3 - 51.

[19] Stein, E.M., and Stein, C.J. (1991): Stock

price distributions with stochastic volatility:

an analytic approach, Rev. Financial

Studies, 4, 727 - 752.

[20] Wiggins, J.B. (1987): Options values under

stochastic volatility: Theory and empirical

estimates, J. Financial Economics, 19, 351 -

372.

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Some Properties of a Cumulative Damage Threshold Crossing Model with Underlying Birth

Process

U. Rizwan and S. Kasthuri

Abstract - In this paper, we describe the cumulative damage random threshold crossing model with underlying birth process and

give the reliability of the system under this model. We present some stochastic properties of the survival function for this model. It

is shown that some partial orderings, namely the stochastic ordering and the failure rate ordering are preserved among the two

systems under consideration. Finally, some open problems related to the presentation of other ageing properties, for our model

are given.

Index Terms – IFR, NBU, PF2, Stochastic Ordering, Failure Rate Ordering, Pure Birth Process.

—————————— ——————————

1. INTRODUCTION

ONSIDER a system subject to shocks

occurring randomly in time. Each shock

deteriorates the system and the damage

accumulates. The system fails when the total

accumulated damage exceeds a certain threshold

𝑎. Suppose 𝜉𝑖(≥ 0) is the amount of damage

caused by the ith

shock and 𝑁(𝑡) ; 𝑡 ≥ 0 is the

number of shocks the system is subject to in the

time interval 0, 𝑡 . Here 𝜉𝑖 and 𝑁 are

independent. Then the probability 𝐻 (𝑡) that the

system will survive beyond time t is

𝐻 (𝑡) = 𝑃(𝑁(𝑡) = 𝑘)𝑃 (𝑘)∞𝑘=0 , 𝑡 ≥ 0, (1.1)

where 𝑃 (𝑘) = 𝑃(𝜉1 + 𝜉2 + ⋯ + 𝜉𝑘 ≤ 𝑎). Shock

models of this kind have been studied by a

number of authors, in which the number of shocks

are governed by a Poisson process. But the

magnitudes of the damages were not assumed to

grow along with time. Ebrahimi [1999] has

assumed that the magnitudes of the shocks vary

with time.

——————————————

U. Rizwan is serving in the Department of Mathematics, Islamiah College, Vaniyambadi, India. E-mail: rizwanmaths@yahoo.com

S. Kasthuri is pursuing Ph.D. degree in Mathematics, Islamiah College, Vaniyambadi, India and is serving in the Department of Mathematics, Auxillium College, Vellore, India. E-mail: kasthuri_aux@yahoo.com

In this paper, we study the cumulative damage

random threshold crossing model in which shocks

occur according to a nonstationary pure birth

process of the following sort : (A-Hameed

[1975]) shocks occur according to a Markov

process; given that k shocks have occurred in

0, 𝑡 , the probability of a shock occurring in

(𝑡, 𝑡 + ∆] is 𝜆𝑘𝜆(𝑡)∆ + 𝜊(∆), while the

probability of more than one shock occurring in

(𝑡, 𝑡 + ∆] is 𝜊 ∆ . Further the damages vary with

time which is different from the existing models.

This is referred to as Pure Birth Shock Model.

Remark 1. In the stationary pure birth process,

given that 𝑘 shocks have occurred in 0, 𝛬 𝑡 , the

probability of a shock occurring in 𝛬 𝑡 , 𝛬 𝑡 +𝜆(𝑡)∆ is of the same form : 𝜆𝑘𝜆(𝑡)∆+𝜊(∆),

where 𝛬 𝑡 = 𝜆 𝑥 𝑡

0𝑑𝑥. It follows that the Pure

Birth Shock Model may be obtained from the

stationary pure birth process by the transformation

𝑡 → 𝛬 𝑡 .

For this shock model, the survival function 𝐻 (𝑡)

in (1.1) can be written as

𝐻 (𝑡) = 𝑆 (𝛬 𝑡 ) and 𝑆 (𝑡) = 𝑧𝑘 𝑡 ∞𝑘=0 𝑃 (𝑘),

where 𝑧𝑘 𝑡 = 𝑃(exactly 𝑘 shocks have occurred

in (0, 𝑡] where 𝜆(𝑡) = 1).

C

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This means that 𝑆 (∙) is the survival function when

the shocks occur according to a stationary birth

process for which the interarrival times between

the shocks number 𝑘 and 𝑘 + 1, 𝑘 = 0, 1,2, . . ., are independent and exponentially distributed with

mean 1/𝜆𝑘 and 1/𝜆𝑘 ∞𝑘=0 = ∞.

In this paper, we consider a model, which is a

more general shock model than that of Esary et al

[1973] and Ebrahimi [1999].

The rest of the paper is organized as follows : In

Section 2, we first give the concepts that are

needed in the ensuing Sections. We also describe

the model and give the reliability of the system

under this model in this Section. Some stochastic

properties of the survival function are given in

Section 3. It is shown in Section 4, that some

partial orderings, namely the stochastic ordering

and the failure rate ordering are preserved among

the two systems under consideration. Finally, in

Section 5, we raise some questions regarding the

preservation of other ageing properties, for our

model.

2. PRELIMINARIES

In this Section, we first give some definitions and

describe the model.

Definition 2.1. The failure rate 𝑟(∙) of a random

variable 𝑇 with distribution function 𝐹(∙) is

defined by

𝑟 𝑡 = 𝑙𝑖𝑚∆01

∆𝑃 𝑡 < 𝑇 ≤ 𝑡 + ∆∣ 𝑇 > 𝑡

=𝑓 𝑡

𝐹 𝑡 , 𝑡 ≥ 0 ,

provided the probability density function 𝑓(∙)

exists, where 𝐹 𝑡 < 1, for all 𝑡 ≥ 0.

Definition 2.2. Let 𝑋 and 𝑌 be two non-negative

random variables with corresponding distribution

functions 𝐹(∙) and 𝐺(∙). Then 𝑋 is said to be

stochastic larger than 𝑌, denoted by 𝑋≥𝑠𝑡𝑌 ,

𝑖𝑓 𝐹 𝑡 ≤ 𝐺 𝑡 , 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 ≥ 0.

Definition 2.3. Let 𝑋 and 𝑌 be two non-negative

random variables with corresponding distribution

functions 𝐹(∙) and 𝐺(∙). Then 𝑋 is said to be

larger than 𝑌 in failure rate order, denoted by

𝑋≥𝐹𝑅𝑌 , 𝑖𝑓 𝐹 (𝑡)/𝐺 (𝑡) is non-decreasing in 𝑡 ≥ 0,

or equivalently, 𝑟𝐹 𝑡 ≤ 𝑟𝐺 𝑡 , for all 𝑡 ≥ 0,

where 𝑟𝐹 ∙ 𝑎𝑛𝑑 𝑟𝐺 ∙ denote the failure rates of

𝐹(∙) and 𝐺(∙).

Definition 2.4. A life distribution 𝐹(∙) and its

survival function 𝐹 = 1 − 𝐹 with support

𝑆 = 𝑡 ∶ 𝐹 > 0 and finite mean 𝜇 = 𝐹 𝑥 𝑑𝑥∞

0

are said to be

(i) increasing failure rate (IFR), if the

conditional survival function 𝐹 𝑥+𝑡

𝐹 𝑡 is

decreasing in t, whenever 𝑥 > 0 and

𝑡 ∊ 𝑆.

(ii) increasing failure rate average (IFRA),

if −𝑙𝑛𝐹 𝑡

𝑡 is increasing in S.

(iii) new better than used (NBU), if

𝐹 𝑥 + 𝑦 ≤ 𝐹 𝑥 𝐹 𝑦 ,

for all 𝑥 ≥ 0 and 𝑦 ≥ 0.

(iv) new better than used in expectation (NBUE), if

𝐹 𝑥 + 𝑦 𝑑𝑦∞

0≤

𝐹 𝑥 𝐹 𝑦 𝑑𝑦∞

0,

for all 𝑥 ≥ 0.

(v) decreasing mean residual life (DMRL), if

1

𝐹 𝑡 𝐹 𝑥 𝑑𝑥

∞

0

is decreasing on S.

(vi) harmonic new better than used in expectation (HNBUE), if

𝐹 𝑥 𝑑𝑥∞

𝑡≤ 𝜇 𝑒𝑥𝑝 −

𝑡

𝜇 , for

𝑡 ≥ 0.

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Definition 2.5. A function 𝑓(∙) is super additive, if

𝑓 𝑡 + 𝑢 ≥ 𝑓 𝑡 + 𝑓 𝑢 , for 𝑡 ≥ 0, 𝑢 ≥ 0.

Definition 2.6. A function 𝑓(𝑥) defined for x in

−∞, ∞ is a P𝑜 lya frequency function of order 2 𝑃𝐹2 if 𝑓 𝑥 ≥ 0 for all x and

𝑓 𝑥1 − 𝑦1 𝑓 𝑥1 − 𝑦2

𝑓 𝑥2 − 𝑦1 𝑓 𝑥2 − 𝑦2 ≥ 0 ,

whenever −∞ < 𝑥1 < 𝑥2 < ∞ and −∞ < 𝑥1 <𝑥2 < ∞.

Suppose that shocks occur randomly in time in

accordance with a pure birth shock process

𝑁(𝑡) ; 𝑡 ≥ 0 as described in Section 1 here

𝑁(𝑡) denotes the number of shocks that have

occurred upto time t. Suppose 𝑇𝑖 , 𝑖 = 1, 2, . . . are

the shock arrival times and 𝑋𝑖 𝑡 − 𝑇𝑖 , 𝑡 ≥ 𝑇𝑖, is

the evolution of the damage to the system from

the ith

shock. Then

𝑆 𝑡 = 𝑋𝑖 𝑡 − 𝑇𝑖 𝑁 𝑡 𝑖=1 , 𝑡 ≥ 0 (2.1)

is the total damage process. It is assumed here that

the processes 𝑋1 𝑡 , 𝑋2 𝑡 , . . . are independent

and all of them have non-decreasing sample paths,

that is, 𝑋𝑖 𝑡 = 𝑋𝑖 (0, 𝑡] , 𝑡 ≥ 0. The above

model is referred to as explosive Poisson shot

noise process. For more details of this process one

may refer to Kl𝑢 ppelberg and Mikosch[1995].

This model may be used in risk analysis, where

𝑁 𝑡 is the number claims in [0, 𝑡], 𝑋𝑖 𝑡 is the ith

claim and 𝑆 𝑡 is the total claim up to time 𝑡.

Suppose that the system fails when 𝑆 𝑡 , in (2.1),

exceeds a known threshold 𝑎. Then the time to

system failure, 𝑇 𝑎 , can be written as

𝑇 𝑎

= 𝑖𝑛𝑓 𝑡 ∶ 𝑆 𝑡 > 𝑎 𝑖𝑓 𝑡 ∶ 𝑆 𝑡 > 𝑎 ≠ ∅

∞ 𝑖𝑓 𝑡 ∶ 𝑆 𝑡 > 𝑎 = ∅

and the reliability, 𝐹 𝑎 𝑡 , of the system is

𝐹 𝑎 𝑡 = 𝑃 𝑇 𝑡 > 𝑡

= 𝑃 𝑆 𝑡 ≤ 𝑎

= 𝑃 𝑋𝑖 𝑡 − 𝑇𝑖 𝑁 𝑡 𝑖=1 ≤ 𝑎 .

Remark 2. When there is no possibility to

determine the threshold 𝑎, then the failure time of

the system is 𝑇 = 𝑖𝑛𝑓 𝑡 ∶ 𝑆 𝑡 > 𝑉 , where 𝑉 is a

positive random threshold with known probability

density function 𝑔𝑉 ∙ and the survival function

𝐹 𝑣 𝑡 𝑔𝑉 𝑣 𝑑𝑣∞

0.

Here, 𝑉 is independent of 𝑆(𝑡).

We assume that for any 𝑛, 𝑋𝑛 𝑡 is non-

decreasing and cadlag. Hence the realizations of

𝑋𝑛 𝑡 are measure defining functions. The process 𝑆(𝑡) ; 𝑡 ≥ 0 is almost surely finite for every

fixed 𝑡 and defines a random measure 𝑆 on Borel

sets.

We now give the reliability, 𝐹 𝑎 𝑡 , of the system

𝐹 𝑎 𝑡 = 𝑃 𝑋𝑖 𝑡 − 𝑇𝑖

𝑁 𝑡

𝑖=1

≤ 𝑎

= 𝑒𝑥𝑝 −𝜆0𝑡 1 + 𝜆0𝜆1 …𝜆𝑘−1∞𝑘=1 ×

0𝑡0𝑢1…0𝑢𝑘−1𝑃𝑖=1𝑘𝑋𝑖𝑢𝑖≤𝑎 𝑒𝑥𝑝𝑗=1𝑘𝜆𝑗−1−𝜆𝑗𝑢𝑗𝑑𝑢𝑘… 𝑑𝑢1

2.2

Here 𝑢𝑖 = 𝑡 − 𝑡𝑖 and 𝑢0 = 𝑡. The number of

shocks occured upto time 𝑡, 𝑁(𝑡) ; 𝑡 ≥ 0 is a

pure birth process with birth rates 𝜆𝑘 . Note here

that when 𝜆𝑖‘s are all equal and equal to 𝜆, then

(2.2) reduces to (2.1) of Ebrahimi [1999] and in

this case, the pure birth process becomes Poisson

process.

Remark 3. When 𝑋 𝑡 , 𝑋1 𝑡 , 𝑋2 𝑡 , . . . are

identical processes and 𝑌𝑖 𝑡 =𝑑 1

𝑡 𝑋 𝑢 𝑑𝑢

𝑡

0, where

=𝑑 denotes equality in distribution, and i = 1, 2, .

then 𝐹 𝑎 𝑡 can be written as

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𝐹 𝑎 𝑡

= 𝑃 𝑌𝑖 𝑡

𝑁 𝑡

𝑖=1

≤ 𝑎

= 𝑒𝑥𝑝 −𝜆0𝑡 + 𝑃 𝑌𝑖 𝑡

𝑘

𝑖=1

≤ 𝑎

∞

𝑘=1

× 𝑃 𝑁 𝑡 = 𝑘

= 𝑒𝑥𝑝 −𝜆0𝑡 + 𝐴𝑘 𝑗

𝑘

𝑗=0

𝑒𝑥𝑝 −𝜆𝑗 𝑡

∞

𝑘=1

× 𝑃 𝑌𝑖 𝑡

𝑘

𝑖=1

≤ 𝑎

(2.3)

where

𝐴𝑘 𝑗

= 𝜆0𝜆1 … 𝜆𝑘−1

𝜆0−𝜆𝑗 𝜆1−𝜆𝑗 . . . 𝜆𝑗−1−𝜆𝑗 𝜆𝑗+1−𝜆𝑗 . . . 𝜆𝑛−𝜆𝑗

(2.4)

Here we note that 𝑆 𝑡 = 𝑌𝑖 𝑡 𝑁 𝑡 𝑖=1 where 𝑆 𝑡

is given by (2.1).

For the survival function 𝐹 𝑎 ∙ defined in (2.2), a

property of pure birth process that is needed in

obtaining the qualitative properties of 𝐹 𝑎 𝑡 ,

proved by Rizwan [2001], is given in the

following result.

Theorem 2.1. Suppose that k events have

occurred during 0, 𝑡 and 𝑘 + 𝑗 events have

occurred during [0, 𝑡 + 𝑥], 𝑡, 𝑥 > 0 in a pure

birth process with rates 𝜆𝑘 . Then the conditional

joint density function of 𝑇1, . . . , 𝑇𝑘+𝑗 , the

successive times of occurrence, given that

𝑁(𝑡) = 𝑘 and 𝑁(𝑡 + 𝑥) = 𝑘 + 𝑗 is given by

𝑓(𝑡1, . . . , 𝑡𝑘 , 𝑡𝑘+1 , . . . , 𝑡𝑘+𝑗 ∣ 𝑁(𝑡) = 𝑘, 𝑁(𝑡 + 𝑥) = 𝑘 + 𝑗)

=

𝜆0𝜆1 … 𝜆𝑘…𝜆𝑘+𝑗−1𝑒𝑥𝑝 −𝜆0 𝑡+𝑥 𝑒𝑥𝑝 − 𝜆𝑖−1−𝜆𝑖 𝑘+𝑗𝑖=1

𝑡+𝑥−𝑡𝑖

𝐴𝑘 𝑣

𝑒𝑥𝑝 −𝜆𝑣𝑡 𝑘𝑣=0 𝐴

𝑗 𝑣

𝑒𝑥𝑝 −𝜆𝑣𝑥 𝑗𝑣=0

(2.5)

where 𝐴𝑛 𝑗

are defined as in (2.4).

3. STOCHASTIC PROPERTIES OF 𝐹 𝑎 𝑡

In this Section, we shall prove some stochastic

properties of the survival function 𝐹 𝑎 𝑡 for the

cumulative damage random threshold crossing

model with underlying birth process. We shall

now prove the following result.

Theorem 3.1. Suppose 𝜆𝑘 ↑, 𝛬 𝑡 be super

additive, 𝑃(𝑘) be NBU and 𝑋 𝑡 , 𝑋1 𝑡 , 𝑋2 𝑡 , . .

. are independent and identical processes. Then

the survival function 𝐹 𝑎 𝑡 in (2.2) is NBU.

Proof. Define

𝑧𝑘 𝑡 = 𝑃 𝑁 𝑡 = 𝑘 , 𝑘 = 0, 1, 2, .. .

Then by Taylor and Karlin [1994], p 303,

𝑧𝑘′ 𝑡 = −𝜆𝑘𝑧𝑘 𝑡 + 𝜆𝑘−1𝑧𝑘−1 𝑡 , (𝑘 ≥ 1),

and 𝑧0′ 𝑡 = −𝜆0𝑧0 𝑡 .

Also, let

𝐹 1𝑎 𝑡 = 𝑧𝑘 𝑡 ∞𝑘=0 𝑃 𝑡 .

Then by Remark 1, we may write 𝐹 𝑎 𝑡 =

𝐹 1𝑎 𝛬 𝑡 . For 𝑥, 𝑡 ≥ 0, using Theorem 2.1, we

may write

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𝐹 1𝑎 𝑡 + 𝑥 = 𝑧𝑘 𝑡

∞

𝑗 =0

∞

𝑘=0

𝑧𝑘 𝑥 𝐴𝑘 𝑣

𝑒𝑥𝑝 −𝜆𝑣𝑡

𝑘

𝑣=0

𝐴𝑗 𝑣

𝑒𝑥𝑝 −𝜆𝑣𝑥

𝑗

𝑣=0

× …

𝑡

𝑡1

𝑡

0

…

𝑡+𝑥

𝑡𝑘+1

𝑡+𝑥

𝑡

𝑡

𝑡𝑘−1

𝑃 𝑋𝑖 𝑡 + 𝑥 − 𝑡𝑖

𝑘+𝑗

𝑖=1

≤ 𝑎

𝑡+𝑥

𝑘+𝑗−1

× 𝑑𝑡𝑘+𝑗 …𝑑𝑡𝑘+1𝑑𝑡𝑘 …𝑑𝑡1 3.1

Let 𝑢𝑖 = 𝑡 − 𝑡𝑖 , 𝑖 = 1,2, … , 𝑘𝑡 + 𝑥 − 𝑡𝑖 𝑖 = 𝑘 + 1, … , 𝑘 + 𝑗

Then (3.1) becomes

𝐹 1𝑎 𝑡 + 𝑥 = 𝑒𝑥𝑝 −𝜆0 𝑡 + 𝑥 𝜆1 … 𝜆𝑘𝜆𝑘+1 …𝜆𝑘+𝑗∞𝑗=0

∞𝑘=0

× …

𝑢1

0

𝑡

0

…

𝑢𝑘+1

0

𝑥

0

𝑢𝑘−1

0

𝑒𝑥𝑝 𝜆𝑖−1 − 𝜆𝑖

𝑘

𝑖=1

𝑢𝑖 + 𝑥 × 𝑒𝑥𝑝 𝜆𝑖−1 − 𝜆𝑖

𝑘+𝑗

𝑖=𝑘+1

𝑢𝑖

𝑢𝑘+𝑗−1

0

× 𝑃 𝑋𝑖 𝑢𝑖 + 𝑥

𝑘

𝑖=1

+ 𝑋𝑖 𝑢𝑖

𝑘+𝑗

𝑖=𝑘+1

≤ 𝑎 × 𝑑𝑢𝑘+𝑗 …𝑑𝑢1

≤ 𝑒𝑥𝑝 −𝜆0𝑡 𝑒𝑥𝑝 −𝜆0𝑥 𝜆1 … 𝜆𝑘𝜆𝑘+1 …𝜆𝑘+𝑗

∞

𝑗 =0

∞

𝑘=0

× …

𝑢1

0

𝑡

0

…

𝑢𝑘+1

0

𝑥

0

𝑢𝑘−1

0

𝑒𝑥𝑝 𝜆𝑖−1 − 𝜆𝑖

𝑘+𝑗

𝑖=𝑘+1

𝑢𝑖

𝑢𝑘+𝑗−1

0

× 𝑃 𝑋𝑖 𝑢𝑖 + 𝑥

𝑘

𝑖=1

≤ 𝑎

× 𝑃 𝑋𝑖 𝑢𝑖

𝑘+𝑗

𝑖=𝑘+1

≤ 𝑎 𝑑𝑢𝑘+𝑗 …𝑑𝑢1

(since 𝑃 𝑘 is NBU)

≤ 𝑒𝑥𝑝 −𝜆0𝑡 𝑒𝑥𝑝 −𝜆0𝑥 𝜆1 … 𝜆𝑘𝜆𝑘+1 …𝜆𝑘+𝑗

∞

𝑗 =0

∞

𝑘=0

× …

𝑢1

0

𝑡

0

…

𝑢𝑘+1

0

𝑥

0

𝑢𝑘−1

0

𝑒𝑥𝑝 𝜆𝑖−1 − 𝜆𝑖

𝑘+𝑗

𝑖=𝑘+1

𝑢𝑖

𝑢𝑘+𝑗−1

0

× 𝑃 𝑋𝑖 𝑢𝑖

𝑘

𝑖=1

≤ 𝑎 𝑃 𝑋𝑖 𝑢𝑖

𝑘+𝑗

𝑖=𝑘+1

≤ 𝑎 𝑑𝑢𝑘+𝑗 …𝑑𝑢1

= 𝐹 1𝑎 𝑡 𝐹 1𝑎 𝑥 ,

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since {𝑋𝑛(𝑡) ; 𝑛 = 1, 2, . . . } is a non-decreasing

process. Since 𝛬(𝑡) is super additive and 𝐹 1𝑎 𝑡 is

increasing, it follows that 𝐹 𝑎 𝑡 is NBU. ■

Remark 4. When {𝑋𝑛(𝑡) ; 𝑛 = 1, 2, . . . } is a

non-identical process, then the last inequality in

the proof of Theorem 3.1 be comes less than, and

in this case, the shocks become increasingly

effective in causing damage.

Remark 5. Assume that shocks occur randomly

in time in accordance with a pure birth process.

Further, the threshold is a positive random

variable 𝑉 with cumulative distribution function

𝐺 ∙ and assume that the process {𝑋𝑛(𝑡) ; 𝑛 = 1, 2, . . . } is independent of the threshold 𝑉. The

survival probability 𝐹 𝑡 for the period [0, 𝑡] is

now given by

𝐹 𝑡

= 𝑒𝑥𝑝 −𝜆0𝑡 1 + 𝜆1 …𝜆𝑘

∞

𝑘=1

× …𝑢1

0

𝑡

0

∞

0

𝑃 𝑋𝑖 𝑢𝑖

𝑘

𝑖=1

< 𝑣

𝑢𝑘

0

× 𝑒𝑥𝑝 𝜆𝑗−1 –𝜆𝑗

𝑘

𝑗 =1

𝑢𝑗 𝑑𝑢𝑘 …𝑑𝑢1 𝑑𝐺 𝑣

= 𝑒𝑥𝑝 −𝜆0𝑡 1 + 𝜆1 …𝜆𝑘

∞

𝑘=1

× …𝑢1

0

𝑡

0

𝑒𝑥𝑝 𝜆𝑗−1

𝑘

𝑗 =1

𝑢𝑘

0

− 𝜆𝑗 𝑢𝑗 𝐺 𝑣 𝑑

𝑑𝑣𝑃 𝑋𝑖 𝑢𝑖

𝑘

𝑖=1

∞

0

< 𝑣 𝑑𝑣 𝑑𝑢𝑘 …𝑑𝑢1

= 𝑒𝑥𝑝 −𝜆0𝑡 1

+ 𝜆1 …𝜆𝑘

∞

𝑘=1

…𝑢1

0

𝑡

0

𝐸 𝐺 𝑋𝑖 𝑢𝑖

𝑘

𝑖=1

𝑢𝑘

0

× 𝑒𝑥𝑝 𝜆𝑗−1 − 𝜆𝑗

𝑘

𝑗 =1

𝑢𝑗 𝑑𝑢𝑘 …𝑑𝑢1

Theorem 3.2. Suppose 𝜆𝑘 ↑, 𝛬 𝑡 is super

additive and 𝐺 𝑣 be NBU. Then 𝐹 𝑎 𝑡 is NBU.

Proof. Let 𝑧𝑘 𝑡 be defined as in Theorem 3.1

and let

𝐹 1 𝑡 = 𝑧𝑘 𝑡

∞

𝑘=0

𝑃 𝑘

Then by Remark 1, we may write

𝐹 𝑎 𝑡 = 𝐹 1 𝛬 𝑡 .

Consider

𝐹 1 𝑡 + 𝑥

= 𝑒𝑥𝑝 −𝜆0 𝑡 + 𝑥

× 𝜆1 … 𝜆𝑘𝜆𝑘+1 …𝜆𝑘+𝑗

∞

𝑗 =0

∞

𝑘=0

× …

𝑢1

0

𝑡

0

…

𝑢𝑘+1

0

𝑥

0

𝑢𝑘−1

0

𝐸 𝐺 𝑋𝑖 𝑢𝑖

𝑘

𝑖=1

𝑢𝑘+𝑗−1

0

+ 𝑥 + 𝑋𝑖 𝑢𝑖

𝑘+𝑗

𝑖=𝑘+1

× 𝑒𝑥𝑝 𝜆𝑖−1 − 𝜆𝑖

𝑘

𝑖=1

𝑢𝑖 + 𝑥

+ 𝜆𝑖−1 − 𝜆𝑖

𝑘+𝑗

𝑖=𝑘+1

𝑢𝑖 × 𝑑𝑢𝑘+𝑗 …𝑑𝑢1

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≤ 𝑒𝑥𝑝 −𝜆0𝑡 𝑒𝑥𝑝 −𝜆0𝑥 𝜆1 … 𝜆𝑘𝜆𝑘+1 …𝜆𝑘+𝑗

∞

𝑗 =0

∞

𝑘=0

× …

𝑢1

0

𝑡

0

…

𝑢𝑘+1

0

𝑥

0

𝑢𝑘−1

0

𝑒𝑥𝑝 𝜆𝑖−1

𝑘+𝑗

𝑖=𝑘+1

𝑢𝑘+𝑗−1

0

− 𝜆𝑖 𝑢𝑖 𝐸 𝑋𝑖 𝑢𝑖

𝑘

𝑖=1

× 𝑋𝑖 𝑢𝑖

𝑘+𝑗

𝑖=𝑘+1

𝑑𝑢𝑘+𝑗 …𝑑𝑢1

= 𝐹 1 𝑡 𝐹 1 𝑥

That is, 𝐹 1 𝑡 is NBU, and since 𝛬(𝑡) is super

additive, it follows that 𝐹 𝑎 𝑡 is NBU. ■

Remark 6. Although 𝐹 𝑎 𝑡 is NBU, it need not

be an IFR.

Theorem 3.3. Suppose 𝜆𝑘 ↑, 𝜆 𝑡 ↑ and

𝑌 𝑡 =1

𝑡 𝑋 𝑢 𝑑𝑢

𝑡

0. Suppose further that

(a) for any t, 𝐺𝑡 𝑢 = 𝑃 𝑌 𝑡 ≤ 𝑢 is PF2 ;

(b) for 𝑡1 ≤ 𝑡2, 𝐺𝑡1 𝑢 /𝐺𝑡2

𝑢 is non-increasing

in 𝑢 ;

(c) for any 𝑘 ≥ 0 and 𝑡1 < 𝑡2,

𝑃 𝑌𝑖 𝑥 + 𝑡1

𝑘

𝑖=1

≤ 𝑎 /𝑃 𝑌𝑖 𝑥 + 𝑡2

𝑘

𝑖=1

≤ 𝑎

is non-decreasing in 𝑥 ; and

(d) for 𝑡1 ≤ 𝑡2, 𝑘1 ≤ 𝑘2,

𝑃 𝑌𝑖 𝑡1 𝑘2𝑖=1 ≤ 𝑎

𝑃 𝑌𝑖 𝑡1 𝑘1𝑖=1 ≤ 𝑎

×𝑃 𝑌𝑖 𝑡2

𝑘2𝑖=1 ≤ 𝑎

𝑃 𝑌𝑖 𝑡2 𝑘1𝑖=1 ≤ 𝑎

≤ 𝑡2

𝑡1 𝑘2−𝑘1

Then 𝐹 𝑎 𝑡 in (2.3) is IFR.

Proof. We need to prove that the determinant

𝐷 = 𝐹 𝑎 𝑥 + 𝑡1 𝐹 𝑎 𝑥 + 𝑡2

𝐹 𝑎 𝑡1 𝐹 𝑎 𝑡2 ≥ 0

for t1 < t2, 𝐹 𝑎 𝑡2 > 0 and 𝑥 ≥ 0. Let 𝑧𝑘 𝑡 be

defined as in Theorem 3.1. Then by (2.3) we have

𝐷 =

𝑧𝑘1 𝑥 + 𝑡1

∞

𝑘1=0

𝑃 𝑌𝑖 𝑥 + 𝑡1

𝑘1

𝑖=1

≤ 𝑎 𝑧𝑘1 𝑥 + 𝑡2

∞

𝑘1=0

𝑃 𝑌𝑖 𝑥 + 𝑡2

𝑘1

𝑖=1

≤ 𝑎

𝑧𝑘2 𝑡1

∞

𝑘2=0

𝑃 𝑌𝑖 𝑡1

𝑘2

𝑖=1

≤ 𝑎 𝑧𝑘2 𝑡2

∞

𝑘2=0

𝑃 𝑌𝑖 𝑡2

𝑘2

𝑖=1

≤ 𝑎

= 𝑧𝑘1

𝑥 + 𝑡1 𝑧𝑘1 𝑥 + 𝑡2

𝑧𝑘2 𝑡1 𝑧𝑘2

𝑡2

𝑘2<∞0≤𝑘1<

𝑃 𝑌𝑖 𝑥 + 𝑡1

𝑘1

𝑖=1

≤ 𝑎 𝑃 𝑌𝑖 𝑥 + 𝑡2

𝑘1

𝑖=1

≤ 𝑎

𝑃 𝑌𝑖 𝑡1

𝑘2

𝑖=1

≤ 𝑎 𝑃 𝑌𝑖 𝑡2

𝑘2

𝑖=1

≤ 𝑎

3.2

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by the basic composition theorem (Karlin, Total

Positivity [1968]).

For t1 < t2, the first determinant is non-negative

(Karlin and Proschan [1960], (Theorem 3) ). From

assumptions (a) to (c), it is easy to verify that

𝑃 𝑌𝑖 𝑥 + 𝑡1 𝑘1+𝑗𝑖=1 ≤ 𝑎

𝑃 𝑌𝑖 𝑥 + 𝑡2 𝑘1+𝑗𝑖=1 ≤ 𝑎

≥𝑃 𝑌𝑖 𝑡1

𝑘1+𝑗𝑖=1 ≤ 𝑎

𝑃 𝑌𝑖 𝑡2 𝑘1+𝑗𝑖=1 ≤ 𝑎

≥𝑃 𝑌𝑖 𝑡1

𝑘1𝑖=1 ≤ 𝑎

𝑃 𝑌𝑖 𝑡2 𝑘1𝑖=1 ≤ 𝑎

(3.3)

It follows from (3.3) that the second determinant

on the right hand side of (3.2) is non-negative and

by Remark 1, the proof is complete. ■

4. SOME PARTIAL ORDERINGS RELATED

TO THE MODEL

In this Section, we compare two systems having

different damage distributions, but

fixed identical threshold.

4.1 Stochastic Orderings

Suppose the i - th shock causes a damage 𝑋𝑖 and

𝑊𝑖 to the first and second systems, respectively

and each system fails if the corresponding

accumulated damage exceeds a common fixed

threshold 𝑎. Let 𝑇1 and 𝑇2 denote the failure

times of the system one and two, respectively.

Then, their respective survival distributions are

𝐹 1𝑎 𝑡 = 𝑃 𝑇1 > 𝑡

= 𝑒𝑥𝑝 −𝜆0𝑡 1 + 𝜆1 …𝜆𝑘

∞

𝑘=1

× …𝑢1

0

𝑡

0

𝑃 𝑋𝑖 𝑢𝑖

𝑘

𝑖=1

≤ 𝑎 × 𝑒𝑥𝑝 𝜆𝑗−1 − 𝜆𝑗

𝑘

𝑗 =1

𝑢𝑗 𝑑𝑢𝑘 …𝑑𝑢1

𝑢𝑘−1

0

4.1

and

𝐹 2𝑎 𝑡 = 𝑃 𝑇2 > 𝑡

= 𝑒𝑥𝑝 −𝜆0𝑡 1 + 𝜆1 …𝜆𝑘

∞

𝑘=1

× …𝑢1

0

𝑡

0

𝑃 𝑊𝑖 𝑢𝑖 ≤ 𝑎

𝑘

𝑖=1

× 𝑒𝑥𝑝 𝜆𝑗−1 − 𝜆𝑗

𝑘

𝑗 =1

𝑢𝑗 𝑑𝑢𝑘 …𝑑𝑢1

𝑢𝑘−1

0

4.2

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We shall now establish the following result.

Theorem 4.1. If 𝑋𝑖 𝑡 ≥ 𝑊𝑖 𝑡 , for 𝑖 = 1, 2, . .. and for each 𝑡 ≥ 0, then 𝑇1≤

𝑠𝑡𝑇2.

Proof. Since for each 𝑡 ≥ 0, 𝑋𝑖 𝑡 ≥ 𝑊𝑖 𝑡 , for

𝑖 = 1, 2, . .., we have

𝑃 𝑋𝑖 𝑡 ≤ 𝑥 ≤ 𝑃 𝑊𝑖 𝑡 ≤ 𝑥 for all 𝑥.

It follows from (4.1) and (4.2) that 𝐹 1𝑎 𝑡 ≤𝐹 2𝑎 𝑡 or 𝑇1≤

𝑠𝑡𝑇2. ■

4.2 Failure Rate Orderings

Suppose X t , X1 t , X2 t , . . . are independent

and W t , W1 t , W2 t , . . . are also independent

and identical processes. Further, suppose that

{𝑋𝑛(𝑡) ; 𝑛 = 1, 2, . . . } and {𝑊𝑛(𝑡) ; 𝑛 = 1, 2, . . . } are indpendent,

𝑌 𝑡 =1

𝑡 𝑋 𝑢 𝑑𝑢

𝑡

0,

𝑍 𝑡 =1

𝑡 𝑊 𝑢 𝑑𝑢

𝑡

0.

Then we have the following result on failure rate

ordering among two systems.

Theorem 4.2. Suppose that the following hold:

(a) for each 𝑡 ≥ 0, 𝐺𝑡(𝑢) = 𝑃(𝑌(𝑡) ≤ 𝑢) and

𝐻𝑡(𝑢) = 𝑃(𝑍(𝑡) ≤ 𝑢) are both PF2 ;

(b) for 𝑡1 < 𝑡2, 𝐺𝑡2(𝑢)/𝐺𝑡1

(𝑢) and 𝐻𝑡2(𝑢)/

𝐻𝑡1(𝑢) are both non-decreasing in 𝑢 ;

(c) for 𝑡 > 0, 𝐻𝑡(𝑢)/𝐺𝑡(𝑢) in non-decreasing in

𝑢; and

(d) for 𝑡1 < 𝑡2

𝑃 𝑌𝑖 𝑡2

𝑘𝑖=1 ≤ 𝑎

𝑃 𝑌𝑖 𝑡1 𝑘𝑖=1 ≤ 𝑎

≥𝑃 𝑍𝑖 𝑡2

𝑘𝑖=1 ≤ 𝑎

𝑃 𝑍𝑖 𝑡1 𝑘𝑖=1 ≤ 𝑎

Then, the failure rate of 𝑇1 is bigger than the

failure rate of 𝑇2, that is, 𝑇1≤𝐹𝑅𝑇2.

Proof. From (2.3), 𝐹 1𝑎 𝑡 and 𝐹 2𝑎 𝑡 can be

written as

𝐹 1𝑎 𝑡 = 𝑃 𝑌𝑖 𝑡

𝑘

𝑖=1

≤ 𝑎

∞

𝑘=0

𝑃(𝑁(𝑡)

= 𝑘) 4.3

𝐹 2𝑎 𝑡 = 𝑃 𝑍𝑖 𝑡

𝑘

𝑖=1

≤ 𝑎

∞

𝑘=0

𝑃(𝑁(𝑡)

= 𝑘) 4.4 We need to prove that the determinant

𝐷1 = 𝐹 1𝑎 𝑡1 𝐹 1𝑎 𝑡2

𝐹 2𝑎 𝑡1 𝐹 2𝑎 𝑡2 ≥

0 4.5

for t1 < t2, 𝐹 2𝑎 𝑡2 > 0. Let 𝑧𝑘 𝑡 be defined as

in Theorem 3.1. Then by (2.3) we have

𝐷1 =

𝑧𝑘1 𝑡1

∞

𝑘1=0

𝑃 𝑌𝑖 𝑡1

𝑘1

𝑖=1

≤ 𝑎 𝑧𝑘1 𝑡2

∞

𝑘1=0

𝑃 𝑌𝑖 𝑡2

𝑘1

𝑖=1

≤ 𝑎

𝑧𝑘2 𝑡1

∞

𝑘2=0

𝑃 𝑌𝑖 𝑡1

𝑘2

𝑖=1

≤ 𝑎 𝑧𝑘2 𝑡2

∞

𝑘2=0

𝑃 𝑌𝑖 𝑡2

𝑘2

𝑖=1

≤ 𝑎

= 𝑧𝑘1

𝑡1 𝑧𝑘1 𝑡2

𝑧𝑘2 𝑡1 𝑧𝑘2

𝑡2

𝑘2<∞0≤𝑘1<

𝑃 𝑌𝑖 𝑡1

𝑘1

𝑖=1

≤ 𝑎 𝑃 𝑌𝑖 𝑡2

𝑘1

𝑖=1

≤ 𝑎

𝑃 𝑌𝑖 𝑡1

𝑘2

𝑖=1

≤ 𝑎 𝑃 𝑌𝑖 𝑡2

𝑘2

𝑖=1

≤ 𝑎

(4.6)

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by the basic composition theorem (Karlin, Total

Positivity [1968], p 17).

For t1 < t2, the first determinant is non-negative

(Karlin and Proschan [1960], (Theorem 3) ).

From assumptions (a) to (c), it follows that the

second determinant on the right side of (4.6) is

non-negative. From (4.5), it is evident that

𝐹 1𝑎 𝑡 /𝐹 2𝑎 𝑡 is non-increasing in 𝑡, which in

turn implies 𝑇1≤𝐹𝑅𝑇2.

■

5. CONCLUSION

Under the cumulative damage threshold crossing

model, we have established some stochastic

properties related to the survival distributions of a

device. Under this model, it is still open to

determine the other stochastic properties viz.

IFRA, NBUE, DMRL HNBUE etc. and other

partial orderings related to the damage

distributions.

REFERENCES

[1] ABDEL-HAMEED, M. and SHIMI, I.N.,

[1978] Optimal Replacement of Damaged

Devices, J. Appl. Prob., 15,153-161.

[2] BARLOW, R.E. and PROSCHAN, F.,

[1965] Mathematical Theory of Reliability,

John Wiley, New York.

[3] EBRAHIMI, N., [1999] Stochastic

Properties of a Cumulative Damage

Threshold Crossing Model, J. Appl. Prob.,

36, 720-732.

[4] ESARY, J.D., MARSHAL, A.W. and

PROSCHAN, F., [1973] Shock Models and

Wear Process, Ann. Prob., 1, 627-649.

[5] FELLER, W., [1965] An Introduction to

Probability Theory and its Applications,

John Wiley and Sons, New York.

[6] KARLIN, S., [1968] Total positivity,

Stanford University Press, PaloAlto.

[7] KARLIN, S. and PROSCHAN, F.,

[1960] P𝑜 lya Type Distributions of

Convolutions, Ann. Math. Stat., 31, 721-

736.

[8] KL𝑈 PPELBERG, C and MIKOSCH, T.,

[1995] Explosive Shot Noise Process with

Applications to Risk Retention, Bernoulli,

23, 125-147.

[9] RIZWAN, U., [2001] Contributions to the

Study of some Stochastic Life Time Models,

Ph.D. Thesis, University of Madras, India.

[10] TAYLOR, H.M. and KARLIN, S., [1994]

An Introduction to Stochastic Modelling,

Academic Press, New York.

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Chemical and Physical Properties of Chitosan Acetate Electrolytic Systems

A. Ayisha Begam, K. Prem Nazeer and Rugmini Radhakrishnan

Abstract - Solid Chitosan Acetate Electrolytic Films were prepared by Grafting the Lithium Iodide salt (0.1 – 0.5 wt %) with Chitosan in

the presence of acetic acid. These films were subjected to Fourier Transform Infrared (FTIR), UV-Visible and XRD spectroscopic

analysis to understand structure, complexation process, optical band gap, type of transition and the shape of the absorption edge. A

systematic study on the suitability of biopolymer electrolyte revealed that the LiI (50 wt %) and chitosan are the most appropriate dopant

and host polymer for the preparation of biopolymer electrolyte respectively.

Index Terms - Chitosan acetate solid electrolyte films – grafting – Structure – Complexation process – Optical Band gap

—————————— ——————————

1. INTRODUCTION

In an attempt to obtain the suitable polymer

electrolytes for electrochemical systems many

new resources have been tried and efforts are

continued to characterize the new materials.

Among the several electrochemical systems, Li+,

Na+ and K

+ ion conducting solid electrolytes have

been developed for high energy density batteries

because of their light weight and high

electrochemical potential (Nazri and Pistoia,

2004; Linden and Reddy, 2002). Unlike the

conventional solid electrolytes, biopolymer

electrolytes are non toxic, degradable, eco friendly

material and can be prepared into flexible thin

films of required size and shape. The flexibility of

polymer electrolytes accommodates the volume

changes of electrodes, which typically occur

during the charge-discharge cycles of the

electrochemical device. It can also act as a

membrane and separator and provide comfortable

path during ion conduction process. Chemically

modified chitosan have been used as a host

polymer to study Solid Polymer Electrolytes(SPE)

——————————————

A. Ayisha Begam is serving in the Department of Physics, Avinashilingam University for Women, Coimbatore, India,

K. Prem Nazeer is serving in the Department of Physics, Islamiah College, Vaniyambadi, India. E-mail : nazeerprem@gmail.com

Rugmini Radhakrishnan served in the Department of Physics, Avinashilingam University for Women, Coimbatore, India

for batteries and the Proton Exchange Membranes

(PEM) for fuel cells (Mohamed et al. 1995,

Subban et al., Morni et al. 1997, 1999). This

paper discusses the chemical and physical

properties of LiI grafted chitosan membrane as

applied to ion conductivity and study its suitability

for biopolymer solid electrolyte battery systems.

2. EXPERIMENTAL

Chitosan-salt complexes were formed by grinding

1 g of chitosan powder (Fluka - medium

molecular weight) with different concentration (10

– 50 wt %) of LiI salts and by mixing this

compound with 50 ml of acetic acid in a 500 ml

glass beaker. This solution was stirred for about 1

hour continuously using a Teflon pellet, which

was rotated with a help of a magnetic stirrer

maintained at room temperature to form 2 % (w/v)

solution. Most of the chitosan salt complexes

dissolved to give a transparent solution

(electrolyte). Minor insoluble solids were

removed using a syringe filter with a pore size of

1micron and the required Chitosan electrolyte

solutions were collected for the preparation of

solid Chitosan electrolyte films.

Film casting technique is employed in the present

study for the preparation of solid biopolymer

electrolyte. The casting of the films was carried

out by pouring the filtered chitosan solution (5ml)

onto optically plane glass moulds (10×10 cm) and

were allowed to dry at room temperature (25 2

C) in a closed atmosphere for 3 days. The dried

films were carefully removed from the mould and

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its edges were clamped onto a well cleaned

optically plane glass plate and finally dried (50C,

24 Hrs) and stored under dry condition. The film

thickness was determined using a universal

length-measuring instrument (TRIMOS,

Switzerland) to an accuracy of 0.110–6

m (the

pressure maintained in the ball contact was 2.47

Pa). Typical thicknesses of ~ 50µm were used for

all the studies. In the present work, FTIR spectra

were recorded by ex situ in the range of 400 –

4000cm-1

using free standing films of biopolymer

electrolytes with a fully computerized Bruker IFS

- 66V spectrometer with 200 scans per spectrum

at 2 cm-1

resolution. The spectral distribution of

transmittance (T) data in the spectral region of

190-1100nm were analyzed at room temperature

using PerkinElmer - Lambada 35 UV-Visible

spectrometer. X-ray diffraction patterns were

recorded using Philips X-ray generator (Model

PW1390) at room temperature with an Ni filter

and CuK radiation( = 1.5418 Å) at 40 kV and

20 mA in the 2 range of 10 to 70 with an

accuracy of 0.02 in 2.

3. RESULTS AND DISCUSSION

3.1 FTIR Spectroscopy Studies on Lithium Iodide

(Li I) Grafted Chitosan Acetate (CHA)

Electrolytes

The FTIR spectra of samples were taken to verify

the occurrence of chitosan - lithium iodide salt

interaction. The small peaks at 1153 cm-1

(C-O-

C vibrations), 900 cm-1

and 647 cm-1

(NH wag

primary and secondary amines) found for pure

chitosan film (Fig 4.1) gradually broadened

and obscured when it is dissolved in acetic

acid (Ayisha Begam et al. (2011)). These

obscured peaks completely disappeared during

complexation process with lithium iodide for

the concentration of 50 wt. % as seen the

Fig 1.

According to Osman and Arof (2003), the

interaction is known to occur between chitosan

and an inorganic salt if there is a shift in

O=C-NHR, NH2 and NH3+ bands from chitosan

Fig. 1 FTIR Spectra of Lithium Iodide grafted

Chitosan Acetate Electrolyte

Scheme 1 Types of possible interaction of Li I

with Chitosan Acetate

spectrum. The bands at 1630 cm-1

and 1568 cm-1

represents acetylated amino group, which is due to

the C=O stretching vibrations of O=C-NHR

(amide I) and NH bending (amide II) respectively

for CHA film. On the addition of Lithium Iodide

(LiI), shift occurred in the carbonyl bands towards

the lower wave numbers. The maximum shift of

the amide I and amide II bands occurred in the

spectra of the samples CHA+ LiI (50 wt. %) is

1615 cm-1

and 1558 cm-1

respectively. All the

observed characteristic bands of chitosan acetate

(CHA) are shifted to lower numbers upon the

addition of LiI salt. The shifting of these bands

particularly depends upon the type of counter

anion and the salt concentration (Muzzarelli

1973). This could be due to the interaction of LiI

salt and nitrogen atom or between the acetic acid

and the nitrogen atom of the chitosan functional

group. The possible H-bonding type of

interaction of LiI salt with N of chitosan acetate is

shown in Scheme 1(a and b).

(c)

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(a) (b) (c)

Fig. 2 FTIR spectra of CHA with (a) 10 wt.% Li I;

(b) 30 wt.% LiI; (c) 50 wt.% LiI in the

region 1400- 1800 cm-1

.

Scheme 2 Interaction of LiI with Chitosan Acetate

Fig.2 depicts the carbonyl and amine bands of

salted CHA with various concentrations of Li I

((a) 10 wt. % (b) 30 wt. % and (c) 50 wt. %). The

amine band has shifted about 10 cm-1

(to 1558

cm-1

) on addition of salt compared with CHA. The

carbonyl band has further shifted by 15 cm-1

(to

1615 cm-1

) compared with CHA. In the spectrum

of the sample containing 10 wt.% salt, the full

width at half maximum (FWHM) for the amine

band is slightly wider than that of the carbonyl

band (the maximum height is taken from the

peaks of the carbonyl and amine bands,

respectively, to their meeting point at 1775 cm-1

).

On addition of 30 wt. % salt, the peak of the

carbonyl band is higher than that of the amine

band and the FWHM for the amine band is still

greater than that of the carbonyl band. The same

situation is still observed in Fig. 2 (c) and it is

inferred that salt-chitosan interaction prefers to

take place at the amine site (Scheme 2).

The band corresponding to OH stretching appears

as broad band around 3300 cm-1

. It is observed

that the broadening increased on the addition of

LiI salt. This could be due to the possible NH…I

or O-Li interaction. These interactions perhaps

lead to band broadening as well. Thus FTIR

studies of samples reveals that interaction is there

between LiI salt and chitosan acetate. These

results show that the chitosan acetate behaves a

successful host polymer for the Li-biopolymer

electrolyte.

3.2 Study of Optical Band Gap, Impurity Levels

and Shape of Absorption Edge in Lithium

Iodide Grafted Chitosan Acetate Electrolyte

The spectral distribution of transmittance (T) data

in the spectral region of 190-1100nm were

analyzed at room temperature using PerkinElmer -

Lambada 35 UV-Visible spectrometer for

Chitosan Acetate and Chitosan grafted with

Lithium Iodide (0.1g –0.5g) electrolyte samples.

Fig. 3 Transmittance spectra of Chitosan Acetate

and Chitosan Acetate grafted with Lithium

Iodide

The average thickness of the electrolyte films

measured was 50μm. Higher transmission in the

higher wavelength region and its decline at

absorption edge was observed for all the samples.

The transmittance spectra of the film indicate that

the films are more than 80 % transparent in the

NIR region, as shown in Fig.3. The transparency

decreases in the visible region due to band to band

absorption.

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Table 1 Optical Parameters of LiI Grafted

Chitosan Acetate Electrolytes

Fig. 4 Plots of (h) ½

versus h - indirect

transition in chitosan acetate

electrolyte

Fig. 5 Plots of (h) ½

versus h - indirect

transition in of 10 wt % Li I grafted

Chitosan electrolyte

Fig. 6 Plots of (h) ½

versus h - indirect

transition in 30 wt % Li I grafted

Chitosan electrolyte

Fig. 7 Plots of (h) ½

versus h - indirect

transition in 50 wt % Li I grafted

Chitosan electrolyte

Chitosan electrolytes are composed of Li I salt

dispersed in a chitosan acetate matrix. The LiI

salt dissociated into ions while complexes with

dissolved chitosan matrix. The chitosan acetate

salt, which produced due to a reaction between the

acetic acid and chitosan (Kaneko, 1997), where

the glucosamine (GlcN) convert into glucosamine

acetate unit (ie. H+ of acetic acid has formed a

dative bond with nitrogen of the chitosan

functional groups) and the disassociated Li I salt

in the polymer are responsible for the formation of

defects in the chitosan electrolytes. According to

Mott and Davis (1979), the width of mobility edge

Sample Type of

Transiti

on

Eg (eV)

Chitosan Acetate Indirect

allowed

1.62; 5.06

10 wt.% LiI

grafted Chitosan

Indirect

allowed

1.5; 4.87

30 wt.% LiI

grafted Chitosan

Indirect

allowed

1.4 ; 4.74

50 wt.% LiI

grafted Chitosan

Indirect

allowed

1.03; 4.64

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depends on the degree of disorder and defects

present in the amorphous structure. Such defect

produces localized states in the forbidden gap. So

the increase of percentage of LiI salt to chitosan

acetate host increases the localized states which

directly affect the decrease in the optical energy

gap of the chitosan acetate as shown in Table 1.

From the optical transmission data analysis, the

absorption co-efficient (α) has been calculated to

evaluate optical band gap and impurity levels.

The frequency dependence of the absorption

coefficient described by an empirical relation α =

A (hν-Eg) p

is used to evaluate the optical band

gap. The frequency dependence of the absorption

coefficient reasonably fit with the above equation

when the value of p = 2. The plots (αhν) 1/2

versus

hν are linear functions, indicating the existence of

indirect and allowed transitions for both Chitosan

Acetate and Chitosan grafted Lithium Iodide

electrolyte films as shown in the Figs. 4 –7.

Extrapolations of linear dependence to zero

absorption coefficient yields the corresponding

optical band gaps Eg as shown in Table 1.

There are two energy gap values as shown in table

1 for each composites of chitosan. The smaller

one may be due to the formation of defects in the

forbidden gap and larger one is due formation of

defects states close to conduction or valence band

of host polymer. As a result, the energy required

to produce a transition from an occupied energy

level to an unoccupied level is decreased, and the

wavelength of the light absorbed become longer.

Thus optical band gap show a bathochromic shift

when the load of LiI salt increases in chitosan

acetate matrix. These defects states are actually

responsible for the increase in the degree of

disorder and ion conduction in the polymer

electrolyte. Similar result was observed for PEO

complexed with NaF based polymer electrolytic

system (Sasikala et al., 2012). These results are in

agreement with those obtained from XRD and

conductivity studies in the present work.

The study of absorption edge in solids were

understood with the help of different theories put

forwarded by Miller and Abraham (1960), Mott,

(1969) and Tauc (1974). Based on these theories

(a)

(b)

(c)

Fig. 8 Plots of α versus hν for (a) 10 Wt%

(b) 30 Wt% and (c) 50 Wt% Li I

grafted CHA electrolytes

(solid lines are exponential

fit and inset linear fit).

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shapes of absorption edges in solids were

classified into three types viz, (i) a Lorentzian

broadened (power law) edge, (ii) an Urbach-rule

exponential edge, and (iii) an abrupt edge

expected for a solid with perfect transitional

periodicity. Normally, the fundamental absorption

edges are Lorentzian in most of the solids due to

the fact that the atomic absorption lines are

broadened by collisions and by radiative damping.

In the lowest part of the absorption edge, the

absorption coefficient varies exponentially with

photon energy, which obeys an exponential law

called Urbach‘s rule (α = α0 exp (hν/Eu)). Here Eu

is a characteristic energy known as band tail or

Urbach energy. However, all these models share

a common fact that the Urbach tail is related to

some kind of disorder in the material and its slope

decreases with increasing disorder. Therefore, the

tail characteristics can be considered as a probe

for the structural study.

The Tauc plot (Fig. 5- 7) can give hint about the

type of absorption edge in LiI grafted CHA. But

the hidden parameters can be studied with the help

of Urbach formula, which was verified by plotting

ln α versus hν for CHA and CHA grafted LiI

electrolyte near the absorption edge as shown in

Figure 8 (a-c). The solid lines are the exponential

fit, fit well with experimental data obeys Urbach‘s

rule. The inset of figure 8 (a-c) shows the slope of

the straight line of these curves yields the value of

band tails (Eu) whose magnitude is 0.42, 0.37 and

0.30 for 10, 30 and 50 Wt% LiI grafted CHA

electrolyte respectively. The gradual lowering of

Eu value indicated the existence of structural

disorder in the host polymer, which transforms

chitosan from semi-crystalline to semi-

amorphous. The results obtained from IR and

XRD spectroscopy in the previous and following

sections respectively were fit well with optical

studies that satisfies the few important parameters

of a good polymer electrolyte.

3.3 Structural Study of LiI Grafted Chitosan

Acetate Solid Electrolyte

X-ray diffraction (XRD) method is used to study

the effect of dissolution of salt on the structure of

the polymer of the salted chitosan acetate. In the

present study, room temperature X-ray diffraction

patterns were recorded using Philips X-ray

generator (Model PW1390) with an Ni filter and

CuK radiation( = 1.5418 Å) at 40 kV and 20

mA in the 2 range of 10 to 70 with an accuracy

of 0.02 in 2.

Fig. 9 XRD spectra of LiI grafted CHA

solid electrolytes

XRD pattern of chitosan acetate films

complexed with different concentration of

Lithium Iodide exhibit an amorphous hump

around 20º as shown in Fig.9. It is evident

from the spectra, that the chitosan acetate – Li I

complex can be concluded as amorphous due the

absence of any characteristic X-ray diffraction

peaks. From the JCPDS card 01-0532, it is

identified that the peak position at 24.85o

(111)

corresponds to maximum intensity reflection of

LiI. The characteristic peak of LiI merged with

characteristic peaks of chitosan acetate even for

lower concentration shows the complete

dissolution of LiI salt in the chitosan acetate

solution as seen in the spectra. The absence of

peaks in the XRD as addition of LiI indicates

strong evidence for the formation of complexation

between the salt and the chitosan polymer.

Puteh et al., (2005) observed a similar result with

Lithium trifluoromethane sulfonimide (LiTFSI) as

doping salt and chitosan as host polymer.

Comparing LiI peak with the peaks of chitosan+

Li I samples, a decrease in the 2Θ value can be

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observed and it is also an indication of

complexation between the chitosan polymer and

the LiI salt. Similar kind of observations was

reported by Morni and Arof (1999) for chitosan-

lithium triflate electrolyte samples. They also

observed that the chitosan remains amorphous

upon the addition of lithium triflate salt fit with

our observation that shows the reduction of

sharpness of the characteristics amorphous

reflection and the broadening of the area

under the curve.

4. CONCLUSION

The investigations on success of CHA as host

polymer and Li I as a most suitable dopant for

electrolytic applications were confirmed with the

help of IR, UV-Vis and XRD spectroscopic

analyses. The IR study showed the chitosan-salt

interaction and the availability of Li+ for the

conduction process. Optical band gap and Urbach

energy results revealed the existence of defects

states, which are actually responsible for the

increase in the degree of disorder and ion

conduction in the polymer electrolyte. XRD

result supports the increased amorphous phase of

CHA in the presence of LiI, which will favour

intra- and inter-chain ion movement and thus

improve the electrical conduction. A complete

study on ionic conductivity will finalize the

suitability of CHA+LiI system as electrolyte for

battery applications.

ACKNOWLEDGEMENT

The authors would like to thank the authorities of

Avinashilingam University for Women,

Coimbatore and Islamiah College, Vaniyambadi

for their support to carry out this work.

REFERENCES

[1] Ayisha Begam, A., Rugmini Radhakrishnan

and Prem Nazeer, K (2011), ‗Study of

Structure-Property Relationship on Sulfuric

Acid Crosslinked Chitosan Membranes‘,

Malaysian Polym. J., 6(1): 27-38.

[2] Kaneko, H; Miura, Y; Kaneko, M and

Tokura, S (1997), Brine, C.J; Sanford, P.A

and Zikakis, J.P (Ed), ‗Advances in chitin

and chitosan‘, Elsevier Applied Science,

UK, 588.

[3] Linden, D and Reddy, T. B (2002),

‗Handbook of Batteries‘, MacGraw Hill

Publishers, London.

[4] Miller, A and Abrahams, E (1960),

‗Impurity conduction at low concentrations‘,

Phys. Rev., 120, 3, 745-755.

[5] Mohamed, N.S., Subban, R.H.Y. and

Arof, A.K (1995). ‗ Polymer batteries

fabricated from lithium complexed

acetylated chitosan‘, J. Power Sources,

56:153-156.

[6] Morni, M.N and Arof, A.K (1999),

‗Chitosan–lithium triflate electrolyte in

secondary lithium cells Journal of Power

Sources‘, 77, 42-48.

[7] Morni, N.M., Mohamed, N.S.and Arof,

A.K(1997).‘ Silver nitrate doed chitosan

acetate films and electrochemical cell

performance‘, Mater. Sci. Eng. B, 45:

140-146.

[8] Mott, N.F (1969), ‗Conduction in non-

crystalline materials III Localized states in a

pseudo gap and near extremities of

conduction and valence bands‘, J. Non-

crystalline Solid, 19, 160, 835-851.

[9] Mott, N.F and Davis, E.A (1979),

‗Electronic Processes in Non – Crystalline

Materials‘, Clarendon Press, Oxford.

[10] Muzzarelli, R.A.A (1973), ‗Natural

chelating polymers‘, Pergamon Press Ltd,

London.

[11] Muzzarelli, R.A.A (1977), ‗Chitin‘,

Pergamon Press, Oxford.

[12] Nazri, G and Gianfranco Pistoia (Eds)

(2004), ‗Lithium batteries-Science and

INTERNATIONAL JOURNAL OF SCIENTIFIC & ENGINEERING RESEARCH, VOLUME 3, ISSUE 11, NOVEMBER -2012 ISSN 2229-5518 73

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Technology‘, Kluwer Academic Publishers,

London.

[13] Osman, Z and Arof, A.K (2003), ‗FTIR

studies of chitosan acetate based polymer

electrolytes‘, Electro chimica Acta, 48, 993-

999.

[14] Puteh, R; Yahya, M.Z.A; Ali, A.M.M;

Sulaiman, M.A and Yahya, R (2005),

‗Conductivity studies on chitosan-based

polymer electrolytes with lithium salts‘,

Indonesian Journal of Physics 16, 1, 17-19.

[15] Sasikala, U; Naveen Kumar, P; Rao,

V.V.R.N and Sharma, A.K (2012),

‗Structrual, Electrical and Parametric

Studies of a PEO based Polymer Electrolyte

for Battery Applications‘, Int. J. Engg. Sci.

& Adv. Tech., 2, 3, 722-730.

[16] Subban, R.H.Y, Arof, A.K.and

Radhakrishna, S(1996).‘Polymer

batteries with chitosan electrolyte mixed

with sodium perchlorate‘, Mater.

Sci.Eng.B, 38:156-1 60.

[17] Tauc, J (1974), ‗Amorphous and Liquid

Semiconductor‘, Plenum press, New York.

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Some Operations and Relations over

Intuitionistic Fuzzy Sets of Root Type

R. Srinivasan

Abstract - In this paper, we define some Operations and relations over Intuitionistic Fuzzy Sets of Root Type and give some properties.

Index Terms - Intuitionistic Fuzzy Set (IFS), Intuitionistic Fuzzy Set of Second Type (IFSST), Intuitionistic Fuzzy Set of Root Type (IFSRT).

—————————— ——————————

1. INTRODUCTION

UZZY 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 IFS and its extension IFSST.

Following the definition of IFS, the authors

introduced the IFSRT. In this paper, we define

some Operations and relations over IFSRT and

state few of their properties.

2. PRELIMINARIES

In this section, we give some definition of various

types of IFS.

Definition: 2.1 Let X be a non empty set. An IFS

A in X is defined as an object of the form.

, ( ), ( ) :A AA x x x x X

where the functions : [0,1]A X and

: [0,1]A X denote the membership and non-

membership function of A respectively and

0 ( ) ( ) 1A Ax x for each .x X

Remark:2.2 An ordinary fuzzy set can also be

generalized as

, ( ),1 ( ) :A Ax x x x X

———————————

R. Srinivasan is serving in the Department of Mathematics, Islamaih

College, Vaniyambadi, India. E-mail: srinivasanmaths@yahoo.com

Definition: 2.3 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 AA x x x x X

where the functions : [0,1]A X and

: [0,1]A X denote the degree of membership

and degree of non-membership functions of A

respectively, and 2 20 [ ( )] [ ( )] 1,A Ax x

for each x X .

Remark: 2.4. It is obvious that for all real

numbers a, b [0,1] if 0 1a b then 2 20 1a b

Definition: 2.5 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 AA x x x x X

where the functions : [0,1]A X and

: [0,1]A X denote the degree of membership

and degree of non-membership functions of A

respectively, and

1 10 ( ) ( ) 1

2 2A Ax x

F

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for each x X

Definition 2.6Let X be a non-empty set. Let A

and B be two IFSRTs such that

, ( ), ( ) :A AA x x x x X

, ( ), ( ) :B BB x x x x X

We define the following relations and operations.

(i) A ⊂B if and only if

( ) ( ) ( ) ( ),A B A Bx x and x x

for all x in X

(ii) A B if and only if

( ) ( ) ( ) ( ),A B A Bx x and x x

for all x in X

(iii) A B if and only if

( ) ( ) ( ) ( ),A B A Bx x and x x

for all x in X

(iv)

,max( ( ), ( )),:

min ( ( ), ( ))

A B

A B

x x xA B x X

x x

(v)

,min( ( ), ( )),:

max ( ( ), ( ))

A B

A B

x x xA B x X

x x

(vi)

, ( ) ( )

( ) ( ), :

( ) ( )

A B

A B

A B

x x x

A B x x x X

x x

(vii) The complement of A is defined by

__

, ( ), ( ) :A AA x x x x X

Definition: 2.7 The degree of non-determinacy

(uncertainty) of an element x X to the IFSRT A

is defined by

2

( ) 1 ( ) ( )A A Ax x x

Definition: 2.8 For every IFSRT A, we define the

following operators.

The Necessity measure on A.

2, ( ), (1 ( )) : .A AA x x x x X

The Possibility measure on A,

2

, 1 ( ) , ( ) : .A AA x x x x X

Definition: 2.9 For every two IFSsRT A and B,

we define the following relations.

(i)

( ) ( ), ,

2@ :

( ) ( )

2

A B

A B

x xx

A B x Xx x

(ii)

, ( ). ( ) ,$ :

( ). ( )

A B

A B

x x xA B x X

x x

It is easy to verify the correctness of the defined

relations.

3. PROPERTIES

In this section, we give some properties.

Proposition: 3.1 For every IFSsRT A and B, we

have

(i) A @ A = A

(ii) A $ A = A

(iii) __ __

@ @A B A B

(iv) __ __

$ $A B A B

Proposition: 3.2 For every IFSsRT 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

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Proposition: 3.3 For every IFSsRT A and B, we

have

(i) ( @ ) @A B A B

(ii) ( $ ) $A B A B

(iii) ( @ ) @A B A B

(iv) ( $ ) $A B A B

4. CONCLUSION

We have made an attempt to establish some

operations and relations over IFSsRT. It is still

open to check whether there exist an IFSRT in

case of the operators already defined on an IFS.

REFERENCES

[1] Atanassov, K.T.(1999), Intuitionistic Fuzzy

Sets, theory and Applications, Springer–

Verlag, New York.

[2] Parvathi, R. and Palaniappan, N. (2004) Some

operations on IFSets 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 Intituitionstic Fuzzy Sets

of Root Type International Journal of

Computational and Applied Mathematics, 4(3)

pp.383-390.

[5] Srinivasan, R. and Palaniappan, N. (2011)

SomeTopological Operators on Intuitionistic

Fuzzy Sets of Root Type, Research methods in

Mathematical sciences, Edited by Dr.

U.Rizwan, 4 pp23-28 India.

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Generalized Hyers-Ulam Stability of the Geometric Mean Functional Equation in Two

Variables

K. Ravi and B.V. Senthil Kumar

Abstract - In this paper, we find the solution and prove the generalized Hyers-Ulam stability of the geometric mean functional equation in two

variables. We also provide counter-example for singular case.

Index Terms - Functional equation, Quadratic functional equation, Generalized Hyers-Ulam stability.

—————————— ——————————

1. INTRODUCTION

N 1940, S. M. Ulam [10] raised the question

concerning the stability of group

homomorphisms. In the year 1950, Aoki [2]

generalized the Hyers theorem for additive

mappings. In the year 1978, Th. M. Rassias [9]

provided a proof of the stability of the linear

mapping by permitting the Cauchy difference to

become unbounded. In the year 1982, J. M.

Rassias [7] proved a similar result when the

unbounded Cauchy difference is bounded by a

product of powers of norms. Since then, the

stability problems of various functional equations

have been extensively investigated by a number of

authors ([3], [5], [6]). The terminology

generalized Hyers-Ulam stability originates from

these historical backgrounds.

Recently, K. Ravi and B. V. Senthil Kumar [8]

investigated the generalized Hyers-Ulam stability

of generalized square root functional equation

1

1 1 1

1

2p p p

i i i j i j

i i j i

p

i i

i

s x x x

s x

(1.1)

——————————————

K. Ravi is serving in the Department of Mathematics, Sacred Heart

College, Tirupattur, India. E-mail: shckravi@yahoo.com

B. V. Senthil Kumar is serving in the Department of Mathematics, C. Abdul Hakeem College of Engineering and Technology, Melvisharam, India. E-mail : bvssree@yahoo.co.in

for arbitrary but fixed real numbers

1 2, ,..., (0,0,...,0)p

so that

1

1

0 ... 1p

p i

i

and :s X Y with X and Y are the sets of non-

negative real numbers. The square root mapping

( )s x x is a solution of the functional equation

(1.1). Moreover, if we substitute

1 2 3 41, ... 0p

and 3 4 ... 0px x x

in equation (1.1), we obtain the 2-dimensional

square root functional equation

2 ( ) ( ).s x y xy s x s y (1.2)

In this paper, we obtain the solution and

investigate the generalized Hyers-Ulam stability

of the geometric mean functional equation in two

variables of the form

2 , 2

( , ) ( , ) ( , ) ( , )

G x u xu y v yv

G x y G x v G u y G u v

(1.3)

which originates from the geometric mean of two

non-negative real numbers x and .y The function

( , )G x y xy

is a solution of the functional equation (1.3). The

above mapping represents the geometric mean of

x and .y

I

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The following Definition 1.1 is useful for the

proof of the main results in the paper.

Definition. 1.1. Let X be the space of non-

negative real numbers. A mapping :G X X

defined by

( , )G x y xy

is called 2-variable geometric mean mapping if it

satisfies the functional equation (1.3). The

functional equation (1.3) is called geometric mean

functional equation.

For convenience, let us denote

( , , , )

2 , 2

( , ) ( , ) ( , ) ( , ).

G x u y v

G x u xu y v yv

G x y G x v G u y G u v

Throughout this paper, let X be the space of non-

negative real numbers.

2. SOLUTION OF FUNCTIONAL EQUATION (1.3)

Theorem 2.1. A mapping :G X X satisfies

(1.3) if and only if there exists an identity mapping

:I X such that

( , ) ( ) ( ),G x y I x I y for all , .x y X

Proof. Let ( , )G x y be a solution of (1.3). Define

1( ) ( , ),G x G x x for all .x X

It is easy to verify that 1G is an identity mapping.

We denote the identity mapping by ( ).I x That is,

1( ) ( ),I x G x for all .x X Hence

( ) ( ) ( , ),I x I y xy G x y

for all , .x y X

Conversely, assume that there exists an identity

mapping :I X such that

( , ) ( ) ( ),G x y I x I y

for all , .x y X Hence

2 , 2

2 2

G x u xu y v yv

I x u xu I y v yv

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

I x I y I x I v

I u I y I u I v

( , ) ( , ) ( , ) ( , )G x y G x v G u y G u v

for all , , , ,x u y v X which completes the proof of

Theorem 2.1.

3. A RELATION BETWEEN

FUNCTIONAL EQUATIONS (1.2) AND (1.3)

Theorem 3.1. Let :s X be a mapping

satisfying (1.2) and :G X X be a mapping

defined by

( , ) ( ) ( )G x y s x s y (3.1)

for all , .x y X Then G satisfies (1.3).

Proof. Replacing ( , )x y by

2 , 2x u xu y v yv

in (3.1) and using (1.2), we obtain

2 , 2G x u xu y v yv

2 2s x u xu s y v yv

[ ( ) ( )][ ( ) ( )]s x s u s y s v ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )s x s y s x s v s u s y s u s v

( , ) ( , ) ( , ) ( , ).G x y G x v G u y G u v

Hence G satisfies (1.3), as desired.

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4. GENERALIZED HYERS - ULAM

STABILLITY FUNCTIONAL EQUATION (1.3)

Theorem 4.1. Let :G X X be a mapping

for which there exists a mapping

: X X X X with the condition

lim4 4 ,4 ,4 ,4 0n n n n n

nx x y y

(4.1)

such that the functional inequality

| ( , , , ) | ( , , , )G x u y v x u y v (4.2)

holds for all , , , .x u y v X Then there exists a

unique 2-variable geometric mean mapping

:g X X R satisfying the functional equation

(1.3) and

1

0

| ( , ) ( , ) |

14 ,4 ,4 ,4

4

i i i i

ii

g x y G x y

x x y y

(4.3)

for all , .x y X The mapping ( , )g x y is defined

by

( , ) lim 4 4 ,4n n n

ng x y G x y

for all , .x y X

Proof. Replacing ( , , , )x u y v by ( , , , )x x y y in (4.2)

and dividing by 4, we obtain

1 1(4 ,4 ) ( , ) ( , , , ).

4 4G x y G x y x x y y

(4.4)

Now, replacing ( , )x y by (4 ,4 )x x in (4.4), dividing

by 4 and summing the resulting inequality with

(4.4), we arrive

2 2

2

1

10

14 ,4 ( , )

4

14 ,4 ,4 ,4 .

4

i i i i

ii

G x y G x y

x x y y

Proceeding further and using induction on a

positive integer ,n we get

1

10

4 4 ,4 ( , )

14 ,4 ,4 ,4

4

n n n

ni i i i

ii

G x y G x y

x x y y

10

14 ,4 ,4 ,4

4

i i i i

ii

x x y y

(4.5)

for all , .x y X In order to prove the convergence

of the sequence

4 4 ,4 ,n n nG x y

replacing ( , )x y by 4 ,4p px y in (4.5) and

multiplying by 4 ,p we find that for 0n p

4 4 ,4 4 4 ,4

4 4 4 ,4 4 ,4

n p n p n p p p p

p n n p n p p p

G x y G x y

G x y G x y

10

14 ,4 ,4 ,4

4

p i p i p i p i

p ii

x x y y

0asp .

This shows that the sequence

4 4 ,4n n nG x y

is a Cauchy sequence. Taking n tending to

infinity in (4.5), we obtain (4.3). To show that g

satisfies (1.3), replacing ( , , , )x u y v by

4 ,4 ,4 ,4n n n nx u y v in (4.2) and multiplying by

4 ,n we obtain

4 4 ,4 ,4 ,4

4 4 ,4 ,4 ,4 .

n n n n n

n n n n n

G x u y v

x u y v

(4.6)

If we suppose n approaches infinity in (4.6), it

follows that g satisfies (1.3) for all , , , .x u y v X

To prove that g is a unique 2-variable geometric

mean mapping satisfying (1.3), let :h X X

be another 2-variable geometric mean mapping

which satisfies (1.3) and the inequality (4.3).

Clearly h and g satisfy (1.3) and using (4.3), we

get

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| ( , ) ( , ) |

4 4 ,4 4 ,4n n n n n

h x y g x y

h x y g x y

4 ,4 4 ,44

4 ,4 4 ,4

n n n n

n

n n n n

h x y G x y

G x y g x y

0

14 ,4 ,4 ,4

2

n i n i n i n i

i

x x y y

(4.7)

for all , .x y X Allow n in (4.7) and using

(4.1), we find that g is the unique such mapping.

Theorem 4.2. Let :G X X be a mapping

for which there exists a function

: X X X X with the condition

1 1 1 1lim 4 4 ,4 ,4 ,4 0n n n n n

nx x y y

(4.8)

such that the functional inequality

| ( , , , ) | ( , , , )G x u y v x u y v (4.9)

holds for all , , , .x u y v X Then there exists a

unique 2-variable geometric mean mapping

:g X X R satisfying the functional equation

(1.3) and

1 1 1 1

0

| ( , ) ( , ) |

4 4 ,4 ,4 ,4i i i i i

i

G x y g x y

x x y y

(4.10)

for all , .x y X The mapping ( , )g x y is defined

by

( , ) lim 4 4 ,4n n n

ng x y G x y

for all , .x y X

Proof. Substituting

( , , , ) , , ,4 4 4 4

x x y yx u y v

in (4.9) and proceeding by similar arguments as in

Theorem 4.1, the proof is complete.

Corollary 4.3. If a mapping :G X X

satisfies the functional inequality

1 1

2 2

4 43 3

4 4

4 4

| ( , , , ) |

| | | | | | | | for p 1,k 0

1| | | | | | | | for , 0

2

| | | | | | | |1

for , 0| | | |4

| | | |

1| | | | | | | | for , 0

2

p p p p

a a b b

p p p p

G x u y v

k x u y v

k x u y v a b k

x u y v

k kx u

y v

k x u y v p k

(4.11)

for all , , , .x u y v X Then there exists a unique 2-

variable geometric mean mapping :g X X

satisfying the functional equation (1.3) and

1

2 22

2

2 2

3

4 44

4

2

| ( , ) ( , ) |

2| | | | for 1

4 4

1| | | | for

24 4

| | | | 1for

42 | | | |4 4

4 1| | | | for

24 4

p p

p

a b

p p

p

g x y G x y

kx y p

kx y a b

x yk

x y

kx y p

for all , .x y X

Proof. If we choose

1

2

3 4 4 4 4

4

| ( , , , ) |

| | | | | | | |

| | | | | | | |

| | | | | | | |

| | | | | | | |

| | | | | | | |

p p p p

a a b b

p p p p

x u y v

k x u y v

k x u y v

x u y vk

x u y v

k x u y v

for all , , , ,x u y v X in Theorems 4.1 and 4.2, it

easy to prove the required results.

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5. COUNTER - EXAMPLE

The following Example 5.1 illustrates the fact that

the functional equation (1.3) is not stable for

1

2p in the fourth inequality of (4.11) in

Corollary 4.3.

Example 5.1. Let : X X be a function

defined by

for , (0,1)( , )

otherwise

xy x yx y

where is a constant, and define a function

:G X X by

0

4 ,4( , ) ,

4

n n

nn

x yG x y

for all , .x y X

Then the function G satisfies the inequality

1 1 1 1

2 2 2 2

| ( , , , ) |

80| | | | | | | |

3

G x u y v

x u y v

(5.1)

for all , , , .x u y v X Therefore there do not exist a

2-variable geometric mean mapping

:g X X and a constant 0 such that 1 1

2 2| ( , ) ( , ) | | | | |g x y G x y x y (5.2)

for all , .x y X

Proof.

0

0

1

( , ) |

4 , 4

4

11

4

4.

3

4

n n

nn

nn

G x y

x y

Hence G is bounded by 4

.3 If

1 1 1 1

2 2 2 21

| | | | | | | | ,4

x u y v

then the left hand side of (5.1) is less than 20

.3

Now, suppose that 1 1 1 1

2 2 2 21

0 | | | | | | | | .4

x u y v

Then there exists a positive integer k such that

1 1 1 1

2 2 2 21

1 1| | | | | | | | .

4 4k kx u y v

(5.3)

Hence 1 1 1 1

2 2 2 21

| | | | | | | |4k

x u y v

implies 1 1 1 1

2 2 2 2 2 24 4 1k k

x u y v

or 1 1 1 1

2 2 2 2 2 2 2 24 4 1, 4 4 1k k k k

x u y v

or

1 1 1 1

2 2 2 2 2 2 2 24 1, 4 1, 4 1, 4 1k k k k

x u y v

or 1 1

2 2

1 1

2 2

1 1

2 2

1 1

2 2

14 ,

2

14 ,

2

14 ,

2

14

2

k

k

k

k

x

u

y

v

and consequently

1 1

1 1

1

1

4 ( ), 4 ( ),

4 ( ), 4 ( ),

4 2 ,

4 2 (0,1).

k k

k k

k

k

x u

y v

x u xu

y v yv

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Therefore, for each value of 0,1,2,..., 1,n k we

obtain

4 ( ), 4 ( ),

4 ( ), 4 ( ),

4 2 ,

4 2 (0,1)

n n

n n

n

n

x u

y v

x u xu

y v yv

and

4 2 ,4 2

4 ,4 4 ,4

4 ,4 4 ,4 0

n n

n n n n

n n n n

x u xu y v yv

x y x v

u y u v

for 0,1,2,..., 1.n k Using (5.3) and the

definition of ,G it is easy to prove the inequality

(5.1) holds true.

We claim that the geometric mean functional

equation (1.3) is not stable for 1

2p in the fourth

inequality of (4.11) of Corollary 4.3.

Assume that there exists a 2-variable geometric

mean mapping :g X X satisfying (5.2).

Therefore, we have

1 1

2 2| ( , ) ( 1) | | | | .G x y x y (5.4)

However, we can choose a positive integer m

with 1.m If 10,4 ,mx then 4 (0,1)n x

for all 0,1,2,..., 1n m and therefore

0

1

0

( , )

4 , 4

4

4

4

( 1)

n n

nn

nm

nn

G x y

x y

xy

m xy

xy

which contradicts (5.4). Therefore, the geometric

mean functional equation (1.3) is not stable for

1

2p in the fourth inequality of (4.11) in

Corollary 4.3.

REFERENCES

[1] J. Aczel and J. Dhombres, Functional

Equations in Several Variables, Cambridge

Univ. Press, 1989.

[2] T. Aoki, On the stability of the linear

transformation in Banach spaces, J.

Math.Soc. Japan, 2(1950), 64-66.

[3] S. Czerwik, Functional Equations and

Inequalities in Several Variables, Wolrd

Scientific Publishing Company, Singapore,

New Jersey, London, 2002.

[4] D.H. Hyers, On the stability of the linear

functional equation, Proc. Nat. Acad. Sci.

U.S.A., 27(1941), 222-224.

[5] D.H. Hyers, G. Isac and Th.M. Rassias,

Stability of Functional Equations in Several

Variables, Birkhauser, Baston, Basel, Berlin,

1998.

[6] S.M. Jung, Hyers-Ulam-Rassias stability of

Functional Equations in Mathematical

Analysis, Hadronic Press Inc. Florida, 1994.

[7] J.M. Rassias, On approximation of

approximately linear mappings by linear

mappings, J. Funct. Anal. 46(1982), 126-130.

[8] K. Ravi and B.V. Senthil Kumar, Rassias

stability of generalized square root functional

equation in several variables, Int. J. Math.

Sci. Engg. Appl.3(III)(2009), 35-52.

[9] Th.M. Rassias, On the stability of the linear

mapping in Banach spaces, Proc. Amer.

Math. Soc. 72 (1978), 297-300.

[10] S.M. Ulam, Problems in Modern

Mathematics, Chapter VI, Wiley-Interscience,

New York, 1964.

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Establishment of an efficient drug from simple hydrocarbon - Thiotepa, and evaluation of similar molecules using

the enzyme Cytochrome

P. K. Mohamed Imran

Abstract - A number of in-vitro and in-vivo studies have been conducted to establish the pharmacological and metabolic activity of the

anti-cancer drug, thiotepa, Triethylenethiophosphoramide (TTP). All have discussed on the efficacy and side effects of the drug.

Some have reported it as an alkylating agent capable of breaking the DNA strands along with some mechanism and others have

established it as an effective inhibitor of an enzyme of Cytochrome family, CYP2B6. The latter study was found to have profound

effect on the metabolic activity of the human body. A theoretical study is established to provide essential insight into the nature of the

ligands as well as the enzyme. Descriptive properties such as Global hardness, softness, Fukui function and electrophilicity index

were computed. Docking was performed using Lamarckian Genetic Algorithm and the interactions of the ligand with the

macromolecule were visualized, correlated with the descriptive properties. This was helpful in establishing the effectiveness of a drug

analogue that might help in deciding the toxicity, side effects or promiscuousness of this type of drug.

Index Terms – in-vitro, in-vivo, TTP, Thiotepa, cyp2b6, Docking.

—————————— ——————————

1. INTRODUCTION

ITH regard to human body, small molecules

are foreign molecules, called as xenobiotics

that attempt to deal with a number of responses.

Some drugs are excreted from the human body

intact. Most drugs, however, need to be modified

structurally to facilitate excretion. The

modification process has been termed drug

metabolism. An ideal drug should reach the site

of action intact, cure the disease before it leaves

the body completing the mission. A drug is

actually something that is not needed by a normal

healthy human body. Hence the study of drug

metabolism should serve two purposes: to

elucidate the function and fate of the drug, and to

manipulate the metabolic process of the potential

drug [1].

1.1 Thiotepa as a promiscuous drug

A very common drug for anti cancer that has been

studied by many researchers is Thiotepa (tri-ethyl-

enethio-phosphoramide). In-vivo Metabolic and

——————————————

P. K. Mohamed Imran is serving in the Department of Chemistry,

Islamiah College , Vaniyambadi, India.

pharmacokinetic activity study revealed that

thiotepa is a moderate inhibitor of cytochrome

P450 3A4 and 2B6 isoenzyme [2]. Thiotepa was

studied clinically over several patients and the

bigger challenge was its clearance from the human

body after metabolism. The role of purified

human Glutathione S-Transferase (GST) on the

formation of mono glutathionyl thiotepa was

studied using HPLC and the results showed that

the aziridine moieties in thiotepa were substrates

for the GST and the conclusion was that in GST

catalyzation of glutathione, conjugation of

thiotepa might prove to be an important factor in

the development of drug [3].

In another intravenous study on women with

metastatic adenocarcinoma of the breast the

researchers found a patterned hyper pigmentation

confined to skin occluded by adhesive containing

material. The results suggested that this

alkylating agent is excreted onto the skin surface

in sweat, accumulates beneath adhesive-

containing bandages and electrocardiogram pads,

and exerts a local toxic effect resulting in

hyperpigmentation [4]. The thiotepa metabolite

triethylene phosphoramide was found to

hydrolyze significantly faster than thiotepa. The

authors established that the cytotoxic effects of

thiotepa were oxygen dependent and might

W

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involve metabolic processes catalyzed by

cytochrome P-450 enzymes [5].

1.2. The enzyme containing heme

Cytochrome P4502B6 [6] is a genetically

polymorphic enzyme that is important in the

metabolism of a number of clinically used drugs

[7]. This is the least studied enzyme in its class.

Cytochromes contain iron bound with four N

atoms attached to protein molecules, a group

called the heme group. The iron ions in

cytochrome are capable of gaining and losing

electrons to produce Fe2+

and Fe3+

, respectively.

Interference with the action of cytochromes is an

important mode of the action of some toxicants.

Cyanide ion, CN–, has a strong affinity for Fe

3+ in

ferricytochrome, preventing it from reverting back

to the Fe2+

form, thus stopping the transfer of

electrons to O2 and resulting in rapid death in the

case of cyanide poisoning [8]. The toxicity of

other similar drugs as the CYP2B6 does not

affectively metabolise was well brought out in a

in-vitro study by Hamaska et al. [9]. This

information can be used for docking studies as

well as more complex calculations.

1.3. The Docking of drug and enzyme

The exact mechanism of action of thiotepa drug

and the inhibition of the cytochrome by this drug

has so far been not yet studied at either the ab-

initio or the DFT level of computation. The

advent of molecular docking studies and the

concept of ‗promiscuousnesses of a drug‘ should

open the discussion on a wider spectrum and help

establish the mechanism by which this drug

works. Another reason for not making a docking

study so far has been the unavailability of the

crystal structure of the enzyme CYP2B6 at the

PDB or any other databank. Though many closer

models such as those of CYP2B4 and CYP2C6

have been successfully achieved and other formats

like sequence, fasta and xml were available, the

Protein Data Bank (PDB) [11] format is the one

that was required for docking purpose. The

crystal structure of CYP2B6 was obtained after a

loop from CYP2C5 was modified [12]. The

binding site consisted of three main pockets

(Fig. 2). Pocket A was the heme pocket formed by

the side chains of Ala298, Thr302, Leu63 and

Phe206.

Fig. 1. The general structure of the molecule

under study. Local properties were

studied for R8 and R19 atoms

Fig. 2. Cytochrome CYP2B6 - heme group at the

centre, which is also the docking site

of the drug. The grid for docking

was chosen around this area.

Genetic algorithms (GAs) are a class of

optimisation methods that are based on various

computational models of Darwinian evolution

[13, 14]. Genetic and evolutionary algorithms can

be used to perform protein–ligand docking [15].

2. THEORETICAL BACKGROUND

The Hohenberg-Kohn theorem [16] is used to

calculate the ground state energy of an atom or

molecule using the electron density as

][)()(][ HKFdrrvrE

where ν(r) is the external potential with

][][][ eeHK VTF

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FHK is the universal Hohenberg–Kohn functional,

which comprises of electronic kinetic energy

functional (T[ρ]) and electron-electron interaction

functional (Vee [ρ]). The first and second partial

derivatives of E[ρ], with respect to the number of

electrons N under the constant external potential

v(r), are defined as the chemical potential (μ) and

the global hardness η of the system, respectively

[17]. The global softness (GS) is the half inverse

of the hardness. The global descriptor of hardness

has been known as an indicator of the overall

stability of the system.

The finite difference approximation [18] for the

computation of chemical potential and hardness, is

given as μ = -(IP + EA)/2 and η = (IP - EA)/2,

respectively, where IP and EA are the first vertical

ionization energy and electron affinity of chemical

species respectively. Using koopmans'

approximation [19] the equations have been

defined as μ = (ELUMO + EHOMO) / 2 and η =

(ELUMO - EHOMO) / 2, where EHOMO and ELUMO are

the orbital energies of highest occupied molecular

orbital and lowest unoccupied molecular orbital of

the N electron system respectively.

AutoDock 4.2 uses free-energy scoring functions

that is based on a linear regression analysis, the

AMBER force field [20], and an even larger set of

diverse protein-ligand complexes with known

inhibition constants. It uses an empirical binding

free energy force field that allows the prediction

of binding free energies, and hence binding

constants, for docked ligands. A general equation

for this is:

ΔG = ΔGvdw + ΔGhbond + ΔGelec +ΔGconform +ΔGtor + ΔGsol

The ΔGs are determined via multiple linear

regression.

3. METHODOLOGY AND COMPUTATIONAL

DETAILS

The molecules were first subjected to molecular

mechanics correction and later optimised using

the MOPAC [21] software at the RM1 level.

These optimized geometries were then used to

arrive at higher basis set geometries such as RHF

3-21G, 3-21 G*, Moller Plasset Second order

(MP2) 6-31, 6-31 d and subsequently at the DFT

level using Double Zeta Valence d+ basis set

(DZV d+) at the Becks' 3 Parameter and Lee Yang

Parr method (B3LYP) using GAMESS [22]. To

obtain electronic population analyses of all the

atoms Mulliken [23], Lowdin [24] and Natural

Population Analysis [25] were done.

The optimized geometry using DZV method was

used for docking purpose. The crystal structure of

Cytochrome CYP2B6 was obtained from Juregen

Pleiss [29]. Water molecules were completely

excluded from the receptor for the study. The

suitable programme selected for the docking study

was AutoDock 4.2 [26] using the MGL Tools

interface [27] that ran on the Cygwin [28], a Unix

like interface running on windows. The target

protein was kept rigid and the ligand was kept

flexible. Kollman charges [29] were assigned to

the ligand and polar hydrogen atoms were added

to the receptor PDB file. The heme protein was

considered as a place for docking or binding.

Ligands were assigned Gasteiger charges [30] by

the program and the non-polar hydrogens were

merged. As AutoDock required pre-calculated

grid maps, one for each atom time, present in the

ligand being docked the same was assigned. The

grid was centered near the heme protein and

closer to F-helix (Fig. 2). For this purpose the

grid box size was kept at 80, 80, 80 Ao (x,y,z).

The dockings were run on a dual core AMD

machine.

4. RESULTS AND DISCUSSION

The general structure of the molecule is given in

Fig. 1. The best docked configurations for the

ligands from the docked results were used to

calculate interaction energies. The binding energy

score is given in Table 6.1 which shows many

favourable values other than for thiotepa itself

(TTP10). Some of them had not docked and

interactions were absent. TTP1 had the highest

score, but it was simply positioned in the gorge of

the receptor without any interaction, at a distance

of about 3.616 Ao from the Fe atom in heme

protein with one of the Cyclopropane (CP) ring

directed towards it. Being non-polar it perhaps did

not favour binding to the enzyme. It showed

minimum negative surface potential (Vs,max= -

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0.0332) from ESP measurements done in the

previous section. With the introduction of the N-

atom the CP ring was oriented away from the

heme and one of the CP ring without N atom was

at a distance of about 4.541 Ao (Fig. TTP2).

Further introduction of the N-atom lead to the

same thing; orientation of the N-atom contained

CP ring away from the heme group , but at a

distance of about 4.384 Ao (Fig. TTP3). The

incorporation of the third N-atom made the

molecule symmetrical and the distance from the

Fe atom to the ligand became 15.162 Ao. The

molecule was found drifted away from the heme

molecule. Another strand of protein was found in

between the ligand and the heme. This was

evident from the fact that TTP3 & 4 had higher

range of ESP values than TTP10 and had lower

binding energy values lower than TTP10 which

had a (Vs,max) value of -0.1, interestingly, as it can

be taken as standard. Some significant

interactions were found for TTP10. The C of

CP was at a distance of about 3.453 Ao with Fe-

heme. The S atom (R19) was at a distance of

about 6.177Ao from Fe-heme and was found

closer to the heme protein residue.

TTP 3,6,11 and 12 were just above TTP10s‘ Vs,max

value. The docking scores were not in that order.

Hence molecular descriptor values can be made

accountable for the observed property. Drift

towards the heme molecule was found for all

excepting TTP1 as all these molecules are

conceptual and for the same there is a reason that

mere inclusion of N and P atoms does not make

the molecule a better place for being consideration

as an effective drug. TTP11 has the maximum

negative surface potential (-0.12), a good surface

potential range (0.1817) and has a good score

from docking too. The Fe-CP distance was about

6.689Ao and the oxygen atom attached to P (P=O)

was at a distance of about 5.502 Ao. To the lower

end of the heme molecule this O had interaction at

a distance of about 4.249Ao and can be considered

for studies. But this molecule had higher

electrophilicity (ω=0.0057 and 0.0672 at MP2 and

DFT respectively) values which would mean that

this molecule might prove to be toxic. This may

be due to the presence of Oxygen atom. The

TTP11 has the highest ω+ values, which has the

fourth lowest LBE value. The molecule 11 with

P=O in the C3 axis may be considered unfit for the

drug study because of this as it accounts for

toxicity. The values change upon introduction of

N atoms at the base (CP rings). TTP 13, 14, 15

and 16 are the molecules having substituents on

them. The range (ESP) is lower for these than

TTP10, Score higher than TTP10 and Vsmax

higher than the TTP10. The Fe-CP distance was

4.844 A, S-Fe length 4.290 Ao and S-C residue

distance was 5.281 Ao. Fluorine atom was directed

away from the heme molecule (Fig. TTP13).

Another interesting interaction was found between

the C to which O is attached within the heme and

the P atom was found at 3.907Ao. The molecule

appears to be in the midst of many interactions.

If the local philicity is considered for atom 19

(R19), the highest ω- value is shown by TTP7,

that has a P atom in it. TTP6 has the highest ω+

value if the local philicity Index is considered for

atom 8 at MP2 by finite method. The LBE

(lowest binding energy) value is the second

lowest. The first five molecules in the series show

no philicity trend either for ω- or ω+. But the LBE

rises up to molecule four, i.e., when the N atoms

are included. The LBE value drops for TTP5,

when P atom is introduced. It can be inferred here

that the inclusion of N atoms in the CP rings has

no effect on the overall philicity of the molecule

and these do not posses any reactivity property.

However the presence of N atoms is not

favourable for docking too. A look at the Surface

charge, Vs,max values using MEP suggests that all

of them are having values lesser values than

TTP10, and in a peculiar trend. For the first five

molecules as the Vs,max is lowered the LBE raises

and when the potential became higher the LBE

values get lowered. The highest Vs,max potential is

shown by TTP1 which has the highest LBE.

The distance between C of CP (on which NO was

substituted) and Fe-heme was 6.203Ao. These

were the maximum interactions found for any

analogue in the series and hence this molecule

may be tried for an in-vitro study to establish its

efficiency. Quite an opposite type of trend is

found for TTP12. Molecules 13, 14 and 15

provide a favorable trend as good binding ligands

if the ω- values are considered.

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TTP1

TTP2

N

TTP3

NN

TTP4

N

NN

TTP10

P(V)

N

N

N

S

TTP13

N

F

P(V)

N

N

S

TTP14

NCl

P(V)

N

N

S

TTP16

N

N

P(V)N

NS

O

Fig. TTPx, where x=the derivative number;

analogues docked (ball & stick, centre)

with CYP2B6 shown as ribbons

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5. CONCLUSION

From the above studies it may be concluded that

for a symmetrical molecule, the heteroatoms at the

principal axis provide a favourable explanation for

change in charges and binding energy upon

docking. That TTP 10 is a good candidate from

the electrophilicity studies, the fluoro and chloro

substituted TTP offer a better ‗performance‘ with

binding energies as far as electrophilicity and

surface charges are considered. The NO

substituent offers a peculiar behaviour and needs

an in-vitro study in view of the biologically

importance of this molecule. This was the only

molecule with maximum closer interactions.

TTP11 and 12 may be considered toxic based on

the electrophilicity values and unfavourable

scores. TTP6 and 15 that have only alkyl group

substituent on R19 and one of the CP rings are

also not favourable. The observation of the

complexes revealed that the alkyl group

substituents are oriented towards the F-heme

protein and those of halogens were quite opposite

to this, i.e. the halo groups oriented away from the

docking site but directed towards the putative

substrate entry. Koopmans' method using MP2

for the electrophilicity offers satisfactory

explanation for these studies, while the DFT fails

to offer plausible explanation of the behaviour of

the molecules of this type.

REFERENCES

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Algorithms; In: Schleyer P von R, Allinger

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Schaefer III H F, Schreiner P R (Ed): The

Encyclopedia of Computational Chemistry,

1998, Chichester, Wiley, p1127–1136.

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1983, 105, 7512.

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[18] Parr R G, Yang W; DFT of atoms and

molecules, 1989, Oxford University Press,

NY.

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Gohlke H, Luo R, Merz K M, Onufriev A,

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Computat. Chem., 2005, 26, 1668-1688.

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Table 1: The LBE and other parameters from the docking of TTP analogues with CYP2B6

SNo Molecule Lowest

Binding

Energy

LBE

Run

Mean

Binding

Energy

No

in

Cluster

Partition

Function

Q

Free

Energy

A

Internal

Energy

U

Entropy

1 TTP1 -5.63 1 -5.51 10 10.09 -1369.74 -5.51 4.58

2 TTP2 -3.81 1 -3.76 3 10.06 -1367.89 -3.65 4.58

3 TTP3 -2.77 9 -2.59 6 10.04 -1366.84 -2.61 4.58

4 TTP4 -2.20 10 -2.20 1 10.03 -1365.95 -1.71 4.58

5 TTP5 -3.24 3 -3.11 10 10.05 -1367.35 -3.11 4.58

6 TTP6 -4.70 10 -4.44 6 10.07 -1368.66 -4.42 4.58

7 TTP7 -4.66 4 -4.52 9 10.08 -1368.75 -4.51 4.58

8 TTP8 -4.30 7 -4.13 8 10.07 -1368.32 -4.08 4.58

9 TTP9 -3.94 9 -3.70 8 10.06 -1367.94 -3.70 4.58

10 TTP10 -3.66 4 -3.57 3 10.06 -1367.67 -3.44 4.58

11 TTP11 -4.40 3 -4.31 10 10.07 -1368.54 -4.31 4.58

12 TTP12 -3.15 7 -3.07 4 10.05 -1367.29 -3.05 4.58

13 TTP13 -3.67 3 -3.48 7 10.06 -1367.66 -3.43 4.58

14 TTP14 -4.16 8 -3.92 4 10.07 -1368.17 -3.93 4.58

15 TTP15 -3.98 2 -3.98 1 10.06 -1367.94 -3.70 4.58

16 TTP16 -4.81 10 -4.53 5 10.07 -1368.50 -4.27 4.58

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Theoretical investigation on the molecular structure of TPD in hole transport layer in

(OLED)

N. Nadeem Afroze and K. Subramani

Abstract - The molecular structure of TPD used as a hole transport layer in organic light emitting diode (OLED). The insertion of hole

blocking and electron injecting layer significantly improves the electroluminescent efficiency and extends the operational life time.

Optimized structure (TPD) neutral molecule in ground and excited states are carried out by Ab initio HF and DFT (B3LYP) method

with 6-311++ G(d,p) as basis set using Gaussian 03 W program. The difference between qualitative agreement with the results for

the neutral molecule in the ground state obtained by density functional theory methods, which were previously reported in the

literature. TPD in the ground state has a pronounced twisted geometry, with both central and peripheral dihedral angles of about 40º.

On the other hand, the structure of excited state is more planar, as the central dihedral angle becomes smaller. We estimated the

reorganization energies associated with the optical transitions, corresponding transition dipole moments, as well as HOMO and

LUMO. .

Index Terms – Optimized structure, TPD, DFT, HOMO, LUMO.

—————————— ——————————

1. INTRODUCTION

N‘- diphenyl-N,N‘-bis(3-methylphenyl)-(1,1‘-

biphenyl)- 4,4‘-diamine (TPD, see Fig. 1) is a

prototypical organic compound used in multi-

layer emitting devices as a hole transporting

material. Its transporting properties were

extensively studied both by experimental and

theoretical methods [1–5]. Recently, more

attention was given to its optical properties, i.e.

absorption and emission behavior [6,7]. It was

found that TPD is also a promising material for

the development of organic lasers, showing

stimulated emission both in neat films and diluted

in polystyrene (PS) [8–12]. Measurements of the

dependence of the signal with the length of the

pump stripe demonstrated that the mechanism

responsible for the amplification process and

narrow emission spectra is amplified spontaneous

emission (ASE). Most organic molecules that

show stimulated emission need to be diluted in a

host matrix in order to show laser activity.

——————————————

N. Nadeem Afroze is pursuing Ph.D. in Chemistry in the Department of Chemistry, Islamiah College, Vaniyambadi, India. E-mail: nadinfo@gmail.com

K. Subramani is serving in the Departmnet of Chemistry, Islamiah College, Vaniyambadi, India.

Few materials, such as various thiophene-based

oligomers [13–15], as well as several spiro-type

materials [16,17], have shown laser action in the

form of neat films. In the case of the

oligothiophenes, their functionalization with

thienyl-S,S-dioxide groups led to high solid-state

PL efficiencies [18] while keeping good chemical

stability. This increase in PL efficiency was due to

the reduction of nonradiative processes in the

functionalized molecules induced by the variation

of both the supramolecular organization in the

solid state and the single molecule properties [19].

CH3

N N

CH3

Figure -1. N,N‘-diphenyl-N,N‘-bis (3-methylphenyl)-

(1,1‘-biphenyl)- 4,4‘-diamine (TPD)

On the other hand, for the spiro-type materials, the

concept of spirolinkage of a chromophore to other

molecular group was used in order to enhance film

quality that has been proven to be a relevant factor

N

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in preventing the PL quenching due to

intermolecular interaction. Very recently, the

usage of tert-butyl groups in sexiphenyl molecules

[20] has served to prevent crystallization of the

dye molecules, so stimulated emission was

observed, as contrary to the simple p-sexiphenyl

molecule. As previously mentioned, TPD has also

shown stimulated emission in neat films. Since

inter chain interactions seems to play a major role

in the luminescence and hence the laser properties

of organic systems [21], the study of the inter

chain effects in TPD would be important to

understand its optical properties. There exist some

density functional theoretical studies of the

geometric structure and energetics of TPD

[22,23], that have been compared with the crystal

structure determined crystallographically [24,25].

However, the main objective of these reports was

to relate the structure of TPD with its transport

properties, since TPD has been extensively used

as a hole transporting material in LED devices.

Concerning the optical properties of TPD, its

optical and lasing properties have not been

reported so far. In this paper, we report results

related to the first-principle investigations on

density functional theory which shows the optical

properties of the neutral TPD molecule.

2. EXPERIMENTAL

The optical properties of a single N,N‘-diphenyl-

N,N‘- bis (3-methylphenyl) - (1,1‘-biphenyl) -

4,4‘ diamine (TPD) molecule , we treat the

electronic ground and excited states. Theoretical

calculations were performed by first principle

method at the Hartree–Fock (HF) or single

excitation CIS level with 6-311++ G basis set,

using Gaussian 03W program package [26].

3. OPTIMIZED GEOMETRIES

The molecular structure of TPD is displayed in

Fig. 1. It consists of a central biphenyl core and

two twisted triphenylamine terminal wings,

extended with functional methyl groups. The main

structural parameters of the geometry optimization

for various TPD states in trans-type geometry

(with tolyl rings on opposite sides) are reported in

Table 1.

Table 1. HF( CIS) optimized geometrical parameters of

TPD in ground and exited states

Rc is the central bond length; Lc the length of N–C

bond with the central biphenyl part; Lt the length

of

N–C bonds with the terminal rings; α the central

dihedral angle; β the dihedral angle for the central

phenyl ring, measured relative to the terminal ring

with the methyl group; γ the dihedral angle for the

central phenyl ring, measured relative to the

terminal ring without the methyl group; ν the

torsion angle of the terminal ring with the methyl

group; µ is the torsion angle of the terminal ring

without the methyl group.

Figure -2. Optimized structure of N,N‘-diphenyl-N,N‘-

bis (3-methylphenyl)-(1,1‘-biphenyl)-

4,4‘-diamine (TPD)

In the case of the neutral molecule in the ground

state, we obtained a twisted geometry that is in

good agreement with previously reported structure

based on density functional theory (DFT) [22,23].

The central biphenyl bond of 1.49A˚ and dihedral

angle of 42.2˚ are comparable with the theoretical

findings of 1.48 A˚ and 33.8˚reported in Ref. [22]

and 34.9˚ reported in Ref. [23].

State

Geometrical parameters

Rc

(A˚)

Lc

(A˚)

Lt

(A˚)

α

(˚)

β

(˚)

γ

(˚)

ν

(˚)

µ

(˚)

Ground 1.49 1.42 1.42 42.2 45.2 45.5 45.6 46.3

Excited 1.42 1.40 1.42 4.6 38.5 38.4 43.6 44.2

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4. RESULTS AND DISCUSSION

The entire set of calculation were performed at

density functional theory (DFT) through the

energetic positions of excitations below the

ionization potential (of 6.25 eV determined by HF

method as an energy of a cation state in the

ground state geometry, comparable with the

experimental value of 6.69 eV [27] ), we see that

the spectra are dominated by the HOMO–LUMO

transition, estimated at 6.61 eV. Focusing on the

ground state and the first excited state, we

calculated the energies of analysed electronic

states for each optimized geometry (see Table 2).

Table 2. Relative energies ( in eV) for ground

and excited states versus geometry

configuration of TPD

According to the Franck–Condon principle, the

shape of the molecule immediately before and

after the electronic transition is practically the

same. The fast electronic transitions are thereafter

followed by slow geometry relaxations towards

the equilibrium geometries of the final electronic

states, see Fig. 2. The larger reorganization

energies correspond to the stronger electron–

vibrational coupling interaction and more

pronounced vibrational structure of related spectra

[28].

5. HOMO, LUMO ANALYSIS

The analysis of the wave function indicates that

the electron absorption corresponds to the

transition from ground to first excited and it is

mainly described by one electron excitation from

the highest occupied molecular orbital (HOMO)

and lowest unoccupied molecular orbital (LUMO)

are very important parameters for quantum

Chemistry. We determine the way the molecule

interacts with other species; hence, they are called

frontier orbital. HOMO in the ground state

geometry is rather delocalized on all benzene

rings (see Figure-3).

Figure -3. HOMO, LUMO

The energies of HOMO and LUMO in the ground

state geometry are EHOMO (g) = – 6.80 eV and

ELUMO (g) = 2.73 eV, respectively. The HOMO-

LUMO gap corresponding to optical spectra,

becomes ELUMO(g) – EHOMO (g) = 9.53 eV.

The energies of HOMO and LUMO in the excited

state geometry are EHOMO(e) = 2.36 eV,

respectively. The HOMO-LUMO gap,

corresponding to PL , is decreased to ELUMO (e) –

EHOMO (e) = 8.56 eV. The absolute energies of 9.53

and 8.56 eV are considerably overestimated

comparing with the previously obtained around

18, 16, and 14A˚ for the, 20 and 30 wt% doped

films. These energy gaps reflect the chemical

activity of the molecule, it is possible to say that

the change of distribution by a change of basis set.

Electronic state Geometry

Ground Excited

Ground 0.0 1.46

Excited 6.61 5.99

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6. CONCLUSION

The optimized geometric structures and energies

of TPD during the optical transitions in the hole

transport layer we presented a semi-empirical

model for TPD absorption and

photoluminescence. Significantly different

molecular shapes in ground and excited state

electronic potentials lead to different spatial

localization of HOMO and LUMO and large

reorganization energies. This is reflected into a

large Stokes shift, estimated to 0.46 eV. Finally,

we generalize that the possible contribution of

TPD, optimized structure with electron hole

transporting layer which aggregates TPD has

optical properties in organic light emitting diodes

(OLED).

ACKNOWLEDGEMENTS

The first author is thankful to Vaniyambadi

Muslim Educational Society for providing

necessary facilities to carry out this research work.

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Sum Labeling for Some Cycle Related Graphs

J. Gerard Rozario, J. Jon Arockiaraj, P. Lawrence Rozario Raj and U. Rizwan

Abstract - A sum labeling is a mapping 𝜆 from the vertices of G into the positive integers such that, for any two vertices

u, v 𝜖 V (G) with labels 𝜆(u) and 𝜆(v), respectively, (uv) is an edge iff 𝜆(u) + 𝜆(v) is the label of another vertex in V (G).

Any graph supporting such a labeling is called a sum graph. It is necessary to add (as a disjoint union) a component to

sum label a graph. This disconnected component is a set of isolated vertices known as isolates and the labeling scheme

that requires the fewest isolates is termed optimal. The number of isolates required for a graph to support a sum labeling

is known as the sum number of the graph. In this paper, we will give optimal sum labeling scheme for some cycle related

graphs.

Index Terms - Sum Labeling, Sum number, Sum graph, Isolates

—————————— ——————————

1. INTRODUCTION

LL the graphs considered here are simple,

finite and undirected. For all terminologies

and notations we follow Harary [3] and graph

labeling as in [2]. Sum labeling of graphs was

introduced by Harary [4] in 1990. Following

definitions are useful for the present study.

Definition 1.1 A Sum Labeling is a mapping 𝜆

from the vertices of G into the positive integers

such that, for any two vertices u, v 𝜖 V (G) with

labels 𝜆(u) and 𝜆(v), respectively, (uv) is an edge

iff 𝜆(u) + 𝜆(v) is the label of another vertex in V

(G). Any graph supporting such a labeling is

called a Sum Graph. Definition 1.2 It is necessary to add (as a disjoint

union) a component to sum label a of isolated

——————————————

J. Gerard Rozario is serving in the Department of Mathematics, St. Joseph’s College of Arts and Science, Cudallore, India. E-mail: rozario.gerard@yahoo.com

J. Jon Arockiaraj is serving in the Department of Mathematics, St. Joseph’s College of Arts and Science, Cudallore, India. E-mail: jonarockiaraj@gmail.com

P. Lawrence Rozario Raj is serving in the Department of Mathematics, St. Joseph College, Trichy, India. E-mail : lawraj2006@yahoo.co.in

U. Rizwan is serving in the Department of Mathematics, Islamiah College, Vaniyambadi, India. E-mail : rizwanmaths@yahoo.com

vertices known as Isolates and the labeling

scheme that requires the fewest isolates is termed

Optimal.

Definition 1.3 The number of isolates required for

a graph G to support a sum labeling is known as

the Sum Number of the graph. It is denoted as

𝜎 𝐺

Definition 1.4 Flower Pot Cracker Graph is

obtained from cycle of m vertices and spider of n

vertices, by joining the center vertex of spider and

any one vertex of the cycle. It is denoted by

FPC(Cm,SP( m21 xm

x2

x1 aaa ,...,, )) where a1 < a2 < ... <

am, x1+x2+...+xm = k and a1x1+a2x2+...+amxm= n-

1. For our convenience, we write FPC(Cm,SP(m21 xm

x2

x1 aaa ,...,, )) as FPC(Cm,SPn) or FPCm,n.

Definition 1.5 Consider n copies of stars

namely 𝐾1,𝑚(1)

, 𝐾1,𝑚(2)

, …… , 𝐾1,𝑚(𝑛)

. Then the graph G =

𝐾1,𝑚(1)

𝐾1,𝑚(2)……𝐾1,𝑚

(𝑛) is the graph obtained

by joining apex vertices of each 𝐾1,𝑚𝑝−1 and 𝐾1,𝑚

𝑝 by

an edge as well as to a new vertex 𝑥𝑝−1 where 2 ≤

𝑝 ≤ 𝑛.

Here G has 𝑛(𝑚 + 2) − 1 vertices and 𝑛(𝑚 +3) − 3 edges.

In this paper, we will prove that FPCm,n , the

graph 𝐾1,𝑚(1)

𝐾1,𝑚(2)……𝐾1,𝑚

(𝑛) are

optimal summable with sum number 1.

A

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2. OPTIMAL SUM LABELING SCHEME

FOR 𝑭𝑷𝑪𝒑,𝒕 AND 𝐾1,𝑚(1)

𝐾1,𝑚(2)……𝐾1,𝑚

(𝑛)

Jeff Ginn [5], gave the sum labeling for unicyclic

graphs with sum number as

𝜎 𝑈𝐶3 = 1 ; 𝜎 𝑈𝐶4 = 3 ; 𝜎 𝑈𝐶𝑛 = 3

𝑓𝑜𝑟 𝑛 ≥ 5 and sum labeling for a special class of

multicyclic graphs. Lawrence Rozario and Gerard

Rozario [6], proved that 𝐹𝑃𝐶𝑝 ,𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑝 >

3 & 𝑡 > 1 is a combination graph. Dani, Nilish A

[1], proved that the graph

𝐾1,𝑚(1)

𝐾1,𝑚(2)……𝐾1,𝑚

(𝑛) is a 3-equitable graph.

In this section, we prove that 𝐹𝑃𝐶𝑝 ,𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑝 ≥

3 & 𝑡 > 1 and 𝐾1,𝑚(1)

𝐾1,𝑚(2)……𝐾1,𝑚

(𝑛) are sum

graphs with sum number 1.

Theorem. 2.1 𝐹𝑃𝐶𝑝 ,𝑡 is optimal summable with

sum number 1 for all 𝑝 ≥ 3 and 𝑡 > 1.

Proof. Let G = 𝐹𝑃𝐶𝑝 ,𝑡 where 𝑝 ≥ 3 𝑎𝑛𝑑 𝑡 > 1.

G has a cycle of p vertices and a spider of t

vertices. G has p + t – 1 vertices. Let v1, v2,

v3,……,vp be the vertices of the cycle where v1 is

the center. vp+1, vp+2…,vp+t-1 be the vertices of

spider. Let ‗m‘ be number of paths in spider and

ni be length of ith

path, where i = 1, 2, 3,…,m.

Define f : V(G) {1,2,3,…, N}

Case (i) p is odd number

Case (i) (a) p =3

For C3, 𝑓(𝑣1) = 1; 𝑓(𝑣2) = 2; 𝑓(𝑣3) = 3 For Spider,

𝑓(𝑣11) = 𝑓(𝑣1) + 𝑓(𝑣3) 𝑓𝑜𝑟 1 ≤ 𝑖 ≤ 𝑚

𝑓 𝑣𝑖2 = 𝑓 𝑣1 + 𝑓 𝑣𝑖1

𝑓 𝑣𝑖𝑗 = 𝑓 𝑣𝑖 𝑗−1 + 𝑓 𝑣𝑖 𝑗−2 ;

3 ≤ 𝑗 < 𝑛𝑖

𝑓 𝑣 𝑖+1 1 = 𝑓 𝑣𝑖𝑗 + 𝑓 𝑣𝑖 𝑗−1 ;

𝑗 = 𝑛𝑖 𝑖 ≠ 𝑚

𝑓 𝑥 = 𝑓 𝑣𝑚 𝑛𝑚 + 𝑓 𝑣𝑚(𝑛𝑚 −1) ,

where x is the isolated vertex.

Hence 𝐹𝑃𝐶3,𝑡 is optimal summable with sum

number 1.

Case (i) (b): p = 5, 7

For C5, C7

𝑓(𝑣1) = 2; 𝑓(𝑣2) = 1; 𝑓(𝑣3) = 3

𝑓(𝑣𝑝) = 𝑓(𝑣1) + 𝑓(𝑣3)

𝑓 𝑣𝑖 = 𝑓 𝑣𝑖−1 + 𝑓 𝑣𝑖−2 ; 4 ≤ 𝑖 𝑝 − 1

For Spider,

𝑓 𝑣11 =

𝑓 𝑣𝑝 + 𝑓 𝑣𝑝−1 𝑖𝑓 𝑓 𝑣𝑝 < 𝑓 𝑣𝑝−1

𝑓 𝑣𝑝−1 + 𝑓 𝑣𝑝−2 𝑖𝑓 𝑓 𝑣𝑝 > 𝑓 𝑣𝑝−1

𝑓𝑜𝑟 1 ≤ 𝑖 ≤ 𝑚

𝑓 𝑣𝑖2 = 𝑓 𝑣1 + 𝑓 𝑣𝑖1

𝑓 𝑣𝑖𝑗 = 𝑓 𝑣𝑖 𝑗−1 + 𝑓 𝑣𝑖 𝑗−2 ;

3 ≤ 𝑗 < 𝑛𝑖

𝑓 𝑣 𝑖+1 1 = 𝑓 𝑣𝑖𝑗 + 𝑓 𝑣𝑖 𝑗−1 ;

𝑗 = 𝑛𝑖 𝑖 ≠ 𝑚

𝑓 𝑥 = 𝑓 𝑣𝑚 𝑛𝑚

+ 𝑓 𝑣𝑚(𝑛𝑚 −1) ,

where x is the isolated vertex.

Hence 𝐹𝑃𝐶𝑝 ,𝑡 is optimal summable with sum

number 1 where p = 5, 7.

Case (i) (c) 𝑝 ≥ 9

For Cp,

𝑓 vp−3

2

= 1;

𝑓 v

p−3

2− 1

= 2;

𝑓 v

p−3

2 + 1

= 3;

𝑓 𝑣 𝑝−3

2− 2

= 5;

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𝑓 𝑣 𝑝−3

2 + 2

= 4

𝑓𝑜𝑟 1 ≤ 𝑖 ≤𝑝−9

2

𝑓 𝑣

𝑝−3

2 + 𝑖 + 2

= 𝑓 𝑣 𝑝−3

2 – 𝑖

+ 𝑓 𝑣 𝑝−3

2 – 𝑖 −1

𝑓 𝑣 𝑝−3

2 – 𝑖 – 2

= 𝑓 𝑣 𝑝−3

2 + 𝑖 +1

+ 𝑓 𝑣 𝑝−3

2 + 𝑖 + 2

𝑓 𝑣𝑝−3 = 𝑓 𝑣1 + 𝑓 𝑣2 ;

𝑓 𝑣𝑝 = 𝑓 𝑣𝑝−3 + 𝑓 𝑣𝑝−4

𝑓 𝑣𝑝−2 = 𝑓 𝑣1 + 𝑓 𝑣𝑝 ;

𝑓 𝑣𝑝−1 = 𝑓 𝑣𝑝−3 + 𝑓 𝑣𝑝−2

For Spider,

𝑓(𝑣11) = 𝑓(𝑣𝑝) + 𝑓 𝑣𝑝−1 𝑓𝑜𝑟 1 ≤ 𝑖 ≤ 𝑚

𝑓 𝑣𝑖2 = 𝑓 𝑣1 + 𝑓 𝑣𝑖1

𝑓 𝑣𝑖𝑗 = 𝑓 𝑣𝑖 𝑗−1 + 𝑓 𝑣𝑖 𝑗−2 ;

3 ≤ 𝑗 < 𝑛𝑖

𝑓 𝑣 𝑖+1 1 = 𝑓 𝑣𝑖𝑗 + 𝑓 𝑣𝑖 𝑗−1 ;

𝑗 = 𝑛𝑖 𝑖 ≠ 𝑚

𝑓 𝑥 = 𝑓 𝑣𝑚 𝑛𝑚 + 𝑓 𝑣𝑚(𝑛𝑚 −1) ,

where x is the isolated vertex.

Hence 𝐹𝑃𝐶𝑝 ,𝑡 is optimal summable with sum

number 1 if p is odd.

Case (ii): p is even number

Case (ii) (a): p = 4

For C4,

𝑓 𝑣1 = 1;

𝑓 𝑣2 = 2;

𝑓 𝑣4 = 3;

𝑓(𝑣3) = 𝑓(𝑣1) + 𝑓(𝑣4)

For Spider,

𝑓(𝑣11) = 𝑓(𝑣2) + 𝑓(𝑣3) 𝑓𝑜𝑟 1 ≤ 𝑖 ≤ 𝑚

𝑓 𝑣𝑖2 = 𝑓 𝑣1 + 𝑓 𝑣𝑖1

𝑓 𝑣𝑖𝑗 = 𝑓 𝑣𝑖 𝑗−1 + 𝑓 𝑣𝑖 𝑗−2 ;

3 ≤ 𝑗 < 𝑛𝑖

𝑓 𝑣 𝑖+1 1 = 𝑓 𝑣𝑖𝑗 + 𝑓 𝑣𝑖 𝑗−1 ;

𝑗 = 𝑛𝑖 𝑖 ≠ 𝑚

𝑓 𝑥 = 𝑓 𝑣𝑚 𝑛𝑚 + 𝑓 𝑣𝑚(𝑛𝑚 −1) ,

where x is the isolated vertex.

Hence 𝐹𝑃𝐶4,𝑡 is optimal summable with sum

number 1.

Case (ii) (b): 𝑝 ≥ 6

For Cp,

𝑓 𝑣1 = 1; 𝑓 𝑣2 = 2;

𝑓 𝑣𝑝 = 3;

𝑓 𝑣3 = 𝑓 𝑣2 + 𝑓 𝑣𝑝

𝑓(𝑣𝑝−1) = 𝑓(𝑣1) + 𝑓(𝑣𝑝)

𝑓𝑜𝑟 4 ≤ 𝑖 ≤𝑝

2

If i is even,

𝑓 𝑣𝑖 = 𝑓 𝑣𝑖−1 + 𝑓 𝑣𝑖−2 − 1

𝑓 𝑣𝑖+ 𝑝−2𝑖+2 = 𝑓 𝑣𝑖+ 𝑝−2𝑖+2 +1 +

𝑓 𝑣𝑖+ 𝑝−2𝑖+2 +2

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If i is odd,

𝑓 𝑣𝑖 = 𝑓 𝑣𝑖−1 + 𝑓 𝑣𝑖−2

𝑓 𝑣𝑖+ 𝑝−2𝑖+2 = 𝑓 𝑣𝑖+ 𝑝−2𝑖+2 +1

+ 𝑓 𝑣𝑖+ 𝑝−2𝑖+2 +2 − 1

𝑓 𝑣 𝑝

2 + 1

= 𝑓 𝑣 𝑝

2 + 𝑓 𝑣

𝑝

2 − 1

For Spider,

𝑓(𝑣11)

=

𝑓 𝑣

𝑝2 + 𝑓 𝑣

𝑝2

+1

𝑖𝑓 𝑓 𝑣 𝑝2 < 𝑓 𝑣

𝑝2

+2

𝑓 𝑣 𝑝2

+1 + 𝑓 𝑣

𝑝2

+2

𝑖𝑓 𝑓 𝑣 𝑝2 > 𝑓 𝑣𝑝+2

𝑓𝑜𝑟 1 ≤ 𝑖 ≤ 𝑚

𝑓 𝑣𝑖2 = 𝑓 𝑣1 + 𝑓 𝑣𝑖1

𝑓 𝑣𝑖𝑗 = 𝑓 𝑣𝑖 𝑗−1 + 𝑓 𝑣𝑖 𝑗−2 ;

3 ≤ 𝑗 < 𝑛𝑖

𝑓 𝑣 𝑖+1 1 = 𝑓 𝑣𝑖𝑗 + 𝑓 𝑣𝑖 𝑗−1 ;

𝑗 = 𝑛𝑖 𝑖 ≠ 𝑚

𝑓 𝑥 = 𝑓 𝑣𝑚 𝑛𝑚

+ 𝑓 𝑣𝑚(𝑛𝑚 −1) ,

where x is the isolated vertex.

Hence 𝐹𝑃𝐶𝑝 ,𝑡 is optimal summable with sum

number 1 where 𝑝 ≥ 6 .

Thus, 𝐹𝑃𝐶𝑝 ,𝑡 is optimal summable with sum

number 1 for all p≥ 3 and t >1.

Illustration 2.1 Sum labeling for 𝐹𝑃𝐶𝑝 ,𝑡 is given

in the figure 2.1

FPC 5,11

FPC 5,10

Figure 2.1

Theorem: 2.2 < 𝐾1,𝑚(1)

𝐾1,𝑚(2)……𝐾1,𝑚

(𝑛)> is

a sum graph with sum number 1.

Proof. Let G = < 𝐾1,𝑚(1)

𝐾1,𝑚(2)……𝐾1,𝑚

(𝑛) >.

The vertex set of

V(G) = {c1, c2,……,cn,

x1,x2,……,xn-1, v11,v12,…,v1m,

v21,v22,…v2m,……,vn1,vn2,… vnm}

where ci is apex vertex of 𝐾1,𝑚(𝑖)

𝑤𝑕𝑒𝑟𝑒 𝑖 =

1,2,3, …𝑛. x1,x2,…,xn–1 be vertices such that ci–1

and ci are adjacent to xi–1, where 2 i n of G and

v11,v12,…,vnm are pendant vertices. Let y be the

isolated vertex.

76

71

2

5 4

13

27116

337

231

11 01

28

29

57

58

115

568

79

16

2

5

4

1

3

81

34

36

70

106108

214

322

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n = 4; m = 5

n = 4; m = 5

Figure 2.2

Define f : V(G) {1,2,3,…, N}

𝑓 𝑐𝑖 = 𝑖 ; 𝑓𝑜𝑟 1 ≤ 𝑖 ≤ 𝑛

Case: (i) n is even

𝑓 𝑥𝑖 = 𝑓 𝑐𝑛−𝑖+2 + 𝑓 𝑐𝑛−𝑖+3 ; 𝑓𝑜𝑟 ( 𝑛 − 1) ≥ 𝑖 ≥ 𝑛

2

𝑓 𝑥𝑖 = 𝑓 𝑥𝑖+1 + 1 ; 𝑓𝑜𝑟 (𝑛 2 − 1) ≥ 𝑖 ≥ 1

𝑓(𝑣11) = 𝑓(𝑥1) + 𝑓 𝑐1

𝑓𝑜𝑟 1 ≤ 𝑖 ≤ 𝑛,

𝑓 𝑣𝑖(𝑗 +1) = 𝑓 𝑣𝑖𝑗 + 𝑓 𝑐𝑖 ; 1 ≤ 𝑗 < 𝑚

𝑓 𝑣 𝑖+1 1 = 𝑓 𝑣𝑖𝑗 + 𝑓 𝑐𝑖 ; 𝑗 = 𝑚 𝑖 ≠ 𝑛

𝑓 𝑦 = 𝑓 𝑣𝑛𝑚 + 𝑓 𝑐𝑛 ,

where y is the isolated vertex.

Hence G is optimal summable with 1 isolated

vertex when n is even.

Case: (ii) n is odd

𝑓 𝑥𝑖 = 𝑓 𝑐𝑛−𝑖+

𝑛−1 2

+ 𝑓 𝑐𝑛−𝑖+

𝑛−1 2

+1 ;

𝑓𝑜𝑟 ( 𝑛 − 1) ≥ 𝑖 ≥(𝑛+1)

2

𝑓 𝑥𝑛−1

2

= 𝑓 𝑥𝑛+1

2

+ 2

𝑓 𝑥𝑖 = 𝑓 𝑥𝑖+1 + 1 ; 𝑓𝑜𝑟 (𝑛−3)

2≥ 𝑖 ≥ 1

𝑓(𝑣11) = 𝑓(𝑥1) + 𝑓 𝑐1

𝑓𝑜𝑟 1 ≤ 𝑖 ≤ 𝑛,

𝑓 𝑣𝑖(𝑗 +1) = 𝑓 𝑣𝑖𝑗 + 𝑓 𝑐𝑖 ; 1 ≤ 𝑗 < 𝑚

𝑓 𝑣 𝑖+1 1 = 𝑓 𝑣𝑖𝑗 + 𝑓 𝑐𝑖 ; 𝑗 = 𝑚 𝑖 ≠ 𝑛

𝑓 𝑦 = 𝑓 𝑣𝑛𝑚 + 𝑓 𝑐𝑛 ,

where y is the isolated vertex.

1 2 3 4

5

1211

8 79

10 13

16

14

18

22

24

20

27

33

36

30

39

43

51

55

47

59

1 2 3 4 5

12 11 9 713

15 16

1417

18

22 24

20

26

28

34 37

31

40

43

51 55

47

59

63

73 78

68

8388

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Thus, with just one isolated vertex we are able to

identify all the edges of G.

Hence, the graph < 𝐾1,𝑚(1)

𝐾1,𝑚(2)……𝐾1,𝑚

(𝑛)>

is a sum graph with sum number 1.

Illustration 2.2 Sum Labeling for

< 𝐾1,5(1)

𝐾1,5(2)𝐾1,5

(3)𝐾1,5

(4)>

and < 𝐾1,5(1)

𝐾1,5(2)𝐾1,5

(3)𝐾1,5

(4)𝐾1,5

(5)>

are shown in figure 2.2

REFERENCES

[1] Dani, Nilish A, Study of some interesting

topics in theory of graphs, Saurashtra

University, 2010.

[2] Gallian J A, A dynamic survey of graph

labeling, The Electronics Journal of

Combinatorics, 16, (2009) DS6.

[3] Harary F, Graph theory, Addison Wesley,

Reading, Massachusetts, 1972.

[4] Harary F, Sum graphs and Difference graphs,

Congress Numerantium, no.72, 101-108,

1990.

[5] Jeff Ginn, The Sum Number of a Unicyclic

Graph, Central Michigan University, NSF

DMS – 0097394, 2002.

[6] P. Lawrence Rozario Raj and J. Gerard

Rozario, Combination Labeling for Some

Cycle and Star Related Graphs, Proceedings

of the Heber International Conference on

Applications of Mathematics and Statistics,

Excel India Publishers, New Delhi, 2012, pp.

500-505.

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Multiple Labeling Approach for Finding Shortest Path with Intuitionistic Fuzzy Arc Length

A.Nagoor Gani and M.Mohammed Jabarulla

Abstract - This paper presents a modified multiple labeling method for finding the shortest path in an intuitionistic fuzzy networks. An

intuitionistic fuzzy shortest path problem on a network in which an intuitionistic fuzzy number instead of a fuzzy number is assigned to

each edge. An algorithm for solving the problem is developed and a shortest path derived from the network. An intuitionistic fuzzy

graphs along with generalizations of algorithms for finding optimal paths with in them have emerged as an adequate modeling tool for

imprecise system. Inputs and outputs of the proposed algorithm are intuitionistic fuzzy numbers. Finally an illustrative numerical

example is given to demonstrate the proposed approach.

Index Terms – Intuitionistic Fuzzy Set (IFS), Intuitionistic Fuzzy Graph (IFG), Shortest Path, Intuitionistic Fuzzy Shortest Path,

Intuitionistic Fuzzy Number (IFN), Trapezoidal Intuitionistic Fuzzy Number(TrIFN), Multiple labeling.

—————————— ——————————

1. INTRODUCTION

HE problem of finding the shortest path from a

specified source node to the other nodes is a

fundamental problem that appears in many

applications. It generates essential information in

transportation, routing and communications

applications.

The fuzzy shortest path problem was first

introduced by Dubois and Prade [2]. Generally

fuzziness can be introduced into the network in a

variety of ways. In Dubois and Prade [2]

discussed the solution to the first problem using

extended sum with Floyd‘s and Ford‘s algorithms

used to solve the problem. Kelin [4] presented by

hybrid multi criteria DP recursion and can find a

path or paths corresponding to the threshold of

membership degree set by a decision maker.

In this paper we propose an intuitionistic fuzzy

number instead of a fuzzy number. An algorithm

is based on the idea that from all the shortest paths

——————————————

A.Nagoor Gani is serving in the Department of Mathematics, Jmala

Mohamed College, Tiruchirapalli, India. E-mail:

ganijmc@yahoo.co.in

M.Mohammed Jabarulla i serving in the Department of Mathematics,

Jmala Mohamed College, Tiruchirapalli, India. E-mail:

m.md.jabarulla@gmail.com

from source node to destination node, the multiple

labeling method can be considered to be a

generalization of Dijkstra‘s algorithm to solve the

simple shortest path problem.

This paper is organized as follows. In Section 2,

we define trapezoidal intuitionistic fuzzy number

and its addition used through out this paper.

Some terminology about networks considered

here is provided. We define an order relation

among intuitionistic fuzzy numbers which is used

in the ranking path distance represented as

intuitionistic fuzzy numbers. In Section 3, we

define a mathematical formulation and an

algorithm for solving intuitionistic fuzzy shortest

path problem is derived on the basis of multiple

labeling method for a multi criteria shortest path.

In Section 4, the proposed algorithm is

numerically evaluated for any networks.

2. PRELIMINARY DEFINITIONS

The concepts of an intuitionistic fuzzy set was

introduced by Atanassov[1] to deal with

vagueness, which can be defined as follows.

2.1 Intuitionistic Fuzzy Set

Let X be an universe of discourse, then an

Intuitionistic Fuzzy Set (IFS) A in X is given by

A = { (x, µA(x), A(x)) / x X }

T

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where the functions µA(x) : X→[0,1] and A(x) :

X→[0,1] determine the degree of membership and

non membership of the element x X,

respectively and for every x X, 0 µA(x) +

A(x) 1. 2.2 Intuitionistic Fuzzy Graph

Let X be an universe, containing fixed graph

vertices and let E X be a fixed set. Construct

the IFS E = { (x, µv(x), v(x)) / x X } where the

functions µv(x):X→[0,1] and v(x):X→[0,1]

determine the degree of membership and non

membership to set E of the element (edge) x X,

respectively and for every x X, such that 0

µv(x) + v(x) 1. 2.3 Intuitionistic Fuzzy Number [19]

Let A = { (x, µA(x), A(x)) / x X } be an IFS, then

we call the pair (µA(x), A(x)) an intuitionistic

fuzzy number.

For convenience, we denote an intuitionistic fuzzy

number by (<a,b,c,d>,<a’,b,c,d’>), where

< a,b,c,d > F(I), < a’,b,c,d’ > F(I), I = [0,1]. 2.4 Trapezoidal Intuitionistic Fuzzy Number [5]

A trapezoidal intuitionistic fuzzy number (TrIFN)

A is denoted by A = {( µA, A) | x R}, where µA

and A are membership and non-membership

functions. The trapezoidal intuitionistic fuzzy

number A is denoted by

' '

1 2 3 4 1 2 3 4, , , , , , ,A a a a a a a a a

where ' '

1 1 2 3 4 4 a a a a a a .The membership

and non-membership functions are as follows.

11 2

2 1

2 3

43 4

4 3

,

1, ( )

,

0,

A

x afor a x a

a a

for a x ax

a xfor a x a

a a

Otherwise

'21 2'

2 1

2 3

'33 4'

4 3

and

,

0, ( )

,

1,

A

a xfor a x a

a a

for a x ax

x afor a x a

a a

Otherwise

Figure . 1 Trapezoidal Intuitionistic Fuzzy Number

' '

1 2 3 4 1 2 3 4, , , , , , ,A a a a a a a a a

2.5 The Additio of Two Trapezoidal Intuitionistic

Fuzzy Number

Let any Trapezoidal Intuitionistic Fuzzy Numbers

' '

1 2 3 4 1 2 3 4, , , : , , , , :A a a a a a a a a

and ' '

1 2 3 4 1 2 3 4, , , : , , , , :B BB b b b b b b b b then

1 1 2 2 3 3 4 4

' ' ' '

1 1 2 2 3 3 4 4

, , ,

: . ,

, , ,

: .

B B

B

a b a b a b a b

A Ba b a b a b a b

2.6 Network Terminology [12]

Consider a directed network G(V, E), consisting of

finite set of nodes V = { 1,2,3,..n } and a set of m

directed edges E V X V. Each edge is denoted

by an ordered pair ( i, j ), Where i, j V and i j.

It is supposed that there is only one directed arc

( i, j ) from i to j. In this network we specify two

nodes, denoted by s and t , which are the source

node and destination node respectively. We

define a path ijP as a sequence

1 1 2 2 -1 -1 { , ( , ), ,..., , ( , ), } si l l l lP i i i i i i i i i j

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of alternating nodes and edges. The existence of

at least one path siP in G(V,E) is assumed for

every node i V – { s }.

ijl denotes the intuitionistic fuzzy number

associated with edge (i,j), corresponding to the

length necessary to traverse ( i, j ) from i to j. We

remark that each length corresponds to the cost ,

the time, etc in practical problems. The

intuitionistic fuzzy distance along the path P,

denoted as ( )d P is defined as ( , )

( ) ij

i j P

d P l

.

2.7 Order Relation

We consider an order relation among intuitionistic

fuzzy numbers using extension principle the IF

MAX and IF MIN are defined as follows

{ , }{ , }

( ) { ( ), ( )}, amax a b br max s t

r Sup min s t

where { , }

max a b

is the membership function of

{ , }.max a b

{ , }{ , }

( ) { ( ), ( )},amin a b br min s t

r Inf max s t

where { , }

min a b is the non-membership function of

{ , }.min a b

Definition. 2.7.1. Let a and b be intuitionistic

fuzzy numbers, then { , }a b min a b a

Definition. 2.7.2. For intuitionistic fuzzy number

a and b ,

[ ] [ ]

[ ] [ ] [0,1]

a b Sup a Sup b

and

Inf a Inf b for any

Definition. 2.7.3. The Order Relation of

and is defined as follows.

(i) For arbitrary intuitionistic fuzzy number

, ( ).a a a reflexivity

(ii) For arbitrary intuitionistic fuzzy numbers

,

( )

a and b a b and b c a c

Antisymmetry

(iii) For arbitrary intuitionistic fuzzy numbers

, ,

( ).

a b and c a b and b c a c

transitivity

Definition. 2.7.4. Let h be a real number in [0,1].

For intuitionistic fuzzy numbers a and b ,

[ ] [ ]

[ ] [ ] [h,1]

ha b Sup a Sup b

and

Inf a Inf b for any

The strict inequality relation h hfor

is also

defined in the same manner as follows:

Definition. 2.7.5. For intuitionistic fuzzy

numbers a and b , and there exist an ha b

[h,1] holding either

[ ] [ ] [ ] [ ] .Sup a Sup b or Inf a Inf b

The following relation holds with regard to

possibility level h.

Lemma. 2.7.6.

'

' 0 1, h hIf a b and h h then a b

Rule. 2.7.7.

Let a and b be trapezoidal intuitionistic fuzzy

number such that ' '

1 2 3 4 1 2 3 4

' '

1 2 3 4 1 2 3 4

( , , , , , , , )

( , , , , , , , )

a a a a a a a a a

and

b b b b b b b b b

then a b

iff the following four inequalities hold

1 1 2 2 3 3 4 4, , a b a b a b and a b .

ha b

iff the following inequalities hold

1 2 1 1 2 1 2 2 3 3

4 3 4 4 3 4

( ) ( ), ,

( ) ( )

a h a a b h b b a b a b

and

a h a a b h b b

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3. MATHEMATICAL FORMULATIONS

We can formulate the intuitionistic fuzzy shortest

path problem in the following linear programming

form (p’):

( , )

j j

( )

1

= 0 , ( 1,2,3..., )

1

0 1 ( , )

ij ij

i j A

ij ji

ij

Min f x l x

Subject to x x

if i s

if i s t i n

if i t

x or for any i j A

where in the objective function means the

addition between trapezoidal intuitionistic fuzzy

numbers.

Definition. 3.1. Let x, yX be two feasible

solution of P’. x dominates y iff ( ) ( )f x f y

holds.

Definition. 3.2. A path stp corresponding to a

non dominated solution x of P’ is called a non

dominated path or pareto optimal path. In other

words, the path stp is a non dominated path or

pareto optimal path iff there exists no other path

'

stp such that '( ) ( ).st std p d p

If we restrict each intuitionistic fuzzy number of

edge length to trapezoidal intuitionistic fuzzy

number, then the problem can be reformulated to

the following multi criteria linear programming

problem with six objective functions.

1 (1)

( , )

2 (2)

( , )

3 (3)

( , )

4 (4)

( , )

'

5 (1)

( , )

'

6 (4)

( , )

( )

( )

( )

( )

( )

( )

ij ij

i j A

ij ij

i j A

ij ij

i j A

ij ij

i j A

ij ij

i j A

ij ij

i j A

Min f x a x

Min f x a x

Min f x a x

Min f x a x

Min f x a x

Min f x a x

subject to x X

where ' '

(1) (2) (3) (4) (1) (4), , , , ,ij ij ij ij ij ija a a a a a are the

elements of trapezoidal intuitionistic fuzzy

number ' '

(1) (2) (3) (4) (1) (2) (3) (4)( , , , , , , , )ij ij ij ij ij ij ij ij ijl a a a a a a a a

In this paper, the proposed problem will be solved

by using the multiple label setting algorithm.

Definition. 3.3. For two trapezoidal intuitionistic

fuzzy numbers

' '

1 2 3 4 1 2 3 4

' '

1 2 3 4 1 2 3 4

( , , , , , , , )

( , , , , , , , ),

a a a a a a a a a

and

b = b b b b b b b b

we say that a is lexicographically smaller than b

if one of the following cases hold

' '

1 1 1 1

' '

1 1 1 1 2 2

' '

1 1 1 1 2 2 3 3

' ' ' '

1 1 1 1 2 2 3 3 4 4 4 4

1. , .

2. , , .

3. , , , .

4. , , , , , .

a b a b

a b a b a b

a b a b a b a b

a b a b a b a b a b a b

4. ALGORITHM

Step-1 Initialization

(i) Assign the label 1[0,( , )] label to node

s, where 0 ( 0,0,0,0 , 0,0,0,0 )TrIFN

(ii) Set the label to temporary one and

initialize the set P to empty as follows:

(1,1) .T and P

Step-2 Label Selection

(i) If ( i = t ), then go to step 7

(ii) Otherwise, among all the temporary labels

determine the lexicographically smallest

one. Let it be the kth

label associated with

node i.

(iii)Discard the pointer (i,k) to this label from

T, and append it to P as follows:

\ ( , ) ( , ).T T i k and P P i k

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Step-3 Label Scanning

(i) If the edge (i,k) forms the cycle then

discard that edge from Temp.

(ii) Otherwise check the node having more

than one edge from a same node among

the permanent edges.

Step-4 If it occurs, the current edge (i,k) preceding

node will be selected and the other edges will be

rejected from the permanent edges.

Step-5 Return to step-2.

Step-6 Optimal Path

Find the optimal path from the permanent P‘ from

s to t.

Step-7 Terminate the execution of the algorithm.

5. NUMERICAL EXAMPLE

As an example to illustrate algorithm, consider a

network with seven nodes and nine directed edges

as shown in the figure – 2.

Figure – 2

In the first step 1, the temporary edge 1[0,( , )]

is assign to source node s = 1. This node is

determined at the first iteration of step 2, and this

edge is set as a permanent edge. From node s =

1, the edges(1,2) and (1,3) are new temporary

edges with [(<25,30,40,55>,<20,30,40,60>) (1,2)]

and [(<21,28,32,50>,<15,28,32,55>) (1,3)]

respectively.

In the second iteration of step 2, the first edge of

node 3 is [(<21,28,32,50>, <15,28,32,55>) (1,3)]

determined as lexicographically smallest among

all the temporary edges. In step 3, the current

edge (1,3) does not form a cycle and sets it as a

permanent one. In the next iteration of step 2,

from this node 3, the node 4 & 5 are temporary

edges with [(<31,42,48,52>,<28,42,48,58>) (3,4)]

and [(<24,35,40,47>,<20,35,40,55>) (3,5)]

respectively. Among all the temporary edges,

node 5 is [(<24,35,40,47>,<20,35,40,55>) (3,5)] is

the lexicographically smallest. In this step , the

current edge (3,5) does not form a cycle and set it

as permanent one. The iterations are repeated

until they reach the destination node 7. Finally we

obtain the shortest path from source node 1 to

destination node 7 is 1 3 5 7 .

REFERENCES

[1] K.Atanassov, ―Intuitionistic Fuzzy Sets‖

Fuzzy sets and System, Volume 20, No. 1,

PP 87- 96, 1986.

[2] Dubois. D and Prade. H, ―Fuzzy Sets and

System‖, Academic Press, New York, 1980.

[3] Shu MH, Cheng CH, Chang JR., ―Using

intuitionistic fuzzy sets for fault-tree

analysis on printed circuit board assembly‖,

Microelectron Reliab 2006;46(12): 2139–

48.

[4] Klein, C. M. 1991 .Fuzzy Shortest Paths,

Fuzzy Sets and Systems 39, 27–41.

[5] V.Lakshmana Gomathi Nayagam,

G.Venkateshwari and Geetha Sivaraman,

―Ranking Of Intuitionistic Fuzzy Numbers‖,

2008 IEEE International Conference on

Fuzzy Systems (FUZZ 2008).

[6] Lawler E. 1976.Combinational

Optimization; Networks and Matroids, Holt,

Reinehart and Winston, New York.

[7] Lin, K. and M. Chen. 1994. The Fuzzy

Shortest Path Problem and its Most Vital

Arcs, Fuzzy Sets and Systems 58, 343–353.

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[8] Liu X. Entropy. 1992. Length measure and

similarity measure of fuzzy sets and their

relations. Fuzzy Sets and Systems,52,305–

18.

[9] Martins EQV. 1984. On a multi-criteria

shortest path problem. European Journal of

Operational Research, 16, 236–45.

[10] Nagoor Gani. A and Mohammed Jabarulla.

M, ―An Intuitionistic Fuzzy Multi Objective

Shortest Path‖, Advances in Fuzzy Sets and

Systems, Volume 7, No. 1, 2010, pp17-26.

[11] Nagoor Gani. A and Mohammed Jabarulla.

M, ―On Searching Intuitionistic Fuzzy

Shortest Path in a Network‖, Applied

Mathematical Sciences, Volume 4, No. 69-

72, 2010.

[12] Nagoor Gani. A and Anusuya. V, ―Shortest

Path on a Network using Fuzzy Number‖,

Advances in Fuzzy Sets and Systems,

Volume 7, No. 1, 2010, pp17-26.

[13] Okada, S. and T. Soper. 2000. A Shortest

Path Problem on a Network with Fuzzy Arc

Lengths, Fuzzy Sets and Systems 109, 129–

140.

[14] Pappis CP, Karacapilidis NI. 1993. A

comparative assessment of measures of

similarity of fuzzy values. Fuzzy Sets and

Systems ,56,171–4.

[15] Ramik J, Rimanek J. 1985. Inequality

relation between fuzzy numbers and its use

in fuzzy optimization. Fuzzy Sets and

Systems,16,123–38.

[16] Tanaka H, Ichihashi H, Asai K. 1984. A

formulation of fuzzy linear programming

problem based on comparison of fuzzy

numbers. Control and Cybernetics 13,185–

94.

[17] Tzung-Nan Chuang, Jung-Yuan Kung, ―A

new approach for the fuzzy shortest path

problem‖, Studies in Computational

Intelligence(SCI) 2.89 100(2005).

[18] Tzung-Nan Chuang, Jung-Yuan Kung, ―A

new algorithm for the discrete fuzzy shortest

path problem in a network‖, Applied

Mathematics and Computation 174 (2006)

660–668.

[19] Wang WJ. 1997. New similarity measures

on fuzzy sets and on elements. Fuzzy Sets

and Systems,85,305–9.

[20] Xinfan Wang, ―Fuzzy Number Intuitionistic

Fuzzy Arithmetic Aggregation Operators‖, -

International Journal of Fuzzy systems,

Volume10, No.2, June 2008.

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The Bivariate Replacement Policy (U,N) for an

Alternative Repair Model

U. Rizwan and Zahiruddeen

Abstract - In this paper, a Bivariate Replacement Policy (U, N) with Negligible Or Non-Negligible (NONN) repair times under

which the system is replaced whenever the cumulative repair time of the system reaches U or the number of failures of the

system exceeds N, whichever occurs first is studied. Explicit expressions for the long run average cost for the model is derived.

Optimality conditions are deduced.

Index Terms - Geometric Process, NONN Repair Times, Optimal Replacement Policy.

—————————— ——————————

1. INTRODUCTION

N this paper, we study a repairable system of a

monotone process model for a one component

multistate degenerative system with (k+1) states

(k- failure states and one working state). Also an

alternative repair model, called the Negligible or

Non-Negligible (NONN) repair times introduced

by Thangaraj and Rizwan [2001] is incorporated

in this model to develop a new repair models. A

replacement policy ( U, N ) is adopted by which

the system will be replaced whenever the

cumulative repair time of the system reaches U;

another replacement policy U but with NONN

repair times, the N policy based on the number of

failures of the system assuming NONN repair

times and a bivariate replacement policy ( U, N )

under NONN repair times, where U is the

cumulative repair time of the system and N is the

number of failures of the system are studied.

Furthermore, explicit expressions for the long-run

average cost of the bivariate policy (U,N) with

NONN repair times is derived.

2. THE MODEL

We make the following assumptions:

——————————————

U. Rizwan is serving in the Department of Mathematics, Islamiah

College, Vaniyambadi, India. E-mail: rizwanmaths@yaoo.com

Zahiruddeen ispursuing in Ph.D. degree in Mathematics at Islamiah

College, Vaniyambadi, India. E-mail : zaheer_nib@yahoo.co.in

At time 0=t , a new system is put into field

use whenever the system fails,it will be

replaced by identical new one,some time later.

The system state at time t , denoted by )(tS is

0,;

( ) =

, ,

= 1,2,....

if the system is

working at time t

S t if the system is in the ith

i type of failure State at time t

i k

Thus the state space is = k0,1,2..... . If the

system fails, then with probability iP , the

system will be in state i , ki 1,2,= and

1=1= i

k

iP

Let iX be the first operating time. For 2n ,

let nX be operating time of the system after

1)( n -st repair, let n be the repair time after

the n -th failure and z be the replacement

time. Now, denote the time of the n th failure

by nt .

Assume that

)(=)( 1 tUtXP ,

and

kitaUitStXP i 1,2,...,=),(=)=)(/( 12 .

In general, for ,1,2,...=;11,2,...,= kinj j

I

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),...,(=)=)(,...,=)(/(11

1111 taaUitSitStXPn

iinnn

where ....1 21 kaaa

Similarly, assume that

kitbVitStYP i 1,2,...,=),(=)=)(/( 11 .

In general, for ,1,2,...=;1,2,...,= kinj j

),...,(=)=)(,...,=)(/(1

11 tbbVitSitStYPn

iinnn

where 0....1 21 kbbb

The working age of the system at time T is

the cumulative life-time given by

1

1 1 1

, <( ) =

, <

n n n n n

n n n n n

t M L M t L MT t

L L M t L M

where i

n

i

n XL 1=

=

andYM i

n

i

n 1=

= 0.== 00 NM

Let r be the reward rate per unit time of the

system when it is operating and C be the

repair cost per unit time the system. Assume

further that the replacement cost comprises of

two parts: one part is the basic replacement

cost R and the other part is the cost

proportional to the length of replacement time

Z at rate pC

The replacement policy ),( NU is used. Let

1U be the first replacement time and in

general for 2n , let nU be the time between

the 1)( n -st replacement and n -th

replacement. Then the sequence

}1,2,=,{ nUn forms a renewal process,

therefore, the inter arrival times between two

consecutive replacement is a renewal cycle.

Let ),( NUC be the long run average cost per unit

time under the bivariate replacement policy

),( NU . By the renewal reward theorem, the long

run average cost per unit time under the

replacement policy ),( NU for a multistate

degenerative system with NONN repair times, is

given by

( , ) =the expected cost incurred in a cycle

C U Nthe expected length of a cycle

)(

)]()(

=

)(

1=

1

1=

)>(

1=

WE

RrcE

zECXrcU

NN

Mn

N

n

n

N

n

pNN

Mn

n

(1)

where is a random variable denoting the

number of failures before the working age of the

system reaches T , W is the length of a cycle and

)( denotes the indicator function. Therefore

10,1,...,= N .

The length of a cycle W under the replacement

policy ),( NU is

( > )

=1

1

( )

=1 =1

= n M NN

n

N N

n n M NN

n n

W X U

X Z

where 10,1,...,= N is the number of failures

before the working age of the system exceeds T.

Now

( )

=1

( )N

n M NN

n

E X

( )

=1

= [( ) / ]N

n M N NN

n

E E X M

)()=/(=1=

0udGuMXE NNn

N

n

U

)())((=1=

0udGXE Nn

N

n

U

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)())(1(=1

1=0

udGppa

Nn

N

n

U

)())(1(=01

1=

udGppa

N

U

n

N

n

)())(1(=1

1=

uGppa

Nn

N

n

(2)

and

]}/)[({=])[( )(

1

1=

)(

1

1=

NUN

Mn

N

n

UN

Mn

N

n

MEEE

)()=/(=1

1=0

udGuME NNn

N

n

U

).(=0

uudGN

U

(3)

Further

( )E W

1

( )

=1 =1

( > )

=1

= [( ) ]

[( ) ] ( )

N N

n n M UN

n n

n M UN

n

E X

E X U E Z

]}=/)[({= )(

1

1=1=

uMXEE NNN

Mn

N

n

n

N

n

( > ) ( > )

=1

[ ] [ ] ( )n M U M UN N

n

E X E U E Z

)(][)(=0

)(

1=

uudGEXE N

U

UN

Mn

N

n

1

( ) ( > )1

=1

( ) [ ] [ ]N

n M U M M Un N N

n

E X E UE

)()())(1(=01

1=

uudGUGppa

N

U

Nn

N

n

(4)

where

1

1=

=

n

nk

n a

Pa

and .=

1

1=

n

nk

n b

Pb

Let j

N

nj

nN XW

1=

=

. Then nNnN WMM = .

Moreover nM and nNW are independent and

),())((=)( 10

ydHytaHtH nNnN

(5)

where )(1 tH nN is the distribution of j

N

njX 1=

.

Since the distribution function of 1nX is

)(=)( taFtH n , equation (5) can be written, by

induction, as )(=)( taGtH n

nNnN . Now

)<<(=][ )<<( nNnn

NMU

nM WMUMPE

)()(=

0udGtdH nnN

uU

U

),())((=

0udGuTaF n

nnN

U

so that equation (4) becomes

)())(1()(=)(1

1=

UGppa

UGUWE Nn

N

n

N

1

10=1

0

( ) ( (1 ) )

( ( )) ( )

NU

N nn

Un

N n n

udG u p pa

G b U u dG u

)())(1()(=)(1

1=0

UGppa

duUGWE Nn

N

n

N

U

1

1=1

0

( (1 ) )

( ( )) ( )

N

nn

Un

N n n

p pa

G b U u dG u

Now equation (1) becomes

( , )C U N

1

1 0=1

10=1

0

10=1

1

1 0=1

(1 ) ( ) ( )

( ) ( (1 ) ) ( )

( )=

( ) ( (1 ) ) ( )

( (1 ) ) ( ( )) ( )

N Un

N n nnn

NU

N Nnn

U

N p

NU

N Nnn

N Un

N n nnn

c p p G b U u dG ua

rUG U c p p dG Ua

r udG u R c

G U du p p G Ua

p p G b U u dG ua

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1

1 0=1

1=1

0

10=1

1

1 0=1

{ ( (1 ) ) ( ( )) ( )

( (1 ) ) ( )}

( )=

( ) ( (1 ) ) ( )

(1 ) ( ( )) ( )

N Un

N n nnn

N

N pnn

U

N

NU

N Nnn

N Un

N n nnn

c p p G b U u dG ua

p p G U R ca

r G u du

G U du p p G Ua

p p G b U u dG ua

The standard minimization procedures can be

adapted to determine the optimal values.

3. CONCLUSION

By considering a repairable system of a monotone

process model of a one component multistate

degenerative system, explicit expressions for the

long-run average cost of the bivariate policy (U,

N) with NONN repair times is derived in this

paper.

REFERENCES

[1] Barlow, R.E, and Proschan, F., (1975)

Statistical Theory of Reliablity and life Testing

Holt, Rinchart and Winston,Inc, NY.

[2] Lam,Y., (1988), A note on the optimal

replacement problem, Advances in Applied

Probability, 20, 479 – 782.

[3] Lam,Y., (1992), Optimal policy for a general

repair replacement model: Discounted reward

case, Communication in Statistics: Theory

and Methods, 8,245 – 267.

[4] Tang, Y and Lam Y, A -shock Maintenance

Model for a Deteriorating System, European

Journal of Operational Research, 168, 541 –

556.

[5] Lam Y., and Zhang Y.L, A -shock

Maintenance of a repairable system,

Computer & Operation Research, 31, 1807 –

1820.

[6] Shantikumar JG., Sumita U, (1983), General

Shock assiciated with correlated renewal

sequences, Journal of Applied Probabilty,

20, 600 –614.

[7] Shantikumar J.G., Sumita U, (1984),

Distribution properties of the system failure

time in a general shock models Advances in

Aplied probability, Journal of Applied

Probabilty, 20, 600 –614.

[8] Thangaraj. V and Rizwan. U, (2001), Optimal

Replacement Policies in Burn-in Process for

an Alternative Repair Model, International J.

Inform. and Management Sciences, 12 (3),

43-56.

[9] Zhang Y. L, (1993), A Geometric process

repair model with good as new preventive

repair, IEEE Transaction on Reliability,

51(2), 223 – 228.

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A Study on Comparison between Fuzzy Assignment Problems using Conversion of

Trapezoidal Fuzzy Numbers in to Triangular Fuzzy Numbers and Conventional Method

S. Manimaran and R. Vasuki

Abstract - Assignment problem is a well-known topic and is used very often in solving problems of engineering and

management science. In this paper trapezoidal fuzzy numbers are considered which are more realistic and general in

nature. The fuzzy assignment problem has been used for ranking the fuzzy numbers. The trapezoidal fuzzy numbers

have been transformed into triangular fuzzy numbers. Then we apply conventional method.

Index Terms – Fuzzy sets, Fuzzy Numbers, Fuzzy Assignment Problem, Fuzzy Ranking.

—————————— ——————————

1. INTRODUCTION

SSIGNMENT Problem (AP) is used

worldwide in solving real world problems. An

assignment problem plays an important role in

industry and other applications. In an assignment

problem, n jobs are to be performed by n persons

depending on their efficiency to do the job. In this

problem cij denotes the cost of assigning the jth

job

to the ith

person. We assume that one person can

be assigned exactly one job; also each person can

do at most one job. The problem is to find an

optimal assignment so that for performing all the

jobs, the total cost is minimum or the total profit is

maximum.

In this paper we investigate a more realistic

problem, namely the assignment problem with

fuzzy costs or times ijc~ . Since the objectives are

to minimize the total cost or to maximize the total

profit, subject to some crisp constraints, the

objective function is considered also as a fuzzy

number. The method to rank the fuzzy objective

values of the objective function by some ranking

method for fuzzy numbers to find the best

alternative. On the basis of this idea the Yager‘s

——————————————

S. Manimaran is serving in the Department of Mathematics, RKM

Vivekananda College, Chennai, India.

R. Vasuki is serving in the Department of Mathematics, S.I.V.E.T.

College, Chennai, India

ranking method has been adopted to transform the

fuzzy assignment problem to a crisp one so that

the conventional solution methods may be applied

to solve the AP.

The idea is to transform a problem with fuzzy

parameters to a crisp version in the LPP form and

to solve it by the simplex method. Other than the

fuzzy assignment problem other applications of

this method can be tried in project scheduling,

maximal flow, transportation problem etc.

2. FUZZY ASSIGNMENT PROBLEM AND

FUZZY RANKING METHOD

In recent years, fuzzy transportation and fuzzy

assignment problems have received much

attention.

Lin and Wen [9] solved the assignment problem

with fuzzy interval number costs by a labeling

algorithm. In the paper by Sakawa et al. [12], the

authors dealt with actual problems on production

and work force assignment in a housing material

manufacturer and a subcontract firm and

formulated two kinds of two-level programming

problems. Applying the interactive fuzzy

programming for two-level linear and linear

fractional programming problems, they derived

satisfactory solutions to the problems and

thereafter compared the results. They examined

actual planning of the production and the work

A

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force assignment of the two firms to be

implemented. Chen [1] proved some theorems

and proposed a fuzzy assignment model that

considers all individuals to have some skills.

Wang [13] solved a similar model by graph

theory. Dubois and Fortemps [3] surveys

refinements of the ordering of solutions supplied

by the max-min formulation, namely the discrimin

partial ordering and the leximin complete

preordering. They have given a general algorithm

which computes all maximal solutions in the sense

of these relations. Different kinds of fuzzy

transportation problems are solved in the papers.

Dominance of fuzzy numbers can be explained by

many ranking methods. Of these, Yager's ranking

method [14] is a robust ranking technique which

satisfies the properties of compensation, linearity

and additivity. We have applied Yager's ranking

technique.

Zadeh [15] in 1965 first introduced fuzzy set as a

mathematical way of representing impreciseness

or vagueness in everyday life.

Fuzzy set. A fuzzy set is characterized by a

membership function mapping the elements of a

domain, space, or universe of discourse X to the

unit interval [0, 1].

A fuzzy set A~

in a universe of discourse X is

defined as the following set of pairs:

A~

= {(x, A~μ (x) : x X}.

Here A~μ : X [0, 1] is a mapping called the

membership function of the fuzzy set A~

and A~μ

(x) is called the membership value of degree of

membership of x X in the fuzzy set A~

. Larger

the value of A~μ (x), stronger is the grade of

membership form in A~

. These membership

grades are often represented by real numbers

ranging from a minimum of 0 to a maximum of 1.

Normal fuzzy set. A fuzzy set A~

of the universe

of discourse X is called a normal fuzzy set

implying that there exist at least one x X such

that A~μ (x) = 1.

Convex fuzzy set. The fuzzy set A~

is convex if

and only if, for any x1, x2 X, the membership

function of A~

satisfies the inequality

A~μ { x1 + (1)x2} min{

A~μ (x1), A

~μ (x2)},

0 1.

For a trapezoidal fuzzy number A(x), it can be

represented by A(a, b, c, d; 1) with membership

function A~μ (x) given by

otherwise.0,

dxc,c)(d

x)(dcxb1,

bxa,a)(b

a)(x

(x)μA~

satisfying the following conditions.

(i) A~μ (x) is a continuous mapping from R to

the closed interval [0, 1];

(ii) A~μ (x) = 0 for all x (, a];

(iii) Strictly increasing and continuous on [a,

b];

(iv) A~μ (x) = 1 for all x [b, c];

(v) Strictly decreasing and continuous on [c, d];

(vi) A~μ (x) = 0 for all x [d, ).

The graphic representation of a trapezoidal fuzzy

number is shown in Figure 1.

Figure 1: Graphical representation of trapezoidal fuzzy

number (a, b, c, d; 1)

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For a triangular fuzzy number A(x), it can be

represented by A(a, b, c; 1) with membership

function A~μ (x) given by

otherwise.0,

dxc,b)(c

x)(cbx1,

bxa,a)(b

a)(x

(x)μA~

satisfying the following conditions.

(x) is a continuous mapping from R

to the closed interval [0, 1];

(x) = 0 for all x (, a];

Strictly increasing and continuous

on [a, b];

(x) = 1 at x = b;

Strictly decreasing and continuous

on [b, c];

(x) = 0 for all x [c, ).

3. THE EXISTING METHOD

The assignment problem can be stated in the form

of n n cost matrix [cij] of real numbers as given

in the following table:

Jobs

1 2 3 j n

1 c11 c12 c13 c1j c1n

2 c21 c22 c23 c2j c2n

Persons I ci1 ci2 ci3 cij cin

N cn1 cn2 cn3 cn4 cnn

Mathematically assignment problem can be stated

as

Minimize

n

1i

n

1j

ijijxcz i = 1, 2, …, n;

j = 1, 2, …, n

subject to

n

ij

j 1

x 1, i 1,2,...,n

(1)

n1,2,...,j1,xn

1i

ij

ijx {0,1},

th

thij

if the i person is assigned 1,

where x the j job

0, otherwise

is the decision variable denoting the assignment of

the person 1 to job j. cij is the cost of assigning

the jth

job to the ith

person. The objective is to

minimize the total cost of assigning all the jobs to

the available persons (one job to one person).

When the costs or time ijc~ are fuzzy numbers,

then the total cost becomes a fuzzy number.

.xc~z~n

1i

n

1j

ijij

Hence it cannot be minimized directly. For

solving the problem we defuzzify the fuzzy cost

coefficients into crisp ones by a fuzzy number

ranking method.

Yager's ranking technique is a robust ranking

technique which satisfies compensation, linearity,

and additivity properties and provides results

which are consistent with human intuition. Given

a convex fuzzy number ,c~ the Yager's Ranking

Index is defined by

,dαcc0.5)c~Y(1

0

U

α

L

α

where U

α

L

α cc is the -level cut of the fuzzy

number .c~

In this paper we use this method for ranking the

objective values. The Yager's ranking index )c~Y(

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gives the representative value of the fuzzy number

.c~ It satisfies the linearity and additivity

property:

If C~

bB~

aA~

and ,F~

tE~

kD~

where a, b, k, t are constants, then we have

)C~

bY()B~

aY(A~

Y and

).F~

tY()E~

kY()D~

Y(

On the basis of this property the fuzzy assignment

problem can be transformed into a crisp

assignment problem in the LPP form. The

ranking technique of Yager is:

If ),V~

Y()U~

Y( then .V~

U~ i.e.,

.U~

}V~

,U~

min{

For the assignment problem (1), with fuzzy

objective function

n

1i

n

1j

ijijxc~z~min

we apply Yager's ranking method (using the

linearity and additive property) to get the

minimum objective value *z~ from the formulation

n

1i

n

1j

ijij

* xc~Yzminimize)z~Y(

Subject to

n1,2,...,i1,xn

1j

ij

(2)

n1,2,...,j1,xn

1i

ij

ijx {0,1},

th

thij

if the i person is 1,

where x assigned the j job

0, otherwise

is the decision variable denoting the assignment of

the person i to job j. ijc~ is the cost of assigning

the jth

job to the ith

person. The objective is to

minimize the total cost of assigning all the jobs to

the available persons (one job to one person).

Since )c~Y( ij are crisp values, this problem (2) is

obviously the crisp assignment problem of the

form (1) which can be solved by the conventional

methods, namely the Hungarian Method or the

Simplex method to solve the LPP form of the

problem. Once the optimal solution x* of Model

(2) is found, the optimal fuzzy objective value *z~

of the original problem can be calculated as

.xc~z~n

1i

n

1j

*

ijij

*

4. NUMERICAL EXAMPLE

Example 4.1. Let us consider a Fuzzy Assignment

Problem with rows representing 4 persons A, B,

C, D and columns representing the 4 jobs Job1,

Job2, Job3 and Job4. The cost matrix ]c~[ ij is

given whose elements are trapezoidal fuzzy

numbers. The problem is to find the optimal

assignment so that the total cost of job assignment

becomes minimum

Solution. In conformation to Model (2) the fuzzy

assignment problem can be formulated in the

following mathematical programming form

Min [ Y(3, 5, 6, 7)x11 + Y(5, 8, 11, 12)x12

+ Y(9, 10, 11, 15)x13 + Y(5, 8, 10, 11)x14

+ Y(7, 8, 10, 11)x21 + Y(3, 5, 6, 7)x22

+ Y(6, 8, 10, 12)x23 + Y(5, 8, 9, 10)x24

+ Y(2, 4, 5, 6)x31 + Y(5, 7, 10, 11)x32

+ Y(8, 11, 13, 15)x33 + Y(4, 6, 7, 10)x34

+ Y(6, 8, 10, 12)x41 + Y(2, 5, 6, 7)x42

+ Y(5, 7, 10, 11)x43 + Y(2, 4, 5, 7)x44 ]

subject to

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x11 + x12 + x13 + x14 = 1,

x11 + x21 + x31 + x41 = 1,

x21 + x22 + x23 + x24 = 1,

x12 + x22 + x32 + x42 = 1, (3)

x31 + x32 + x33 + x34 = 1,

x13 + x23 + x33 + x43 = 1,

x41 + x42 + x43 + x44 = 1,

x14 + x24 + x34 + x44 = 1,

xij {0, 1}.

Now we calculate Y(3, 5, 6, 7) by applying the

Yager's Ranking Method. The membership

function of the trapezoidal number (3, 5, 6, 7) is

otherwise.0,

7x6,1

x)(76x51,

5x3,2

)3(x

(x)μA~

The -cut of the fuzzy number (3, 5, 6, 7) is

U

α

L

α cc = (2 + 3, 7 )

for which

11

1L U

α α0

1

0

1

0

Y(c ) Y(3,5,6,7)

0.5 c ,c dα

0.5(2α 3 7 α)dα

0.5(α 10)dα

5.25.

Proceeding similarly, the Yager's ranking indices

for the fuzzy costs ijc~ are calculated as:

12 13 14

21 22 23

24 31 32

33 34 41

42 43 44

Y(c ) 9, Y(c ) 11, Y(c ) 8.5,

Y(c ) 9, Y(c ) 5.25, Y(c ) 9,

Y(c ) 8, Y(c ) 4.25, Y(c ) 8.25,

Y(c ) 11.75, Y(c ) 6.75, Y(c ) 9,

Y(c ) 5, Y(c ) 8.25, Y(c ) 4.5.

We replace these values for their corresponding

ijc~ in (3), which results in a conventional

assignment problem in the LPP form. We solve it

by using LINGO 9.0 to get the following optimal

solution

* * * *

13 22 31 44

* * * * * *

11 12 14 21 23 24

* * * * * *

32 33 34 41 42 43

x x x x 1,

x x x x x x 0,

x x x x x x 0,

with the optimal objective value 25.25)z~Y( *

which represents the optimal total cost. In other

words the optimal assignment is

A 3, B 2, C 1, D 4.

The fuzzy optimal total cost is calculated as

13 22 31 44c c c c

(9,10,11,15) (3,5,6,7)

(2,4,5,6) (2,4,5,7)

= (16, 23, 27, 35).

Also we find that

25.25.35)27,2,Y(16,)z~Y( *

5. CONVENTIONAL (HUNGARIAN) METHOD

Convert the trapezoidal fuzzy numbers into

triangular fuzzy numbers. Then apply

conventional method to find the optimal solution

in fuzzy environment

6. CONCLUSION

A conventional method is proposed to solve the

fuzzy assignment problem with trapezoidal fuzzy

numbers in real life situations. To illustrate the

proposed method a numerical example is solved

and the results are compared with the

conventional method. If there is no uncertainity

about the cost then the conventional method gives

the same result as in the fuzzy assignment

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problem with trapezoidal fuzzy numbers. The

proposed method of solving assignment problem

is simple in nature and applicable to any kind of

assignment problems.

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Computational and Applied Mathematics,

Vol. 197, 2006, pp. 233244.

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defuzzification methods based on area

compensation, Fuzzy Sets and Systems, 82,

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[7] G.J. Klir and B. Yuan, Fuzzy sets and fuzzy

logic (Theory and applications), Prentice-

Hall, International Inc., 1995.

[8] H.W. Kuhn, The Hungarian method for the

assignment problem, Naval Research

Logistics Quarterly, Vol. 2, 1955, pp.

8397.

[9] C.J. Lin and U.P. Wen, A labeling algorithm

for the fuzzy assignment problem, Fuzzy

Sets and Sytsems, Vol. 142, 2004, pp.

373391.

[10] T.S. Liou and M.J. Wang, Ranking fuzzy

number with integral values, Fuzzy Sets and

Systems, Vol. 50, 1992, pp. 247255.

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Mathematical Modeling, Vol. 33, 2009, pp.

39263935.

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European Journal of Operational Research,

Vol. 135, 2001, pp. 142157.

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problem, Fuzzy Math., Vol. 3, 1987, pp.

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3, Vol. 8, 1975. pp. 199249, Vol. 9, 1976,

pp. 4358.

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Generalized Partial Sums on Infinite Series of Rational Factorial Functions

M.Maria Susai Manuel, G.Britto Antony Xavier, G.Dominic Babu and K.Srinivasan

Abstract - In this paper, the authors obtain some significant formulae for the general partial sums on infinite series of rational

functions of the generalized polynomial factorial using the inverse of generalized difference operator of th

n kind n

for the positive

integer n and for the positive real . Suitable examples are provided to illustrate the main results.

Index Terms - Generalized difference operator, generalized polynomial factorial, generalized factorial, partial sums.

—————————— ——————————

1. INTRODUCTION

N number theory, some applications, like sum

of the n th powers of an arithmetic progression,

the sum of the products of n consecutive terms of

an arithmetic progression and the sum of an

arithmetic-geometric progression are developed in

[2] using the generalized difference operator

defined as

( ) = ( ) ( )u k u k u k .

Using , Generalized Bernolli‘s polynomials

( , )nB k are established in [4, 5]. Qualitative

behaviors, like rotatory, spiral, boundedness,

recessive and dominant properties of the

generalized Ricatti‘s equation

( ) ( ) ( ) ( ) = ( ) ( ), ( )p k u k p k u k q k u k k

——————————————

M. Maria Susai Manuel is serving in the Department of Science and Humanities, R.M.D. Enngineering College, Kavaraipettai, Tamil Nadu, India. E-mail: manuelmsm_03@yahoo.co.in

G. Britto Antony Xavier is serving in the Department of Mathematics, Sacred Heart College, Tirupattur, India.

G. Dominic Babu is serving in the Department of Mathematics, Sacred Heart College, Tirupattur, India.

K. Sriivasan is serving in the Department of Mathematics, Sacred Heart College, Tirupattur, India.

for integers and are developed in [3, 6, 7,

8]. Sums and sum of partial sums of higher

powers, products of consecutive terms of an

arithmetic progression are established in [9]. As

infinite partial sums on product of terms of

rational functions of are not yet developed in the

literature, in this paper, we derive the formula for

infinite partial sums on product of terms of

rational functions of polynomial factorial and

factorials. Here we assume the notations:

( ) ={ , , 2 , }j j j j , 1( ) = ( )j j ,

X and X are the upper integer part and

integer part of X respectively and (0, ) .

2. PRELIMINARIES

In this section, we present some basic definitions

and some results which will be useful for the

subsequent discussion.

Definition 2.1 [2] The generalized difference

operator on real valued function

( ), [0, )u k k is defined as

( ) = ( ) ( ).u k u k u k (1)

and the generalized difference operator of the thr

kind is defined as

I

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( ) = ( ( ( ( )))).r

r times

u k u k (2)

Definition 2.2 For (1)n and (0, ) , the

generalized positive polynomial factorial and

reciprocal polynomial factorial are respectively

defined as

( ) = ( )( 2 )...( ( 1) )nk k k k k n (3)

and

( )

1 1= .

( )( 2 )...( ( 1) )n k k k k nk (4)

(4) is generalized reciprocal factorial when

=k

n

.

Definition 2.3 [2] For ( )u k the inverse operator

1 is defined as

1( ) = ( ), ( ) = ( ) ,jif v k u k then v k u k c (5)

where jc is a constant for all

( ), =k

k j j k

.

In general 1 ( 1)( ) = ( ( ))n nu k u k for

(2)n .

Lemma 2.4 [11] If 1 ( ) = 0lim

k

u k

and

=k

j k

, then

1

=1

( ) | = ( ).k

r

u k u k r

(6)

Theorem 2.5 If ( ) = 0limr

k

u k

for =1,2, ,r m

and [ , )k m , then

( 1)

=

( 1)( ) | = ( ).

( 1)!

mm

k

r m

ru k u k m r

m

(7)

Proof. The proof follows by taking 1 on (6) for

( 1)m times and applying (6).

3. MAIN RESULTS AND APPLICATIONS

Here, we present 1 of certain rational functions

to find the sum and partial sums on infinite series

of generalized rational factorial and polynomial

factorial functions.

Theorem 3.1 Let , (1)m n , ( 1)m n ,

0nk . Then

( 1)

( )=

( ) ( )

( 1) 1

( 1)! ( )

1= .

( 1) ( )

m

nr m

m m n m

r

m k m r

n k m

(8)

Proof. From the Definitions 2.1 and 2.3, we

obtain

1

( ) ( 1)

1 1| = , 2.

( 1) ( )kn n

nk n k

(9)

The proof follows by taking 1 on (9) for

( 1)m times and (7).

The following example illustrate Theorem 3.1 for

m=3.

Example 3.2 For 4n , [3 , )k , (0, )

and = 3m , equation (8) becomes

(2)

( ) (3) 3 ( 3)=3

( 1) 1= .

(2)!( ( 2) ) ( 1) ( 3 )n nr

r

k r n k

(10)

In particular, = 2k , = 0.5, = 4n in (10), we

obtain

4

1 3 1=

2 1.5 1 0.5 2.5 2 1.5 1 6 0.5

.

Theorem 3.3 For 0k and (2)( ) 0k r ,

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2 2

(2)=1

2 (2 1) 1= .

( )r

k r

kk r

(11)

Proof. From (1) and (5), we find

1

2 2(2)

2 1= .

( )

k

kk

(12)

The proof follows by applying (6) on (12) for the

limit k to .

The following is an illustration of Theorem 3.3.

Example 3.4 By taking = 0.4 in (11), we get

2 2

(2)=1

0.4

2 (2 1)(0.4) 1=

(0.4)( (0.4))r

k r

kk r

In particular, when =1k ,

2 2 2 2 2 2

2 1(0.4) 2 3(0.4) 2 5(0.4) 1= .

0.4(1.4) (1) (1.8) (1.4) (2.2) (1.8)

Theorem 3.5 For 0k and (2)( ) 0k r ,

( 1)

=1 (2)

( 1) 1= .

( ) 2 2

k r kr

k r

k r k

(13)

Proof. From (1) and (5), we have

1

(2)

2 1=

( ) 2 2

k k

k

k k

(14)

The proof follows by (6) and (14).

The following is the illustration for Theorem 3.5.

Example 3.6 Taking = 0.5 in (13), we obtain

( 1)(0.5) 0.5

=1 (2) (0.5) 0.50.5

( 1)(0.5) 1= .

2( (0.5)) 2

k r kr

k r

kk r

In particular when = 2k ,

2 2.5 3

0.5 0.5 0.5

3 3.5 4 1= .

16(2.5)(2)2 (3)(2.5)2 (3.5)(3)2

Theorem 3.7 For [ , )k , (0, ) and

(2)2(2 3 ) 0k

(2)

=1 2

1 1=

(2 )2(2 (2 1) )r kk r

(15)

Proof. From (1) and (5), we have

1

(2)2

1 1=

(2 )2(2 3 ) kk

(16)

The proof follows by (6), (16) and = 0jc as

.k

The following is the illustration for Theorem 3.7.

Example 3.8 Taking = 0.5 in (15), we find

(2) (2) (2)0.5 0.5 0.5

1 1 1

(2 1.5) (2 2.5) (2 3.5)

1=

(2 0.5)

k k k

k

In particular when =1.3k , we get

1 1 1 1=

4.1 3.6 5.1 4.6 6.1 5.6 3.1

.

Theorem 3.9 For [ , )k , (0, ) and

(2)( 2 ) 0k

=0 (2)

2( ) 5 1=

( 2 ) 3 ( )3

k r kr

k r

k r k

(17)

Proof. From (1) and (5), we find

1

(2)

2 5 1=

( 2 ) 3 ( )3

k k

k

k k

(18)

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The proof follows by (6), (18) and = 0jc as

k .

The following is the illustration for Theorem 3.9.

Example 3.10 Taking = 0.3 in (17), yields

0.3 0.6

(2) (2)0.3 0.30.3 0.3

0.9

(2) 0.3 0.30.3

2 1.5 2 2.1

( 0.6) 3 ( 0.9) 3

2 2.7 1= .

( 1.2) 3 ( 0.3)3

k k

k k

k k

k k

k

k k

In particular when =12k ,

41 42

43 40

25.5 26.1

12.6 12.3 3 12.9 12.6 3

26.7 1= .

13.2 12.9 3 12.3 3

Theorem 3.11 For [ , )k , (0, ) and

(4)( 4 ) 0k

(4) (3)

=0

3 (3 7 )=

( 4 ) 6 ( 3 )r

k r k

k r k

(19)

Proof. From (1) and (5), we get

1

(4) (3)

3 3 7=

( 4 ) 6 ( 3 )

k k

k k

(20)

The proof follows by (6), (20) and = 0jc as

k .

The following is the illustration for Theorem 3.11.

Example 3.12 When = 2.7 , (19) becomes

(4) (4) (4)2.7 2.7 2.7

(3)2.7

8.1 10.8 13.5

( 10.8) ( 13.5) ( 16.2)

3 18.9=

16.2( 8.1)

k k k

k k k

k

k

Theorem 3.13 For [ , )k , (0, ) and

(2)2(2 ) 0k

1=0 (2)

2

1=

3 (2( ) ) 4(3) (2 )

k r kr

k r

k r k

(21)

Proof. From (1) and (5), we get

1

1(2)2

( ) 1=

3 (2 ) 4(3) (2 )

k k

k

k k

(22)

The proof follows by (6), (22) and = 0jc as

.k

The following is the illustration for Theorem 3.13.

Example 3.14 Taking = 0.2 in (21), we arrive

0.2

(2) (2)0.2 0.20.4 0.4

0.41

(2)0.2 0.20.4

0.2 0.4

3 (2 0.2) 3 (2 0.6)

( 0.6) 1=

3 (2 1) 4(3) (2 0.2)

k k

k k

k k

k k

k

k k

Theorem 3.15 For [ , )k , (0, ) and

( ) 0

k

k

=0

( ) 1=

( )

k r kr

r

k r

k r k

(23)

Proof. From (1) and (5), we get

1 =

( )

kk

k

k

kk

(24)

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The proof follows by (6), (24) and = 0jc as

.k

The following is to illustrate Theorem 3.15.

Example 3.16 Taking =1.2 in (23), we get

1.2 2.4

1.2 1.21.2 1.2

3.6

1.2 1.221.2 1.2

1.2

( 1.2) 1.2( 2.4)

( 2.4) 1=

1.2 ( 3.6)

k k

k k

k k

k k

k

k k

In particular, when = 3k the above series

becomes

(4) (5) 2 (6) (3)1.2 1.2 1.2 1.2

3 4.2 5.4 1=

4.2 (1.2)5.4 1.2 6.6 3a

Theorem 3.17 For [ , )k , (0, ) and

(2)( 4 ) ( ) 0

k

k k

2 2

=0(2)

( 2 ) 3

( 4 ) ( )

1=

( 3 )

k rr

r

k

k r

k r k r

k k

(25)

Proof. From (1) and (5), we find

2 2

1

(2)

(( 2 ) 3 )=

( 4 ) ( ) ( 3 )

k k

k k

k

k k k k

(26)

The proof follows by (6), (26) and = 0jc as

k

The following example illustrates Theorem 3.17.

Example 3.18 When =1.7 , (25) yields

2 2

1.7

1.7(2)1.7 1.7

2 2

2(1.7)

1.7(2)1.7 1.7

1.71.7

( 2(1.7)) 3(1.7)

( 4(1.7)) ( 1.7)

( 3(1.7)) 3(1.7)

1.7( 5(1.7)) ( 2(1.7))

1=

( 3(1.7))

k

k

k

k

k k

k

k k

k k

.

Theorem 3.19 For [ , )k , (0, ) and

2 2 (2)( 2 ) ( ) 0

k

k k

3 3

=02 2 (2)

( )

(( ) 2 ) ( )

k rr

r

k r

k r k r

2 2

1= .

(( ) 2 )

k

k k

(27)

Proof. From (1) and (5), we obtain

3 31

2 2 (2) 2 2

( )=

( 2 ) ( ) (( ) 2 )

k k

k k

k

k k k k

(28)

The proof follows by (6),(28) and = 0jc as

k

The following is the illustration for Theorem 3.19.

Example 3.20 Taking = 3.5 in (27), we obtain

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3 3

3.5

3.52 2 (2)3.5 3.5

3 3

2(3.5)

3.52 2 (2)3.5 3.5

3 3

3(3.5

2 2 2 (2)3.5 3.5

3.5

( 2(3.5) ) ( 3.5)

( 3.5) 3.5

3.5(( 3.5) 2(3.5) ) ( 2(3.5))

( 2(3.5)) 3.5

3.5 (( 2(3.5)) 2(3.5) ) ( 3(3.5))

k

k

k

k

k k

k

k k

k

k k

)

3.5

3.52 23.5

1=

(( 3.5) 2(3.5) )

k

k k

In particular, when = 9k , above series becomes

3 3

2 2 (2) (4)3.5 3.5

3 3

2 2 (2) (5)3.5 3.5

3 3

2 2 2 (2) (6)3.5 3.5

2 2 (3)3.5

9 3.5

(9 2(3.5) ) 12.5

12.5 3.5

3.5(12.5 2(3.5) ) 16

16 3.5

3.5 (16 2(3.5) ) 19.5

1=

(5.5 2(3.5) )9

Theorem 3.21 For [ , )k , (0, ) ,

(2)( ( 1) ) ( ) 0

k

k m k

2

=0(2)

(( ) ( 1)( ) )

( ( 1) ) ( )

1= .

( )

k rr

r

k

k r m k r

k r m k r

k m k

(29)

Proof. From (1) and (5), we obtain

21

(2)

(( ) ( 1) )=

( ( 1) ) ( ) ( )

k k

k k

k m k

k m k k m k

(30)

The proof follows by (6), (30) and = 0jc as

k

The following are to illustrate Theorem 3.21.

Example 3.22 Taking = 1, = 0.8m in (29)

2 2

0.8(3)0.8 0.8

2 2

0.8

0.8(3)0.8 0.8

2 2

2(0.8)

0.82 (3)0.8 0.8

0.80.8

0.8

( 0.8)

( 0.8) 0.8

0.8( 2(0.8)) ( 0.8)

( 2(0.8)) 0.8

0.8 ( 3(0.8)) ( 2(0.8))

1=

( 0.8)

k

k

k

k

k

k k

k

k k

k

k k

k k

.

In particular, when = 2k

2 2 2 2

(3) (3) (3) (4) (3)0.8 0.8 0.8 0.8 0.8

2 0.8 2.8 0.8 1=

2.8 2 0.8 3.6 2.8 1.2 2

.

The following theorem gives the formula for

finding sum of infinite second partial sums of

reciprocals of products of consecutive terms of an

arithmetic progression.

Theorem 3.23 For the positive integer 4n ,

[3 , )k and (0, ) ,

(2)

( )=3

3 ( 3)

( 1) 1

2! ( )

1= ,

( 1)( 2)( 3) ( 3 )

nm

n

m

k m

n n n k

(31)

In general for ( 1), [ , )n r k r and

(0, ) ,

( 1)

( )=

( ) ( )

( 1) 1

( 1)! ( )

( 1)= .

( 1) ( )

r

nm r

r

r r n r

m

r k m

n k r

(32)

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Proof. The proof follows from (8) and constants

' = 0ijc s as k .

The following theorem gives the formula for

finding sum of infinite fifth partial sums of

reciprocals of products of consecutive terms of an

arithmetic progression.

Theorem 3.24 Let 7n , [6 , )k and

(0, ) . Then

(5)

( ) (6) 6 ( 6)=1

( 5) 1 1= .

(5)! ( ( 1) ) ( 1) ( 6 )n nt

t

k t n k

(33)

Proof. The proof follows by (8) and (31).

The following example illustrates Theorem 3.24.

Example 3.25 Substituting = 3.6, = 7n in (33),

we find

(7) (7) (7)3.6 3.6 3.6

6 (1)3.6

1 6 21

( ) ( 3.6) ( 2(3.6))

1=

6!3.6 ( 21.6)

k k k

k

In particular, when = 23k , above series becomes

(7) (7) (7) 63.6 3.6 3.6

1 6 21 1=

23 26.6 30.2 6!3.6 (1.4) .

REFERENCES

[1] R.P Agarwal, Difference Equations and

Inequalities, Marcel Dekker, New York,

2000.

[2] M.Maria Susai Manuel, G.Britto Antony

Xavier and E.Thandapani, Theory of

Generalized Difference Operator and Its

Applications, Far East Journal of

Mathematical Sciences, 20(2) (2006), 163 -

171.

[3] M.Maria Susai Manuel, G.Britto Antony

Xavier and E.Thandapani, Qualitative

Properties of Solutions of Certain Class of

Difference Equations , Far East Journal of

Mathematical Sciences, 23(3) (2006), 295-

304.

[4] M.Maria Susai Manuel, G.Britto Antony

Xavier and E.Thandapani, Generalized

Bernoulli Polynomials Through Weighted

Pochhammer Symbols, Far East Journal of

Applied Mathematics, 26(3) (2007), 321-333.

[5] M.Maria Susai Manuel, A.George Maria

Selvam and G.Britto Antony Xavier, On the

Solutions and applications of Some Class of

Generalized Difference Equations, Far East

Journal of Applied Mathematics, 28(2)

(2007), 223 - 241.

[6] M.Maria Susai Manuel, A.George Maria

Selvam and G.Britto Antony Xavier, Rotatory

and Boundedness of Solutions of Certain

Class of Difference Equations, International

Journal of Pure and Applied Mathematics,

33(3) (2006), 333-343.

[7] M.Maria Susai Manuel and G.Britto Antony

Xavier, Recessive, Dominant and Spiral

Behaviours of Solutions of Certain Class of

Generalized Difference Equations,

International Journal of Differential

Equations and Applications, 10(4) (2007),

423-433.

[8] M.Maria Susai Manuel, A.George Maria

Selvam and G.Britto Antony Xavier, Regular

Sink and Source in terms of Solutions of

Certain Class of Generalized Difference

Equations, Far East Journal of Applied

Mathematics, 28(3) (2007), 441 - 454.

[9] M.Maria Susai Manuel, G.Britto Antony

Xavier and V.Chandrasekar, Generalized

Difference Operator of the Second Kind and

Its Application to Number Theory,

International Journal of Pure and Applied

Mathematics, 47(1) (2008), 127 - 140.

[10] M.Maria Susai Manuel, G.Britto Antony

Xavier, V.Chandrasekar and R.Pugalarasu,

On Generalized Difference Operator of Third

Kind and its Applications to Number

Theory,International Journal of Pure and

Applied Mathematics 53(1) (2009),69-82.

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[11] M. Maria Susai Manuel, Adem Kilicman, G.

Britto Antony Xavier,R. Pugalarasu3and D.

S. Dilip On the solutions of second order

generalized difference equations Advances in

Difference Equations 2012, (acepted

2012:105).

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Solution of a Conjecture on Skolem Mean Graph

of stars K1, K1, K1, m K1, n

V. Balaji, D.S.T. Ramesh and V. Maheswari

Abstract – In this paper, we prove the conjecture that the four stars K1, K1, K1, m K1, n is a skolem mean graph

if |m–n| < 4 + 2 for = 2,3,4, . . .; m = 2,3,4, . . . and ≤ m < n.

Index Terms – Skolem mean graph and star.

—————————— ——————————

--

1. INTRODUCTION

LL graphs in this paper are finite, simple and

undirected. Terms not defined here are used

in the sense of Harary [5]. In [1], skolem mean

labeling was focused an assignment of label to the

vertices x V with distinct elements f(x) from 1,

2, . . . , p in such a way that when the edge e = uv

is labeled with

2

f(v)f(u) if f(u) + f(v) is even and

2

1f(v)f(u) if f(u) + f(v) is odd

then the edges get distinct labels from the set {2,3,

…, p} and it was proved that any path is a skolem

mean graph, if m ≥ 4, K1, m is not a skolem mean

graph and the two stars K1, m K1, n is a skolem

mean graph if and only if |m–n| ≤ 4. In [2], it was

proved that the three star K1, K1, m K1, n is a

skolem mean graph if

——————————————

V. Balaji is serving in the Department of Mathematics, Sacred Heart

College, Tirupattur, India. E-mail: pulibala70@gmail.com

D.S.T. Ramesh is serving in the Department of Mathematics,

Margoschis College, Nazerath, India. E-mail : dstramesh@gmail.com

V. Maheswari is serving in the Department of Mathematics,

Manonmaniam Sundaranar University, Tirunelveli, India.

|mn| = 4+ for = 1, 2, 3, . . . ;

m = 1, 2, 3, . . . ;

n = + m + 4 and

≤ m < n ;

the three star K1, K1, m K1, n is not a skolem

mean graph if

|mn| > 4 + for = 1, 2, 3, . . . ;

m = 1, 2, 3, . . . ;

n ≥ + m + 5 and

≤ m < n ;

the four star K1, K1, K1, m K1, n is a

skolem mean graph if

|mn| = 4 + 2 for = 2, 3, 4, . . . ;

m = 2, 3, 4, . . . ;

n = 2 + m + 4 and

≤ m < n ;

the four star K1, K1, K1, m K1, n is not a

skolem mean graph if

|mn| > 4 + 2 for = 2, 3, 4, . . . ;

m = 2, 3, 4, . . . ;

n ≥ 2 + m + 5 and

≤ m < n ;

the four star K1,1 K1, 1 K1, m K1, n is a

skolem mean graph if

|mn| = 7 for m = 1, 2, 3, . . . ;

n = m + 7 and

1 ≤ m < n

and the four star K1,1 K1, 1 K1, m K1, n is not

a skolem mean graph if

A

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|mn| > 7 for m = 1, 2, 3, . . . ;

n ≥ m + 8 and 1 ≤ m < n.

In [3], the condition for a graph to be skolem

mean is that p ≥ q + 1. In [4], We proved a

conjecture that the three stars K1, K1, m K1, n

is a skolem mean graph if |mn| < 4+ for

= 1, 2, 3, . . . ; m = 1, 2, 3, . . . and ≤ m < n .

2. MAIN THEOREMS

Definition 2.1.0 The four star is the disjoint

union of K1, a, K1, b, K1, c and K1, d . It is denoted

by K1, a K1, b K1, c K1, d.

Theorem 2.1.1 If ≤ m < n, the four star

K1, K1, K1, m K1, n

is a skolem mean graph if |mn| < 4 + 2 for =

2, 3, 4, . . . ; m = 2, 3, 4, . . . .

Proof.

Case : 1 Consider the graph

G = K1, K1, K1, m K1, n.

Let ≤ m < n where n = 2 + m + 3 for = 2, 3,

4, . . . and m = 2, 3, 4, . . . . Let us take the case

that |mn| < 4 + 2 for = 2, 3, 4, . . . ; m = 2, 3,

4, . . . and n = 2 + m + 3. We have to prove that

G is a skolem mean graph.

Therefore, the graph

G = K1, K1, K1, m K1, n

where n = 2 + m + 3 for = 2, 3, 4, . . . and m

= 2, 3, 4, . . . .

Let

{u} {ui: 1 i }, {v} {vj: 1 j },

{w} {wk: 1 k m} and {x} {xh: 1 h n}

be the vertices of G. Then G has 2 + m + n + 4

vertices and 2 + m + n edges.

We have

V(G) = {u, v, w, x} {ui: 1 i }

{vj: 1 j } {wk: 1 k m}

{xh: 1 h n}.

The required vertex labeling f: V(G) {1, 2, 3, 4,

. . . , 2 + m + n + 4} is defined as follows:

The corresponding edge labels are as follows:

The edge label of

uui is i + 3 for 1 i ;

vvj is + j + 3 for 1 j ;

wwk is 2 + k + 4 for 1 k m and

xxh is 2 2 5

2

h m n for 1 h n – 1.

The edge label of xxn is 2 + m + n + 4. Hence

the induced edge labels of G are distinct. Hence

the graph G is a skolem mean graph.

Case : 2 Consider the graph

G = K1, K1, K1, m K1, n.

Let ≤ m < n where n = 2 + m + 2 for = 2, 3,

4, . . . and m = 2, 3, 4, . . . . Let us take the case

that |mn| < 4 + 2 for = 2, 3, 4, . . . ; m = 2, 3,

4, . . . and n = 2 + m + 2. We have to prove that

G is a skolem mean graph.

Therefore, the graph

G = K1, K1, K1, m K1, n

where n = 2 + m + 2 for = 2, 3, 4, . . . and

m = 2, 3, 4, . . . .

Let

{u} {ui: 1 i }, {v} {vj: 1 j },

{w} {wk: 1 k m} and {x} {xh: 1 h n}

be the vertices of G. Then G has 2 + m + n + 4

vertices and 2 + m + n edges.

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We have

V(G) = {u, v, w, x} {ui: 1 i }

{vj: 1 j } {wk: 1 k m}

{xh: 1 h n}.

The required vertex labeling

f: V(G) {1, 2, 3, 4, . . . , 2 + m + n + 4}

is defined as follows:

The corresponding edge labels are as follows:

The edge label of

uui is i + 2 for 1 i ;

vvj is + j + 2 for 1 j ;

wwk is 2 + k + 3 for 1 k m and

xxh is 2 2 6

2

h m n for 1 h n –2.

The edge label of xx n-1 is 2 + m + n +3 and the

edge label of xxn is 2 + m + n + 4. Hence the

induced edge labels of G are distinct. Hence the

graph G is a skolem mean graph.

Case : 3 Consider the graph

G = K1, K1, K1, m K1, n.

Let ≤ m < n where n = 2 + m + 1 for = 2, 3,

4, . . . and m = 2, 3, 4, . . . . Let us take the case

that |mn| < 4 + 2 for = 2, 3, 4, . . . ; m = 2, 3,

4, . . . and n = 2 + m + 1. We have to prove that

G is a skolem mean graph.

Therefore, the graph

G = K1, K1, K1, m K1, n

where n = 2 + m + 1 for = 2, 3, 4, . . . and

m = 2, 3, 4, . . . .

Let

{u} {ui: 1 i }, {v} {vj: 1 j },

{w} {wk: 1 k m}

and {x} {xh: 1 h n}

be the vertices of G. Then G has 2 + m + n + 4

vertices and 2 + m + n edges.

We have

V(G) = {u, v, w, x} {ui: 1 i }

{vj: 1 j } {wk: 1 k m}

{xh: 1 h n}.

The required vertex labeling f: V(G) {1, 2, 3, 4,

. . . , 2 + m + n + 4} is defined as follows:

The corresponding edge labels are as follows:

The edge label of

uui is i + 2 for 1 i ;

vvj is + j + 2 for 1 j ;

wwk is 2 + k + 3 for 1 k m and

xxh is 2 2 7

2

h m n for 1 h n –1.

The edge label of xxn is 2 + m + n + 4. Hence

the induced edge labels of G are distinct. Hence

the graph G is a skolem mean graph.

Case : 4 Consider the graph

G = K1, K1, K1, m K1, n.

Let ≤ m < n where n = 2 + m for = 2, 3, 4, .

. . and m = 2, 3, 4, . . . . Let us take the case that

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|mn| < 4 + 2 for = 2, 3, 4, . . . ; m = 2, 3, 4, . .

and n = 2 + m. We have to prove that G is a

skolem mean graph.

Therefore, the graph

G = K1, K1, K1, m K1, n

where n = 2 + m for = 2, 3, 4, . . . and

m = 2, 3, 4, . . . .

Let

{u} {ui: 1 i },

{v} {vj: 1 j },

{w} {wk: 1 k m} and

{x} {xh: 1 h n}

be the vertices of G. Then G has 2 + m + n + 4

vertices and 2 + m + n edges.

We have

V(G) = {u, v, w, x} {ui: 1 i }

{vj: 1 j } {wk: 1 k m}

{xh: 1 h n}.

The required vertex labeling

f: V(G) {1, 2, 3, 4, . . . , 2 + m + n + 4}

is defined as follows:

The corresponding edge labels are as follows:

The edge label of

uui is i + 1 for 1 i ;

vvj is + j + 2 for 1 j ;

wwk is 2 + k + 3 for 1 k m and

xxh is 2 2 8

2

h m n for 1 h n –1.

The edge label of xxn is 2 + m + n + 4. Hence

the induced edge labels of G are distinct. Hence

the graph G is a skolem mean graph.

REFERENCES

[1] V. Balaji, D. S. T. Ramesh and A.

Subramanian, Skolem Mean Labeling,

Bulletin of Pure and Applied Sciences, vol.

26E No. 2, 2007, 245 – 248.

[2] V. Balaji, D. S. T. Ramesh and A.

Subramanian, Some Results On Skolem Mean

Graphs, Bulletin of Pure and Applied

Sciences, vol. 27E No. 1, 2008, 67 – 74.

[3] J. A. Gallian, A Dynamic Survey of Graph

Labeling, The Electronic Journal of

combinatorics 16(2009), # DS6.

[4] V. Balaji, Solution of a Conjecture on Skolem

Mean Graph of stars K1, K1, m K1, n

International Journal of Mathematical

Combinatorics, vol.4, 2011, 115 – 117.

[5] F. Harary, Graph Theory, Addison – Wesley,

Reading Mars., (1972).

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A Geometric Process Repair Model for a Cold Standby epairable System With Imperfect Delay

Repair under T policy

P.Govindaraju and U.Rizwan

Abstract - In this paper, a cold standby repairable system with imperfect delay repair consisting of two dissimilar components and one repairman is studied. Assume that both component 1 and component 2 after repair are not as good as new, and the deterioration of the system is stochastic. Under these assumptions, using a geometric process, we consider a replacement policy T based on the number of failures of component 1 under which the system is replaced when the number of failures of component 1 reaches T. Our problem is to determine an optimal replacement policy T such that the average cost rate (i.e. the long-run average cost per unit time) of the system is minimized. The explicit expression of the average cost rate of the system is derived. Index Terms - Geometric process; Renewal process; Imperfect delay repair; Replacement policy; Renewal reward theorem.

—————————— ——————————

1. INTRODUCTION

T the earlier stage, many replacement models for

a one-component repairable system with one

repairman (i.e. simple repairable system) assumed that

the system after repair is as good as new. This is a

perfect repair model. However, this assumption is not

always true. In practice, most repairable systems are

deteriorative due to the ageing effect and the

accumulative wear. Barlow and Hunter (1960) first

presented a minimal repair model in which the

minimal repair does not change the age of the system.

Brown and Proschan (1983) first reported an imperfect

repair model in which the repair is perfect repair with

probability p or minimal repair with probability

p1 . For a deteriorating repairable system, it seems

reasonable that the successive working times of the

system after repair may become shorter and shorter

while the consecutive repair times of the system after

failure may become longer

——————————————

U. Rizwan is serving in the Department of Mathematics, Islamiah

College, Vaniyambadi, India. E-mail: rizwanmaths@yaoo.com

P. Govindaraju is serving in the Department of Mathematics, Islamiah

College, Vaniyambadi, India. E-mail : govindrajmaths69@gmail.com

and longer. Ultimately, it cannot work any longer,

neither can it be repaired. For such a stochastic

phenomenon, Lam (1988) first introduced a geometric

process repair model to approach it. Under this model,

he studied two kinds of replacement policies for a

simple repairable system, one based on the working

age T of the system and the other based on the

number of failures N of the system. Because the

geometric process is a special monotone process,

Stadje and Zuckerman (1990) introduced a general

monotone process repair model to generalize Lam‘s

work. Finkelstein (1993) presented a general repair

model based on a scale transformation after each repair

to generalize Lam‘s work. Stanley (1993) considered a

repair and replacement model for a system with a

random magnitude of shock at each failure and

proposed a replacement policy based on a random

threshold. In order to improve the reliability, raise the

availability or reduce the cost, a two-component

redundant system is often employed. Based on Lam

(1988), Zhang et al (2006) applied the geometric

process repair model to a two-component cold standby

repairable system with one repairman. Assume that

each component after repair is not as good as new.

Under this assumption, using a geometric process, they

studied two kinds of replacement policies, one based

on the working age T of component 1 under which the

system is replaced when the working age of

component 1 reaches T, and the other based on the

A

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number of failures N of component 1 under which the

system is replaced when the number of failures of

component 1 reaches N. For example, the maintenance

of the computer after failure needs definite waiting

time such that the repairman can just arrive at the

locale; the system after failure cannot be repaired

immediately because the repairman is taking a

vacation or because the repair is complicated to need

inviting a high-tech repairman or other. These will

cause a delay repair time. Thus, the purpose of this

paper is to consider a cold standby repairable system

consisting of two dissimilar components and one

repairman in which when any component fails,

sometimes the repair is delayed and sometimes the

repair is immediate, and the repair is called imperfect

delay repair. For such a cold standby repairable

system, using a geometric process, we consider a

replacement policy T based on the number of failures

of component 1 under which the system is replaced

when the number of failures of component 1 reaches T

. Our problem is to determine an optimal replacement

policy T such that the average cost rate of the system

is minimized. The explicit expression of the average

cost rate of the system is derived.

2. DESCRIPTION OF THE MODEL

For ease of reference, we first state the definitions of

stochastic order and geometric process as follows.

Definition 1 A random variable X is said to be

stochastically smaller than another random variable

Y , if )>()>( YPXP , for all real . It is

denoted by YX st . A stochastic process

1,2,=,nX n is said to be stochastically

increasing, if 1 nstn XX , for 1,2,=n .

Definition 2 A Markov process 1,2,=,nX n with

state space 0,1,2, is said to be stochastically

monotone, if

1 1 1 2| = | = ,n n st n nX X i X X i

1 20 .for any i i

Clearly, the stochastically monotone concept for a

Markov process is defined for a Markov process and is

based on the transition probabilities from one state to

another state, conditioning on the former state.

However, the stochastically monotone concept for a

stochastic process defined here is for a general process

and is based on the conditional distribution of two

successive random variables in the process.

Definition 3 A stochastic process 1,2,=,nX n is

a geometric process, if there exist a constant 0>a

such that 1,2,=,1 nXa n

n forms a renewal

process. The number a is called the ratio of the

geometric process.

If 1<<0 a , then the GP is stochastically increasing;

if 1>a , the GP is stochastically decreasing and if

1=a , the GP will reduce to a renewal process. If

1=][ 1XE and

11

=][nn

aXE

Definition 4 An integer valued random variable N is

said to be a stopping time for the sequence of

independent random variables ,, 21 XX , if the event

nN = is independent of ,, 21 nn XX , for all

1,2,=n .

We study such a two-component cold standby

repairable system with imperfect delay repair by

making the following assumptions 81 AA .

A1 At the beginning, the two components are

both new, and component 1 is in a

working state while component 2 is in a

cold standby state.

A2 Whenever the system fails in any of the

failure states, it will be repaired. The

system will be replaced by an identical

one some times later.

A3 When the two components in the system

are both good, one is working and the

other is under cold standby. Whenever the

working one breaks down, the repair is

delayed with probability p or immediate

with probability p1 . At the same time,

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the standby one begins to work. The

repair discipline is ‖first in first out‖.

Whenever the repair of the failed one is

completed, it either begins to work again

or become under cold standby. If a

component fails during the repair of the

other, it must wait for repair and the

system is down.

A4 Assume that the time interval between the

completion of the 1)( n th repair and the

completion of the n th repair of

component i is called the n th cycle of

component 1,2,=1,2;=, nii . Let )()( , i

n

i

n YX and )(i

nZ be respectively the

working time, the repair time and the

delay repair time of component i in the n

th cycle, 1,2,=1,2;=, nii Obviously,)()( , i

n

i

n YX and )(i

nZ are respectively a

sequence of nonnegative random

variables, we define GF , and H

respectively as a distribution for )()( , i

n

i

n YX

and 1,2,=1,2;=,,)( niiZ i

n. The

distributions of )(i

nX and )(i

nY are

respectively denoted by

)(=)();(=)( 1=)(1)( tbGtGtaFtF n

i

i

n

n

i

i

n

where 1,<1,0,1,2,=0, ii bnt and

assume that

1,2.=0,>1

=][0,>1

=][ )(

1

)(

1 iYEXEi

ii

Thus

1,2,=1,2.;=)( niX i

n

and 1,2,=1,2.=)( niY i

n

are, respectively, a stochastically

decreasing geometric process with the

ratio ai and a stochastically increasing

geometric process with the ratio bi. And

assume that 1,2,=1,2.=,)( niZ i

n is a

sequence of nonnegative random variables

with same distribution function

0>),( ttH and assume that

1=][ )(i

nZE .

A5 The survival time )(i

nX , the repair time ( )i

nY and )(i

nZ , )1,2,=( n all are

independent random variables.

A6 Assume that the replacement policy T

based on the number of failures of

component 1 is used. The system will be

replaced sometime by a new and identical

one, and the replacement time is

negligible.

A7 Assume that any component in the system

cannot produce the working reward during

cold standby, and no cost is incurred

during waiting for repair and delay repair.

A8 Assume that the repair cost rate of

component i is 1,2)=()( ic i

r while the

working reward rate of two components is

same c . And the replacement cost of the

system is C .

3. THE REPLACEMENT POLICY T

In this section, we will consider the replacement

policy T based on the number of failures of

component 1. Because the two components appear

alternately in the system, when the failure number

of component 1 reachesT , then component 2 may

be in the cold standby state or in the repair state or

in the delay repair state. Naturally, a practicable

replacement policy T should be that component 1

is not repaired any more when the failure number

of component 1 reaches T ; while component 2

works until failure in the N th cycle. Thus, the

renewal point under the policy T is established.

Let 1 be the first replacement time of the system,

and 2)( nn be the time between the 1)( n cth

replacement and the n th replacement of the

system under policy T : Obviously },,{, 21

forms a renewal process, and the interarrival time

between two consecutive replacements is called a

renewal cycle.

Our problem is to determine an optimal

replacement policy T such that the average cost

rate of the system is minimized. Let )(TC be the

average cost rate of the system with imperfect

delay repair under policy T . Thus, according to

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renewal reward theorem (see, for example Ross

[19]), we have

t( ) =

t

he expected cost incurred in a renewal cycleC T

he expected length of a renewal cycle

The working age T of the system at time t is the

cumulative life-time given by

1

1 1 1 1

<=

<

n n n n n

n n n n n

t M L M t L MT

L L M t L M

where i

n

i

n XL 1=

= and i

n

i

n YM 1=

= and

0== 00 ML .

Let 1T be the first replacement time and in general

for ,2,3,= n let nT be the time between the

1)( n -st replacement and the n -th replacement.

Thus the sequence ,, 21 TT constitutes a renewal

process. Thus a cycle is completed, if a

replacement is done. By the theory of renewal

reward process, the long-run average cost per unit

time is given by

e( ) =

e

xpected cost incurred in a cycleC T

xpected length of a cycle

,

)(

)(

=1

1=1=

1=

1

1=

ZEYEXE

XREZEcRYcE

i

i

i

i

ii

i

pii

i

(1)

where is a random variable which denotes the

number of failures in time T .

Since is also a stopping time with respect to the

-fields 1,2,=>,,,,< 21 XXX , by

Wald‘s equation, we have

.)(

=1

1=1=

n

n

n

i

i a

TFXE

(2)

where )(nF is the n -fold convolution of )(F

with itself and

.)(

=1=

1

1=n

n

n

i

i b

TGYE

(3)

where )(nG is the n -fold convolution of )(G

with itself.

Using equations (2) and (3) in equation (1), we

obtain on simplification that

12=1 =2

1=1 =1

( ) ( )

( ) =( ) ( )

n npn n

n n

n n

n nn n

G T F Tc R c r r

b aC T

F T G T

a b

(4)

where =)(ZE .

4. CONCLUSION

In this paper, using geometric process repair

model, we studied a cold standby repairable

system consisting of two dissimilar components

and one repairman with imperfect delay repair

under T policy. We have also derived the long-run

average cost under this policy.

REFERENCES

[1] Barlow, R.E. and Hunter, L.C., (1960)

Optimum preventive maintenance policy.

[2] Brown, M. and Proschan, F., (1983)

Imperfect repair. J. Appl. Prob. 20, 851-859.

[3] Lam, Y., (1988) Geometric processes and

replacement problem. Acta Math. Appl. Sin.

4(4), 366-377.

[4] Lam, Y., (1990) A repair replacement

model. Adv. Appl. Prob. 22, 494-497.

[5] Lam, Y., (1991) An optimal repairable

replacement model for deteriorating

systems. J. Appl. Prob. 28, 843-851.

[6] Lam, Y., (2007) The Geometric Process and

Its Applications. World Scientific

Publishing CO. Pte. Ltd., Singapore.

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[7] Leung, K.N.G., (2005) A Note on a

Bivariate Optimal Replacement Policy for a

Repairable System, Engineering

Optimization, 38, 621 -625.

[8] Finkelstein, M. S., (1993) A scale model of

general repair. Microelectron. Reliab. 33,

41-44.

[9] Stadje, W. and Zuckerman, D., (1990)

Optimal strategies for some repair

replacement models. Adv. Appl. Prob. 22,

641-656.

[10] Stanley, A. D. J., (1993) On geometric

processes and repair replacement problems.

Microelectron. Reliab. 33, 489-491.

[11] Zhang, Y.L., (1994) A bivariate optimal

replacement policy for a repairable system.

J. Appl. Prob. 31, 1123-1127.

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District-wise Component Analysis of the Growth of Cotton Production in Tamil Nadu

R. Meenakshi

Abstract: This paper analyzes the components of cotton production in the districts of Tamil Nadu during the pre and post reform

periods.

Index Terms - Component Analysis, Cotton Production, Reform Periods.

—————————— ——————————

1. INTRODUCTION

N attempt is made here to analyze the

components of cotton production in districts

of Tamil Nadu state in pre and post reform

periods. The agricultural development and the

prosperity of rural masses in Tamil Nadu state

mainly depend upon sound agricultural base.

Cotton production by and large can be increased

by increasing area under cultivation, double

cropping and raising yield of cotton by the

application of new agricultural technology and by

the reorganization of institutional factors.

The favourable resource-base complemented by

suitable modern technologies facilitated to

increase cotton output and yield in the state. The

total cotton production in Tamil Nadu state was

estimated at 225448 (in bales of 170 kg / lint)

which contributes around 11 percent of all India

cotton production in 2009 – 2010.

In recent years many attempts have been made to

assess the growth of cotton production of Tamil

Nadu state. So a quantitative assessment of the

growth of cotton and the contribution of

components to the cotton production in this state

assumes significance.

R. Meenakshi is serving as Associate Professor and Head of the

Department of Economics, Sri Sarada College for Women, Salem, India.

Accordingly an attempt has been made in the

present study to apply the seven-factor model with

a view to identify the contribution of the

components to cotton production in districts of

Tamil Nadu state and state as a whole in pre and

post reform periods.

2. METHODOLOGY AND DATA

The basic data comprises year-wise information

on area, production yield and price of cotton and

the total cropped area in districts of Tamil Nadu

state and they are obtained from ‗Season and Crop

Reports‘ published by Tamil Nadu Government

for the pre-reform period 1971-72 to 1989-90 and

post-reform period 1990-91 to 2009-10. The

growth of cotton production for the purpose of

decomposition analysis in pre-reform period has

been compiled as a change in production of

current period (taken as an average of the last

three years) 1987-88 through 1989-90 over the

base period ( taken as an average of first three

years) 1971-72 through 1973-74 for each time

period and in the post-reform period the data have

been compiled as a change in production of

current period (average of the last three years)

2007-08 through 2009-10 over the base period

(average of first three years) 1990-91 through

1992-93. The changes in components have also

been similarly computed on the basis of three-year

averages of base and current years. Constant price

weights have been assigned to cotton crop based

on the three year average of farm harvest prices of

1971-72, 1972-73 and 1973-74 in pre-reform

period and 1990-91, 1991-92 and 1992-93 in post-

reform period.

A

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The seven component elements of decomposition

analysis are

(i) area

(ii) yield rate

(iii) cropping pattern

(iv) area and yield

(v) area and cropping pattern

(vi) yield and cropping pattern

(vii) area, yield and cropping pattern.

The following is the equation of the model.

Pt = P0 = (At – A0)

0

( )

n

i io io

t

n

i io it io

t

W C Y

A W C Y Y

0

0

( )

( ) ( )

n

i io it io

t

n

t i io it io

t

A W Y C C

A A W Y C C

0

0

( ) ( )

(C )(Y )

n

t i it io io

t

n

i it io it io

t

A A W Y Y C

A W C Y

)Y (Y )C (C WAA ioitioiti

n

t0t

where

Pi = production in the current period

Po = production in the base period

Wi = weight used for each crop

Cio = cropping pattern in the base year

Cit = cropping pattern in the current period

Yio = yield level in the base year

Yit = yield level in the current period

Ao = gross cropped are in the base year

At = gross cropped area in the current year

Further Pt and Po were equated as under.

Pt = ititi

n

tt Y C WA

P0 = ioioi

n

t0 Y C WA

Each component of the equation was multiplied

by 100 and divided by the total components under

consideration so as to obtain a percentage growth

rate of each component. These percentage would

help one to directly assess the share of each

component in the growth rate of cotton

production.

The study addresses itself to the main issues viz,

A) Identifying the components that have

contributed to the increase in cotton

production in districts of Tamil Nadu state.

B) Analyzing the factors which have

contributed to the cotton production and

quantifying the relative contribution of

each component in the growth of cotton in

districts of Tamil Nadu state and state as a

whole.

3. RESULTS AND DISCUSION

Different variables i.e. acreage, yield, change in

crop pattern for cotton crop in districts of Tamil

Nadu state in pre and post reform periods were

estimated in tables with a view to identify their

relative contribution to production. Substituting

these variables in the decomposition model, the

contribution of different factors i.e. area, yield,

cropping pattern towards cotton production is thus

illustrated for all districts and state as a whole.

The contribution of components to the cotton

production is shown in district-wise analysis the

pre-reform period and post-reform period.

4. PRE-REFORM PERIOD

4.1 CHENGALPATTU- MGR

Table I presents a contribution of different

components to production of cotton in this region

in the pre-reform period. The study results in the

table reveal that the only factor contributing to the

highest increase in cotton production was yield

(111.23%). Acreage, I order interaction between

area and crop pattern and the I order interaction

between yield and crop pattern also account for

positive effects of cotton production. The

remaining factors do not account for increase in

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production and have destabilizing effects on

cotton growth.

4.2 SOUTH ARCOT

Table I reveals that as far as the influence of crop

pattern (37.64%) is concerned, it is responsible for

the maximum increase in cotton production in

South Arcot in pre-reform period. What is

surprising in this district is that all components

resulted in a positive production growth. The

analysis reveals that the whole increase in cotton

production in South Arcot region was brought by

all components though the positive entries of crop

pattern, yield and I order interaction between yield

and crop pattern are more than the positive entries

of the remaining components of cotton

production. 4.3 NORTH ARCOT – AMBEDKAR –

THIRUVANNAMALAI SAMBUVARAYAR

In this region, the data provided show that crop

pattern contributed positively and significantly

towards increase in cotton production followed by

I order interaction between area and crop pattern

(25.80%). The yield factor and the I order

interaction between yield and crop pattern have

also added a positive contribution towards cotton

crop production. A small decrease in production

was noticed by I order interaction between area

and yield, acreage and II order interaction

between area, yield and crop pattern.

4.4 SALEM – DHARMAPURI

The results indicate that the pure effect of crop

pattern (47.05%) contributed positively and

significantly towards increase in cotton production

in Salem and Dharmapuri. The next component

that has resulted in a positive increase in

production was I order interaction between yield

and crop pattern (23.35%). Similar to South Arcot

district, the contribution of all components in

Salem – Dharmapuri had added a positive effect

towards increase in cotton production. 4.5 COIMBATORE - PERIYAR

It may be recalled from Table I that as far as the

influence of acreage (58.61%) is concerned, it is

responsible for the significant increase in cotton

production followed by the significant

contribution of crop pattern (54.62%) for cotton

production. The influence of yield accounts for

reduction in crop production. The first order

interaction terms namely (i) interaction between

area and yield (ii) yield and crop pattern had also

added a positive contribution towards cotton

production. The other interaction terms had a

destablishing effect on production. 4.6 TIRUCHIRAPPALLI – PUDUKOTTAI

As regards cotton production in Tiruchirappalli –

Pudukottai region is concerned it may be noted

that the highest positive entry of the component is

crop pattern (96.11%). Factors like acreage and I

order interaction between area and crop pattern

showed positive contribution towards production.

All other components added negative effect

towards cotton crop production. 4.7 THANJAVUR

With regard to Thanjavur district, the contribution

of crop pattern in positive (138.43%) and there is

every reason to believe that this component plays

an important role in increasing the production of

cotton. The first order interaction between area

and yield and second order interaction between

area, yield and crop pattern are also positive but

negligible towards increase in production. The

remaining components are found to be negative

and are not responsible for cotton growth is this

region. 4.8 MADURAI – DINDIGUL

Considering the importance of increase in cotton

production in Madurai – Dindigul, it may be noted

that components like crop pattern (64.25%), yield

(22.94%) and I order interaction between yield

and crop pattern (21.59%) do contribute positively

and significantly towards this crop. The

contribution of the remaining components is

found to be negative and seemed to be the source

of instability in cotton production. 4.9 RAMANATHAPURAM – KAMARAJAR –

PASUMPON MUTHURAMALINGAM

It may be seen from Table I that the crop pattern

(52.81%) and I order interaction between yield

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and crop pattern (51.42%) seemed to be the main

source for the significant increase in cotton

production in this area. To a certain extent yield

had also showed a positive effect towards

production. The remaining components failed to

contribute towards increase in cotton crop

production.

4.10 TIRUNELVELI – CHIDAMBARANAR

Table I shows that the I order interaction between

yield and crop pattern (58.19%) contributed

positively and significantly towards increase in

cotton production in this area. The influence of

yield (49.31%) and crop pattern (26.82%) had

added a positive contribution towards this crop.

As against this acreage, I order interaction terms

namely (i) between area and crop pattern (ii)

between area and yield and second order

interaction between area, yield and crop pattern

had a destablishing effect production.

4.11 TAMIL NADU

The state accounted for the maximum increase in

cotton crop production through yield component.

The study results in Table 3.11 and Figure 3.11

reveal that the other factors contributing to the

increase in cotton production were I order

interaction between area and crop pattern and II

order interaction between area, yield and crop

pattern. The remaining components did not

account for increase in production and had a

destablising effect on cotton growth.

The overall result shows that the significant

increase in cotton production is achieved by the

influence of crop pattern followed by yield

component.

Considering the importance of cotton production

in post-reform period an in-depth study is again

taken up for the districts of Tamil Nadu and state

as whole. Different variables i.e. acreage, yield,

crop pattern and their interaction terms were

estimated in the following Table I with a view to

identify their relative contribution to cotton

production.

5. POST-REFORM PERIOD

5.1 CHENGALPATTU- CHENNAI –

KANCHEEPURAM - THIRUVALLUR

Table II shows that as for as the influence of crop

pattern (81.42%) is concerned, it is responsible for

the significant increase in cotton production.

Contrary to this, yield component (-26.84%) did

not account for increase in production and had a

destablishing effect on cotton growth. The

positive entries of acreage followed by I order

interaction between yield and crop pattern had

contributed to the large share of the increase in

production. Again I order interaction between area

and yield, I order interaction between area, yield

and crop pattern appear to be positive components

contributing more or less equal percentage

towards increase in cotton production. With the

exception of yield and I order interaction between

area and crop pattern, other factors are responsible

for increase in cotton production.

5.2 SOUTH ARCOT – CUDALLORE –

VILUPPURAM

The percentage contribution of different variables

to total increased cotton production in this region

is shown in Table II. The study results reveal that

the factor contributing to the highest increase in

cotton production was acreage (80.47%) followed

by yield (69.26%). The I order interaction terms

namely (i) interaction between area and crop

pattern (ii) yield and crop pattern account for a

positive effect towards production. The remaining

components failed to account for increase in

production and had destablising effects on cotton

growth.

5.3 NORTH ARCOT – VELLORE –

THIRUVANNAMALAI SAMBUVARAYAR

Table II shows that acreage (87.64%) is responsible for

the significant increase in cotton production followed

by yield component (23.08%). The positive

contribution of acreage and yield resulted in the

positive contribution of I order interaction between

area and yield. All other components are negative and

had contributed to the decrease in cotton crop

production.

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5.4 SALEM – NAMAKKAL

The results indicate that the pure effect of yield

(123.17%) contributed positively and significantly

towards increase in cotton production. The crop

pattern had also added a positive effect towards

increase in production. Due to this the first order

interaction between yield and crop pattern is

found to be positive and accounted for the growth

of cotton. Acreage I order interaction terms (i)

between area and crop pattern (ii) area and yield

and II order interaction between area, yield and

crop pattern do have a more destabilizing

influence on cotton production. 5.5 DHARMAPURI – KRISHNAGIRI

The data provided in Table II show that all the

major components namely acreage, yield and crop

pattern when considered independently contribute

towards increase in cotton production and among

these three components acreage (55.87%)

accounted for the maximum increase in cotton

crop followed by yield (51.06%). All the I order

interaction terms failed to contribute to the growth

of cotton production. However the second order

interaction between area, yield and crop pattern

had added a positive contribution towards cotton

production. 5.6 COIMBATORE – THIRUPPUR

With regard to Coimbatore district, the

contribution of crop pattern is positive (104.78%)

and there is every reason to believe that this

component plays an important role in increasing

the production of cotton. Acreage and yield failed

to contribute towards increase in production. Due

to this, the first order interaction between acreage

and yield was also negative revealing

destabilizing effect on cotton crop. The remaining

components accounted for the positive increase in

production. 5.7 ERODE

In Erode district crop pattern (107.43%) plays an

important role in influencing cotton crop.

Contrary to crop pattern, yield component does

have a more destabilizing influence on this crop.

The positive entry of crop pattern overweighs the

negative entry of yield. As a result the I order

interaction between yield and crop pattern is

found to be positive and had added a positive

contribution towards cotton production.

Components like acreage, I order interaction

between area and yield are also positive in this

region. The remaining components accounted for

decrease in production due to their negative

entries in cotton production.

5.8 TIRUCHIRAPPALLI – KARUR –

PERAMBALUR – ARIYALUR

In this region both crop pattern (97.54%) and

yield (24.05%) seemed to be the main source for

the significant increase in cotton production. Due

to this, the I order interaction between yield and

crop pattern had added a positive contribution

towards cotton production. Acreage is found to be

negative. It is because of the negative entry of

acreage, components like I order interaction

between area and crop pattern, area and yield, II

order interaction between area, yield and crop

pattern are responsible for decrease in cotton

production. 5.9 PUDUKOTTAI

With respect to Pudukottai district, the

contribution of crop pattern (123.28%) is positive

and significant towards increase in cotton

production. Components like acreage and I order

interaction between area and yield are also

positive and hence they are responsible for

increase in production. All other components are

negative and seemed to be the source of instability

in cotton crop.

5.10 THANJAVUR – THIRUVARUR –

NAGAI QUAID-E-MILLETH –

NAGAPATTINAM

The results indicate that crop pattern (98.17%)

contributed significantly towards increase in

cotton production followed by yield. Along with

these components, the positive values of I order

interaction between area and crop pattern, I order

interaction between area and yield had added an

increase in cotton production. The remaining

components are found to be negative and hence

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failed to contribute towards increase in production

of cotton crop. 5.11 MADURAI – THENI – DINDIGUL

Considering the importance of cotton production

in this area, crop pattern (103.53%) does

contribute significantly towards increase in

production. Next to this I order interaction

between yield and crop pattern and acreage and

crop pattern contribute positively towards cotton.

Again the entry of I order interaction between area

and yield had added positive increase in

production. All other components failed to

increase cotton crop production.

5.12 RAMANATHAPURAM – VIRUDHUNAGAR – KAMARAJAR – PASUMPON MUTHURAMALINGAM – SIVAGANGAI

The data provided in Table II reveal that all the

major components namely acreage, yield and crop

pattern when considered independently contribute

towards increase in production and among these

components, crop pattern accounted for the

maximum increase in cotton production (89.80%)

followed by yield (28.83%) and acreage (23%).

All I order interaction terms failed to contribute to

the cotton growth. But the II order interaction

between area, yield and crop pattern had added a

positive effect towards increase in production. 5.13 THIRUNELVELI – THOOTHUKUDI

The study results indicate that crop pattern

(98.97%) contributed positively towards increase

in cotton production followed by acreage.

Contrary to these two components, yield is found

to be negative and had a destabilizing effect

towards production. Again I order interaction

between area and crop pattern, II order interaction

between area, yield and crop pattern failed to

contribute towards the growth of production of

cotton. The remaining components contribute

towards increase in cotton production. 5.14 TAMIL NADU

For the state as a whole the increase in cotton

production was mainly due to the influence of

crop pattern (97.15%) followed by acreage

(32.53%). With these two components the I order

interaction between area and yield and the

interaction between yield and crop pattern account

for increase in production. All the remaining

components had negative entries and hence they

were not responsible for cotton production

growth.

The overall result shows that the significant

increase in cotton production is achieved by the

influence of crop pattern in pre-reform period and

yield in the post-reform period.

6. CONCLUSION

Considering the importance of cotton production

in Tamil Nadu state an in-depth study is now

necessary to identify different constraints to the

productivity and then efforts should be made for

enhancing cotton productivity to meet the

domestic requirements of cotton in Tamil Nadu

state. In Tamil Nadu state, cotton is grown under

risky conditions by resource poor farmers. Hence

technologies for this unfavourable environment

and packages of practices to suit these farmers

must be developed. Rainfed cotton crop research

is to be a priority.

To improve the cotton production the following

measures may be considered.

Better cultural practices and the use of tractors

instead of bullocks in seed-bed preparation

after initial watering of land is positively

related to yields.

Farm characteristics including the farmer‘s

education is an important dimension affecting

the difference between below and above

average yields.

A positive link between the farmer‘s visit to

extension agency for advice and his end-

season cotton output is not as apparent as

commonly hypothesized.

Attempts to identify a common and robust

pattern of spray timings applicable for the

entire area.

Minimization of inflator trends in factor prices

is very essential.

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Assuring remunerative output prices, through

effective support price mechanism is the need

of the hour.

Though cotton crop has received attention to a

certain extent by the policy makers in recent

years, in view of the present need, more

efforts have to be focused on research and

development in the cotton crop.

Liberalized import substitution and export

promotional policies may be encouraged.

Greater degree of specialization and

professionalism are to be inducted in expanding

the production of cotton if Tamil Nadu state is to

emerge as the leading producer of cotton in India.

ACKNOWLEDGEMENTS

The Author is highly thankful to the project fellow

Ms. S. Deepalakshmi for data collection.

REFERENCES

[1] A.V.K. Sastri, ―Relative contribution of Area and

Yield to Increased Production of Wheat during the

First Plan‖, Agricultural Situation in India, Vol.

XV, No. 5, August, 1960, 481-486.

[2] B.S. Minhas, ―Rapporteur‘s Report on

Management of Agricultural Growth‖, Indian

Journal of Agricultural Economics, Vol. XXI, No,

4, Oct.-Dec. 1966.

[3] B.S. Minhas and A. Vaidyanathan, ―Growth of

Crop Output in India, 1951-54 to 1956-61: An

Analysis of component Element‖, Journal of

Indian Society of Agricultural Statistics, Vol.

XVII, No. 2, Dec., 1965, 230-252.

[4] J.R. Anderson, Peter BR. Hazell and L. Evans,

―Variability in cereal Yield: Implications for

Agricultural Research and Policy‖, Summary

Proceedings of IFPRI/DSE Workshop on cereal

Yield Variability, Feldanfing. Germany, Nov.,

1986, 26-29.

[5] S. Narender, et al, ―District-wise Measurement and

Decomposition of the Growth of Agricultural

Output in Andhra Pradesh‖, Agricultural Situation in

India, April, 1989, 3-7.

[6] Peter B.R. Hazell, ―Instability in Indian Foodgrain

Production‖, Research Report 30, IFPRI,

Washington DC, U.S.A. 1982.

[7] R. Swarup and B.K. Sikka ―Agricultural

Development in Himachal Pradesh‖, Agricole

Publishing company, 1983, 64-68.

[8] S. Mahendradev, ―Growth and Instability in

Foodgrains Production: An

Inter-State Analysis‖, Economic and Political

Weekly. Sept., 26, 1987, pp A82-A92,

[9] Suresh Pal and A.S. Sirohi, ―Sources of Growth

and Instability in Indian Crap Production-A

Decomposition Analysis‖, Agricultural Situation

in India, Feb., 1989, 933-936.

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TABLE I

DISTRICT- WISE PERCENTAGE CONTRIBUTION OF DIFFERENT VARIABLES TO

TOTAL INCREASED COTTON PRODUCTION IN PRE-REFORM PERIOD

(1971 - 1972 to 1989 - 1990) IN TAMIL NADU

DISTRICT 1 2 3 4 5 6 7 Total

1. Chengalpattu-MGR 0.09 111.23 -11.87 0.34 -0.03 0.19 -0.05 100.00

2. South Arcot 5.13 31.87 37.64 2.96 2.51 18.43 1.46 100.00

3.

North Arcot

(Ambedkar-

Thiruvannamalai-

Sambuvarayar)

-3.54 1.49 69.91 25.80 -0.55 10.91 -4.02 100.00

4. Salem –

Dharmapuri 6.60 1.25 47.05 12.35 3.27 23.35 6.13 100.00

5. Coimbatore –

Periyar 58.61 -5.99 54.62 -8.99 0.99 0.92 -0.16 100.00

6. Tiruchirappalli –

Pudukottai 12.07 -8.83 96.11 18.68 -1.72 -13.66 -2.65 100.00

7. Thanjavur -1.60 -1.18 138.43 -22.13 0.19 -16.32 2.61 100.00

8. Madurai – Dindigul -3.38 22.94 64.25 -3.19 -1.14 21.59 -1.07 100.00

9.

Ramanathapuram

(Kamarajar-

Pasumponmuthu

Ramalingam)

-4.26 22.97 52.81 -9.53 -4.14 51.42 -9.28 100.00

10. Tirunelveli -

Chidambaranar -4.97 49.31 26.82 -5.85 -10.78 58.19 -12.72 100.00

11. Tamil Nadu -228.40 466.24 -70.18 9.12 -60.58 -18.62 2.42 100.00

Source : Compiled by the Author.

1.Area 4. I order interaction between area and crop pattern

2.Yield 5. I order interaction between area and yield

3.Crop Pattern 6. I order interaction between yield and crop pattern

7. II order interaction between area, yield and crop pattern

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TABLE II

DISTRICT- WISE PERCENTAGE CONTRIBUTION OF DIFFERENT VARIABLES TO TOTAL INCREASED

COTTON PRODUCTION IN POST-REFORM PERIOD (1990 - 1991 to 2009 - 2010) IN TAMIL NADU

DISTRICT 1 2 3 4 5 6 7 Total

1.

Chengalpattu-

Chennai-

Kancheepuram-

Thiruvallur

29.74 -26.84 81.42 -28.58 9.42 25.79 9.05 100.00

2.

South Arcot –

Cuddalore-

Villupuram

80.47 69.26 -56.09 21.56 -26.63 18.56 -7.13 100.00

3.

North Arcot-

Vellore-

Thiruvannamalai-

Sambuvarayar

87.64 23.08 -13.39 -10.38 17.90 -2.73 -2.12 100.00

4. Salem –Namakkal -27.89 123.17 22.84 -5.84 -31.48 25.78 -6.59 100.00

5. Dharmapuri-

Krishnagiri 55.87 51.08 19.99 -6.46 -16.49 -5.90 1.91 100.00

6. Coimbatore-

Thiruppur -0.84 -17.70 104.78 0.62 -0.11 13.17 0.08 100.00

7. Erode 54.66 -106.61 107.43 -33.24 32.98 64.84 -20.06 100.00

8.

Tiruchirappalli-

Karur-Perambalur-

Ariyalur

-18.70 24.05 97.54 -16.36 -4.03 21.03 -3.53 100.00

9. Pudukottai 15.93 -13.38 123.28 -14.14 1.53 -11.86 -1.36 100.00

10.

Thanjavur-

Thiruvarur-Nagai-

Quid-e-milleth-

Nagapattinam

-7.15 20.54 98.17 6.26 1.31 -17.98 -1.15 100.00

11. Madurai-Theni-

Dindigul 27.66 -48.09 103.53 -19.19 8.90 33.37 -6.18 100.00

12

Ramanathapuram-

Virdhunagar-

Kamarajar-

Pasumpon

Muthuramalingam

–Sivagangai

23.00 28.83 89.80 -17.91 -5.75 -22.45 4.48 100.00

13. Tirunelveli-

Thoothukudi 17.91 -11.94 98.97 -15.57 1.88 10.38 -1.63 100.00

Tamil Nadu 32.53 -32.24 97.15 -16.28 5.40 16.14 -2.70 100.00

Source : Compiled by the Author

1. Area 4. I order interaction between area and crop pattern

2. Yield 5. I order interaction between area and yield

3. Crop Pattern 6. I order interaction between yield and crop pattern

7. II order interaction between area, yield and crop pattern

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History of Algebra

U. Rizwan

Abstract - This paper presents a review of the algebra and presents some facts.

Index Terms – Algebra, Geometry.

—————————— ——————————

The Arabic word for restoration, al-jabru, is the

root of the word algebra.

The history of algebra goes way back in time, but

its importance is unparalleled by any other branch

of mathematics. The word Algebra literally

means the re-union of broken parts based on the

origins of Arabic language. It was first used

around 830 AD by Arab scholars.

The history of algebra began in ancient Egypt and

Babylon, where people learned to solve linear (ax

= b) and quadratic (ax2 + bx = c) equations, as

well as indeterminate equations such as x2 + y

2 =

z2, whereby several unknowns are involved. The

ancient Babylonians solved arbitrary quadratic

equations by essentially the same procedures

taught today. They also could solve some

indeterminate equations.

The Alexandrian mathematicians Hero of

Alexandria and Diophantus continued the

traditions of Egypt and Babylon. But

Diophantus's book Arithmetica is on a much

higher level and gives many surprising solutions

to difficult indeterminate equations. This ancient

knowledge of solutions of equations in turn found

a home early in the Islamic world, where it was

known as the science of restoration and balancing.

In the 9th century, the Arab mathematician Al-

Khwarizmi wrote one of the first book on algebras

in Arabic language, a systematic expose of the

basic theory of equations,with examples and

proofs.

——————————————

U. Rizwan is serving in the Department of Mathematics, Islamiah

College, Vaniyambadi, India. E-mail: rizwanmaths@yaoo.com

By the end of the 9th century, the Egyptian

mathematician Abu Kamil had stated and proved

the basic laws and identities of algebra and solved

s complicated problems as finding x, y, and z

such that x + y + z = 10, x2 + y

2 = z

2, and

xz = y2.

Ancient civilizations wrote algebraic expressions

using only occasional abbreviations, but by

medieval times Islamic mathematicians were able

to talk about arbitrarily high powers of the

unknown x, and work out the basic algebra of

polynomials (without yet using modern

symbolism). This included the ability to multiply,

divide, and find square roots of polynomials as

well as a knowledge of the binomial theorem. The

Persian mathematician, astronomer, and poet

Omar Khayyam showed how to express roots of

cubic equations by line segments obtained by

intersecting conic sections, but he could not find a

formula for the roots. A Latin translation of Al-

Khwarizmi's Algebra appeared in the 12th

century. In the early 13th century, the great Italian

mathematician Leonardo Fibonacci achieved a

close approximation to the solution of the cubic

equation x3 + 2x

2 + cx = d. Because Fibonacci had

traveled in Islamic lands,he probably used an

Arabic method of successive approximations.

Early in the 16th century, the Italian

mathematicians Scipione del Ferro, Niccolò

Tartaglia, and Gerolamo Cardano solved the

general cubic equation in terms of the constants

appearing in the equation. Cardano's pupil,

Ludovico Ferrari, soon found an exact solution to

equations of the fourth degree, and as a result,

mathematicians for the next several centuries tried

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to find a formula for the roots of equations of

degree five, or higher. Early in the 19th century,

however, the Norwegian mathematician Niels

Abel and the French mathematician Evariste

Galois proved that no such formula exists.

An important development in algebra in the 16th

century was the introduction of symbols for the

unknown and for algebraic powers and operations.

As a result of this development, Book III of La

geometrie (1637), written by the French

philosopher and mathematician René Descartes,

looks much like a modern algebra text.

Descartes's most significant contribution to

mathematics, however, was his discovery of

analytic geometry, which reduces the solution of

geometric problems to the solution of algebraic

ones. His geometry text also contained the

essentials of a course on the theory of equations,

including his so-called rule of signs for counting

the number of what Descartes called the true

(positive) and false (negative) roots of an

equation. Work continued through the 18th

century on the theory of equations, but not until

1799 was the proof published, by the German

mathematician Carl Friedrich Gauss, showing that

every polynomial equation has at least one root in

the complex plane (see Number: Complex

Numbers).

By the time of Gauss, algebra had entered its

modern phase. Attention shifted from solving

polynomial equations to studying the structure of

abstract mathematical systems whose axioms were

based on the behavior of mathematical objects,

such as complex numbers, that mathematicians

encountered when studying polynomial equations.

Two examples of such systems are algebraic

groups (see Group) and quaternions, which share

some of the properties of number systems but also

depart from them in important ways. Groups

began as systems of permutations and

combinations of roots of polynomials, but they

became one of the chief unifying concepts of

19th-century mathematics. Important

contributions to their study were made by the

French mathematicians Galois and Augustin

Cauchy, the British mathematician Arthur Cayley,

and the Norwegian mathematicians Niels Abel and

Sophus Lie. Quaternions were discovered by

British mathematician and astronomer William

Rowan Hamilton, who extended the arithmetic of

complex numbers to quaternions while complex

numbers are of the form a + bi, quaternions are of

the form a + bi + cj + dk.

REFERENCES

[1] Donald R. Hill, Islamic Science and

Engineering (Edinburgh University Press,

1994).

[2] Ziauddin Sardar, Jerry Ravetz, and Borin Van

Loon, Introducing Mathematics (Totem

Books, 1999).

[3] George Gheverghese Joseph, The Crest of the

Peacock: Non-European Roots of

Mathematics (Penguin Book, 2000).

[4] John J O'Connor and Edmund F Robertson,

History Topics: Algebra Index. In Mac Tutor

History of mathematics (University of St

Andrews, 2005)

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Some More Filtering Techniques For The Removal of Speckle Noise From Medical

Images

Gnanambal Ilango and B. Shanthi Gowri

Abstract - In Image Processing, removal of noise from the medical images is very challenging. Many filtering techniques have

been introduced to reduce noise in medical images. The speckle noise is commonly found in medical images. This paper

proposes twelve more filtering techniques for the removal of speckle noise from medical images. The quality of the enhanced

images is measured by the statistical quality measures: RMSE and PSNR.

Index Terms - Digital topological neighbourhood, Brain cancer, Speckle noise, RMSE, PSNR.

—————————— ——————————

1. INTRODUCTION

IGITAL image analysis plays a vital role in

medical imaging like magnetic resonance

imaging, ultra sound imaging, X-ray and

computed tomography. Departure of the ideal

signal is usually referred to as noise. The noises in

such digital images arise during image acquisition

and/or transmission. The data dropout noise is

generally called as speckle noise. Speckle noise is

a multiplicative noise that degrades the visual

evaluation in medical imaging. Speckle noise

suppression plays a very essential role in

diagnosis. The image acquisition devices need

despeckling techniques for medical imaging in

routine clinical practice. Image filtering is an

important technique used for the detection and

removal of noise from the digital images. Median

filter has been introduced by Turkey in 1970[10].

It is a non-linear filter used for smoothing the

images. Sudha et al recommends a novel

thresholding algorithm for denoising speckle noise

in ultrasound images with wavelets[9].

——————————————

Gnanambal Ilango is serving in the Department of Mathematics,

Government Arts College, Coimbatore, India. E-mail :

gnanamilango@yahoo.co.in

B. Shanthi Gowri is pursuing Ph.D. degree course in Mathematics,

Government Arts College, Coimbatore, India and is working at Sri

Krishna College of Engineering and Technology, Coimbatore, India. E-

mail : shanthigowrib@gmail.com

An improved adaptive median filtering method for

denoising impulse noise reduction was carried out

by Mamta Juneja et al [4]. Thangavel et al

showed that the M3-filter had performed better

than Mean, Median, Max, Min and various other

filters [8]. The hybrid max filter which

performs significantly better than many other

existing techniques for removal of speckle

noise was shown

Gnanambal et al [1].In this experimental work, we

took the Magnetic resonance image of the brain

cancer with speckle noise, and applied different

filtering techniques for noise removal. The quality

of the de-noised image is measured by the

statistical quality measures: RMSE and PSNR.

This work is organized as follows: In Section 2,

basic definitions are given. Section 3 discusses the

various filtering techniques for de-noising the

speckle noise in Medical images. Section 4 deals

with the proposed new filtering techniques for de-

noising the speckle noise in MRI. Section 5

discusses the comparison of results of various new

filtering techniques. Section 6 concludes the

experimental results.

2. BASIC DEFINITIONS

Definition 2.1 [7] A digital image is a function

𝑓: 𝑍 𝑋 𝑍 → [0,1,2, … . 𝑁 − 1] in which 𝑁 − 1 is

a positive whole number belonging to the natural

D

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interval [1,256]. The functional value of ‗f‘ at any

point p(x, y) is called the intensity or gray level of

the image at that point and it is denoted by f(p).

Definition 2.2 [7] Let X be an array of points

having positive integer coordinates (x, y), where

1 ≤ 𝑥 ≤ 𝑚, 1 ≤ 𝑦 ≤ 𝑛. The neighbourhood of a

point 𝑝 ∈ 𝑋 is a subset of X which contains an

open set containing p. It is denoted by N (p).

Definition 2.3 [6] The 4-neighbours of a point

p(x, y) are its four horizontal and vertical

neighbours 𝑥 ± 1, 𝑦 and 𝑥, 𝑦 ± 1 . The point

p(x, y) together with its 4-neighbours is called the

4 - neighbourhood of the point p. It is denoted by

𝑁4 𝑝 .

Definition 2.4 [6] The 8-neighbours of a point

p(x, y) consists of its 4-neighbours together with

its four diagonal neighbours (𝑥 + 1, 𝑦 ± 1)

and(𝑥 − 1, 𝑦 ± 1). The point p(x, y) together with

its 8-neighbours is called the 8-neighbourhood of

the point p. It is denoted by N8 (𝑝).

Definition 2.5 [1] The LT-neighbours of a point

p(x, y) consists of the neighbours 𝑥 − 1, 𝑦 + 1

and (𝑥 + 1, 𝑦 − 1). The point p(x, y) together

with its LT-neighbours is called the LT-

neighbourhood of the point p. It is denoted

by 𝐿3(𝑝).

Definition 2.6 The 6-neighbours of a point

p(x, y) consists of its 4-neighbours together

with the neighbours (𝑥 − 1, 𝑦 + 1) and

(𝑥 + 1, 𝑦 − 1). The point p(x, y) together with its

6-neighbours is called the 6-neighbourhood of the

point p. It is denoted by 𝑁6(𝑝).

Definition 2.7 The 12-neighbours of a point

p(x, y) consists of its 8-neighbours together with

the neighbours (𝑥 ± 2, 𝑦) and(𝑥, 𝑦 ± 2). The

point p(x, y) together with its 12-neighbours is

called the 12-neighbourhood of the point p. It is

denoted by 𝑁12(𝑝).

3. SOME EXISTING FILTERING

TECHNIQUES In this section, we provide the definitions of some

existing filters. The image processing function in

a spatial domain can be expressed as g(p) = 𝛾(f(p),

where 𝛾 is the transformation function, f(p) is the

pixel value (gray level value) of the point p(x,y)

of input image and g(p) is the pixel value of the

corresponding point of the processed image.

3.1 MEAN FILTER [3]

Mean filter is a simple linear filter, intuitive and

easy to implement method of smoothing images.

This filter reduces the amount of intensity

variation between one pixel and the next. It is

often used to reduce noise in images. In mean

filter, the pixel value of a point p is replaced by

the mean of pixel values of 8-neighbourhood of a

point ‗p‘. The operation of this filter can be

expressed as:

𝑔 𝑝 = 𝑚𝑒𝑎𝑛{ 𝑓 𝑝 , 𝑤𝑕𝑒𝑟𝑒 𝑝 𝜖 𝑁8 𝑝 }. 3.2 MEDIAN FILTER [1] The median filter plays an important role in image

processing and vision. It is useful for reducing salt

and pepper noise in an image. In median filter, the

pixel value of a point p is replaced by the median

of pixel values of 8-neighbourhood of a point ‘p‘.

The operation of this filter can be expressed as:

𝑔 𝑝 = 𝑚𝑒𝑑𝑖𝑎𝑛{𝑓 𝑝 , 𝑤𝑕𝑒𝑟𝑒 𝑝 𝜖 𝑁8 𝑝 }.

3.3 CENTER WEIGHTED MEAN FILTER [5]

Center weighted mean filter is a filter that gives

more weight to the center pixel. This weight

corresponds to the size of the kernel. If the kernel

is 3x3, then the total number of pixels that will be

converting to array for performing sorting will be

11. This filter is defined as

𝑔 𝑝 = 𝑚𝑒𝑎𝑛{ 2 ◊ 𝑓 𝑝𝑐 , 𝑓 𝑝 𝑤𝑕𝑒𝑟𝑒 𝑝 ∈ 𝑁8 𝑝 },

where ◊ represents replication operator and 𝑓(𝑝𝑐)

is gray level value of the center pixel.

3.4 CENTER WEIGHTED MEDIAN FILTER [5]

Center weighted median filter is a filter that gives

more weight to the center pixel. This weight

corresponds to the size of the kernel. If the kernel

is 3x3, then the total number of pixels that will be

converting to array for performing sorting will be

11. This filter is defined as

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𝑔 𝑝 = 𝑚𝑒𝑑𝑖𝑎𝑛{ 2 ◊ 𝑓 𝑝𝑐 , 𝑓 𝑝 𝑤𝑕𝑒𝑟𝑒 𝑝 ∈ 𝑁8 𝑝 }

where ◊ represents replication operator and 𝑓(𝑝𝑐)

is gray level value of the center pixel.

4. PROPOSED NEW FILTERING

TECHNIQUES

In this section, we will provide the definition of

proposed filtering techniques. These filters are not

yet applied by the researchers for the removal of

speckle noise in Magnetic resonance image.

4.1 N6 MEAN FILTER

In the N6 Mean filter, the pixel value of a point p

is replaced by the mean of the pixel values of

6-neighbourhood of a point ‗p‘. The operation of

this filter can be expressed as

𝑔 𝑝 = 𝑚𝑒𝑎𝑛{ 𝑓 𝑝 , 𝑤𝑕𝑒𝑟𝑒 𝑝 𝜖 𝑁6 𝑝 }.

4.2 N6 MEDIAN FILTER

In the N6 Median filter, the pixel value of a point

p is replaced by the median of the pixel values of

6-neighbourhood of a point ‗p‘. The operation of

this filter can be expressed as

𝑔 𝑝 = 𝑚𝑒𝑑𝑖𝑎𝑛{ 𝑓 𝑝 , 𝑤𝑕𝑒𝑟𝑒 𝑝 𝜖 𝑁6 𝑝 }.

4.3 N12 MEAN FILTER

In the N12 Mean filter, the pixel value of a point p

is replaced by the mean of the pixel values of 12-

neighbourhood of a point ‗p‘. The operation of

this filter can be expressed as

𝑔 𝑝 = 𝑚𝑒𝑎𝑛{ 𝑓 𝑝 , 𝑤𝑕𝑒𝑟𝑒 𝑝 𝜖 𝑁12 𝑝 }.

4.4 N12 MEDIAN FILTER

In the N12 Median filter, the pixel value of a point

p is replaced by the median of the pixel values of

12-neighbourhood of a point ‗p‘. The operation of

this filter can be expressed as

𝑔 𝑝 = 𝑚𝑒𝑑𝑖𝑎𝑛{ 𝑓 𝑝 , 𝑤𝑕𝑒𝑟𝑒 𝑝 𝜖 𝑁12 𝑝 }.

4.5 L3 MEAN FILTER

In the L3 Mean filter, the pixel value of a point p is

replaced by the mean of the pixel values of

LT-neighbourhood of a point ‗p‘. The operation of

this filter can be expressed as

𝑔 𝑝 = 𝑚𝑒𝑎𝑛{ 𝑓 𝑝 , 𝑤𝑕𝑒𝑟𝑒 𝑝 𝜖 L𝟑 𝑝 }.

4.6 L3 MEDIAN FILTER

In the L3 Median filter, the pixel value of a point p

is replaced by the median of the pixel values of

LT-neighbourhood of a point ‗p‘. The operation of

this filter can be expressed as

𝑔 𝑝 = 𝑚𝑒𝑑𝑖𝑎𝑛{ 𝑓 𝑝 , 𝑤𝑕𝑒𝑟𝑒 𝑝 𝜖 L𝟑 𝑝 }.

4.7 CENTER WEIGHTED N6 MEAN

(CWN6 MEAN) FILTER Center Weighted N6 Mean filter is a filter that

gives more weight to the center pixel of the

6- neighbourhood of a point p. This filter is

defined as

𝑔 𝑝 = 𝑚𝑒𝑎𝑛{ 2 ◊ 𝑓 𝑝𝑐 , 𝑓 𝑝 𝑤𝑕𝑒𝑟𝑒 𝑝 ∈ 𝑁6 𝑝 },

where ◊ represents replication operator and 𝑓(𝑝𝑐)

is gray level value of the center pixel. 4.8 CENTER WEIGHTED N6 MEDIAN

(CWN6 MEDIAN) FILTER

Center Weighted N6 Median filter is a filter

that gives more weight to the center pixel of

the 6- neighbourhood of a point p. This filter is

defined as

𝑔 𝑝 = 𝑚𝑒𝑑𝑖𝑎𝑛{ 2 ◊ 𝑓 𝑝𝑐 , 𝑓 𝑝 𝑤𝑕𝑒𝑟𝑒 𝑝 ∈ 𝑁6 𝑝 },

where ◊ represents replication operator and 𝑓(𝑝𝑐)

is gray level value of the center pixel.

4.9 CENTER WEIGHTED N12 MEAN

(CWN12 MEAN) FILTER

Center Weighted N12 Mean filter is a filter that

gives more weight to the center pixel of the 12-

neighbourhood of a point p. This filter is defined

as

𝑔 𝑝 = 𝑚𝑒𝑎𝑛{ 2 ◊ 𝑓 𝑝𝑐 , 𝑓 𝑝 𝑤𝑕𝑒𝑟𝑒 𝑝 ∈ 𝑁12 𝑝 },

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where ◊ represents replication operator and 𝑓(𝑝𝑐)

is gray level value of the center pixel. 4.10 CENTER WEIGHTED N12 MEDIAN

(CWN12 MEDIAN) FILTER

Center Weighted N12 Median filter is a filter

that gives more weight to the center pixel of

the 12- neighbourhood of a point p. This filter is

defined as

𝑔 𝑝 = 𝑚𝑒𝑑𝑖𝑎𝑛{ 2 ◊ 𝑓 𝑝𝑐 , 𝑓 𝑝 𝑤𝑕𝑒𝑟𝑒 𝑝 ∈ 𝑁12 𝑝 },

where ◊ represents replication operator and 𝑓(𝑝𝑐)

is gray level value of the center pixel. 4.11 CENTER WEIGHTED L3 MEAN

(CWL3 MEAN) FILTER

Center Weighted L3 Mean filter is a filter that

gives more weight to the center pixel of the

LT- neighbourhood of a point p. This filter is

defined as

𝑔 𝑝 = 𝑚𝑒𝑎𝑛{ 2 ◊ 𝑓 𝑝𝑐 , 𝑓 𝑝 𝑤𝑕𝑒𝑟𝑒 𝑝 ∈ 𝐿3 𝑝 },

where ◊ represents replication operator and 𝑓(𝑝𝑐)

is gray level value of the center pixel. 4.12 CENTER WEIGHTED L3 MEDIAN

(CWL3 MEDIAN) FILTER Center Weighted L3 Median filter is a filter that

gives more weight to the center pixel of the

LT neighbourhood of a point p. This filter is

defined as

𝑔 𝑝 = 𝑚𝑒𝑑𝑖𝑎𝑛{ 2 ◊ 𝑓 𝑝𝑐 , 𝑓 𝑝 𝑤𝑕𝑒𝑟𝑒 𝑝 ∈ 𝐿3 𝑝 },

and ◊ represents replication operator and 𝑓(𝑝𝑐) is

gray level value of the center pixel.

5. EXPERIMENTAL RESULT ANALYSIS

AND DISCUSSION The proposed filtering techniques have been

implemented using MATLAB7.0. The

performance of all the twelve new filtering

techniques are analyzed and discussed. We use

statistical tools Root Mean Square Error (RMSE)

and Peak Signal to Noise Ratio (PSNR) to

evaluate the enhancement of Magnetic resonance

images.

𝑅𝑀𝑆𝐸 = 𝑓 𝑖, 𝑗 − 𝑔 𝑖, 𝑗

2

𝑚𝑛

𝑃𝑆𝑁𝑅 = 20 log10 255

𝑅𝑀𝑆𝐸

Here 𝑓(𝑖, 𝑗) is pixel value of original MRI, g (i, j)

is the pixel value of enhanced MRI and m and n

are the total number of pixels in the horizontal and

vertical dimensions of the image. If the value of

RMSE is low and the value of the PSNR is high,

then the enhancement approach is better. The

MRI of brain cancer [11] with speckle noise and

filtered MRI of brain cancer image obtained by

the proposed filters are compared and analyzed.

Table-1 shows the RMSE and PSNR values of

different proposed filters for MRI brain cancer

image corrupted by speckle noise with variances

0.015, 0.025, 0.035, 0.045, 0.055

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Fig1. Shows the images obtained by applying the

proposed filtering techniques for the brain cancer

MRI with speckle noise of variance 0.015.

Chart1. Shows the analysis of RMSE and PSNR

values of brain cancer image corrupted by speckle

noise of variance 0.015.

6. CONCLUSION

In this work, we have introduced twelve new

filtering techniques for removal of speckle noise

from medical images. To demonstrate the

performance of the proposed techniques, the

experiments have been conducted on brain cancer

MRI. The performance of speckle noise removal

by proposed filtering techniques is measured

using quantitative performance measures such as

RMSE and PSNR. The experimental results

indicate that the CWN6 Mean, CWN6 Median,

CWN12 Mean, CWN12 Median, CWL3 Mean and

CWL3 Median are comparatively better than N6

Mean, N6 Median, N12 Mean, N12 Median, L3

Mean and L3 Median filters respectively and

CWL3 Median filter performs significantly better

than the other proposed filters as well as better

than the Hybrid max filter proposed by

Gnanambal et al[1].

REFERENCES

[1] Gnanambal Ilango and R. Marudhachalam,

New hybrid filtering techniques for removal

of speckle noise from ultrasound

medical images, Scientia Magna, vol. 7, No.

1, (2011), 38-53.

[2] Gnanambal Ilango and R. Marudhachalam,

New hybrid filtering techniques for removal

of Gaussian noise from medical images,

ARPN Journal of Engineering and Applied

Sciences,Vol 6, No. 2, (2011), 8-12.

[3] R. Gonzalez and R. Woods, Digital Image

Processing, Adison-Wesley, New

York,1992

[4] Mamta Juneja and Rajni Mohana, An

improved Adaptive Median Filtering

Method for Impulse Noise Detection,

International Journal of Recent Trends in

Engineering, No.1, (2009), 274-278.

[5] R. Marudhachalam and Gnanambal Ilango,

Center Weighted Hybrid Filtering

Techniques for denoising of medical

images, World Congress on Engineering and

Technology, (2011), 542-545.

[6] R. Klette and A. Rosenfeld, Digital

Geometry, Kaufmann, San Francisco, 2004.

[7] A. Rosenfeld, Digital Topology, American

Mathematical Monthly, 86 (1979), 621-630

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[8] K. Thangavel, R. Manavalan, and

I. Laurence Aroquiaraj, Removal of Speckle

noise from Ultrasound Medical Image based

on Special Filters: Comparative Study,

International Conference on Graphics,

Vision and Image Processing, (2009), 25-32.

[9] S. Sudha, G. R. Suresh and R. Sukanesh,

Speckle noise reduction in Ultrasound

Images by Wavelet thresholding based on

Weighted variance, International journal of

Comp. Theory and Engg, No.1, (2009), 7-12

[10] J. W. Turkey,Nonlinear (nonsuperposable)

methods for smoothing data, Proc. Congr.

Rec. EASCOM‘74, 1974, 673-681

[11] MRI brain cancer image taken from

www.google.com – images – brain – cancer

847 600 x 45

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Comprehensive Study on Various Types of Stegnographic Schemes and Possible

Steganalysis methods for various Cover Carrier like Image, Text, Audio and Video

H. Faheem Ahmed and U. Rizwan

Abstract - Steganalysis is a very challenging field because of the scarcity of knowledge about the specific characteristics of the cover

media (an image, an audio or video file) that can be exploited to hide information and detect the same. The approaches adopted for

steganalysis also sometimes depend on the underlying steganography algorithm(s) used.

Index Terms – Stegnography, Steganalysis.

—————————— ——————————

1. STEGNOGRAPHY

HE word steganography literally means

covered writing as derived from Greek

steganós (covered) and graptos (writing). Using

steganography, one can embed a secret message

inside a piece of unsuspicious information and

send it without anyone knowing of the existence

of the secret message. As the field of

steganography has progressed, people have

become increasingly interested in being able to

detect these hidden messages inside media. The

field of steganalysis has emerged to meet this

need. Steganalysis can be defined as, ―the art and

science of detecting steganography‖. The main

goals of steganalysis are to detect steganography

and to detect what method (or piece of software)

was used to hide the information.

Steganography and cryptography are closely

related. Cryptography scrambles messages so they

cannot be understood. Steganography on the

other hand, will hide the message so that they

cannot be seen.

Fig.1 Types of Steganography

——————————————

U. Rizwan is serving in the Department of Mathematics, Islamiah College, Vaniyambadi, India. E-mail: rizwanmaths@yaoo.com

H. Faheem Ahmed is pursuing Ph.D. degree in Computer Science,Islamiah College, Vaniyambadi, India.

Fig. 2. A block diagram of a generic steganographic

system

2. IMAGE STEGNOGRAPHIC TECHNIQUES

The various image steganographic techniques are:

(i) Substitution technique in Spatial Domain: In

this technique only the least significant bits of the

cover object is replaced without modifying the

complete cover object. It is a simplest method for

data hiding but it is very weak in resisting even

simple attacks such as compression, transforms,

etc.

1) Data Hiding by LSB: least-significant-bit

2) Data Hiding by MBPIS: The Multi Bit

Plane Image Steganography (MBPIS).

3) Data Hiding by MBNS: Multiple-Based

Notational System based on human vision

sensitivity (HVS).

T

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4) Data Hiding by QIM: Quantization index

modulation

5) Data Hiding by PVD: The pixel-value

differencing

6) Data Hiding by GLM : Gray level

modification

(ii)Transform domain technique: The various

transform domains techniques are Discrete Cosine

Transform (DCT), Discrete Wavelet Trans- form

(DWT) and Fast Fourier Transform (FFT) are

used to hide information in transform coefficients

of the cover images that makes much more robust

to attacks such as compression, filtering, etc.

(iii) Spread spectrum technique: The message is

spread over a wide frequency bandwidth than the

minimum required bandwidth to send the

information. The SNR in every frequency band is

small. Hence without destroying the cover image

it is very difficult to remove message completely.

(iv) Statistical technique: The cover is divided

into blocks and the message bits are hidden in

each block. The information is encoded by

changing various numerical properties of cover

image. The cover blocks remain unchanged if

message block is zero.

(v) Distortion technique: Information is stored by

signal distortion. The encoder adds sequence of

changes to the cover and the decoder checks for

the various differences between the original cover

and the distorted cover to recover the secret

message. Some common Image Steganography

Technique in Spatial and Transform Domain

[146] has been discussed below.

3. IMAGE STEGNANALYSIS

Image steganography algorithms are more often

based on an embedding mechanism called Least

Significant Bit (LSB) embedding. Each pixel in an

image is represented as a 24-bitmap value,

composed of 3 bytes representing the R, G and B

values for the three primary colors Red, Green and

Blue respectively. A higher RGB value for a pixel

implies larger intensity. For instance, a pixel p

represented as FF FF FF is composed of all of

these three primary colors at their maximum

intensity and hence the color represented by this

pixel is white. LSB embedding exploits the fact

that changing the least significant bit of each of

the three bytes of a pixel would produce only a

minor change in the intensity of the color

represented by the pixel and this change is not

perceptible to the human eye. For example,

changing the color values of pixel p to FE FE FE

would make the color darker by a factor of 1/256.

Steganography algorithms based on LSB

embedding differ on the pattern of modification a

modification of randomly chosen pixels or

modification restricted to pixels located in certain

areas of the image. Images can be represented in

different formats, the three more commonly used

formats are: GIF (Graphics Interchange Format),

BMP (Bit Map) and JPEG (Joint Photographic

Exchange Group). Each of these image formats

behaves differently when a message is embedded

in it. Accordingly, there exist different image

steganalysis algorithms for each of these three

image formats.

4. PALETTE IMAGE STEGNANALYSIS

Palette image steganalysis is primarily used for

GIF images. The GIF format supports up to 8 bits

per pixel and the color of the pixel is referenced

from a palette table of up to 256 distinct colors

mapped to the 24-bit RGB color space. LSB

embedding of a GIF image changes the 24-bit

RGB value of a pixel and this could bring about a

change in the palette color (among the 256 distinct

colors) of the pixel. The strength of the

steganographic algorithm lies in reducing the

probability of a change in the palette color of the

pixel and in minimizing the visible distortion that

embedding of the secret image can potentially

introduce. The steganalysis of a GIF stego image

is conducted by performing a statistical analysis of

the palette table vis-à-vis the image and the

detection is made when there is an appreciable

increase in entropy (a measure of the variation in

the palette colors).The change in entropy is

maximal when the embedded message is of

maximum length.

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5. RAW IMAGE STEGNANALYSIS

The Raw image steganalysis technique is

primarily used for BMP images that are

characterized by a lossless LSB plane.LSB

embedding on such images causes the flipping of

the two grayscale values. The embedding of the

hidden message is more likely to result in

averaging the frequency of occurrence of the

pixels with the two gray-scale values. For

example, if a raw image has 20 pixels with one

gray-scale value and 40 pixels with the other gray-

scale value, then after LSB embedding, the count

of the pixels with each of the two gray-scale

values is expected to be around 30. It is based on

the assumption that the message length should be

comparable to the pixel count in the cover image

(for longer messages) or the location of the hidden

message should be known (for smaller messages).

6. JPEG IMAGE STEGNANALYSIS

JPEG is a popular cover image format used in

steganography. Two well-known Steganography

algorithms for hiding secret messages in JPEG

images are: the F5 algorithm and Outguess

algorithm. The F5 algorithm uses matrix

embedding to embed bits in the DCT(Discrete

Cosine Transform) coefficients in order to

minimize the number of changes to a message.

7. GENERIC IMAGE STEGNANALYSIS

ALGORITHMS

The generic steganalysis algorithms, usually

referred to as Universal or Blind Steganalysis

algorithms, work well on all known and unknown

steganography algorithms. These steganalysis

techniques exploit the changes in certain innate

features of the cover images when a message is

embedded. The focus is on to identify the

prominent features of an image that are monotonic

and changes statistically as a result of message

embedding. The generic steganalysis algorithms

are developed to precisely and maximally

distinguish these changes. The accuracy of the

prediction heavily depends on the choice of the

right features, which should not vary across

images of different varieties.

8. EVALUATION OF STEGNANALYSIS

TOOLS

In order to evaluate the steganalysis tools, it is

essential that the whole process is forensically

sound to ensure the validity of the findings.

Therefore, the following are the steps that will be

followed throughout the process: 1.Obtain the

steganographic and steganalysis tools 2. Verify

the tools (to ensure the tools is doing what it

claims) 3. Obtain cover images, and generate

MD5 hashes 4. Apply steganalysis on cover

images, and generate MD5hashes 5. Generate

steganographic images, and generate MD5hashes

6. Apply steganalysis on the steganographic

image, and generate MD5 hashes In each of the

steps where the cover images or the

steganographic images are involved, MD5 hashes

have been used to verify whether the image has

changed in any sense.

9. CONCLUSION

In this paper, we have analyzed the steganalysis

algorithms available for Image Steganography. In

summary, each carrier media has its own special

attributes and reacts differently when a message is

embedded in it. Therefore, the steganalysis

algorithms have also been developed in a manner

specific to the target stego file and the algorithms

developed for one cover media are generally not

effective for a different media. This paper would

cater well to providing an overview of the

steganalysis algorithms available for images.

Image steganalysis algorithms can be classified

into two broad categories: Specific and Generic.

The Specific steganalysis algorithms are based on

the format of the digital image (e.g. GIF, BMP

and JPEG formats) and depend on the underlying

steganography algorithm used. The Generic image

steganalysis algorithms work for any underlying

steganography algorithm, but require more

complex computational and higher-order

statistical analysis. The audio steganalysis

algorithms exploit the variations in the

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characteristic features of the audio signal as a

result of message embedding. Audio steganalysis

algorithms that detect the discontinuities in phase

(as a result of phase coding), variations in the

amplitude (as a result of Echo hiding) and the

changes in the perceptual and non-perceptual

audio quality metrics as a result of message

embedding have been proposed. The video

steganalysis algorithms that utilize the temporal

redundancies at the frame level and inter-frame

level have been observed to be more effective

than algorithms based on spatial redundancies.

Nevertheless, video steganalysis algorithms that

simultaneously exploit both the temporal and

spatial redundancies have also been proposed and

shown to be effective. In summary, each carrier

media has its own special attributes and reacts

differently when a message is embedded in it.

Therefore, the steganalysis algorithms have also

been developed in a manner specific to the target

stego file and the algorithms developed for one

cover media are generally not effective for a

different media. This paper would cater well to

providing an overview of the steganalysis

algorithms available for the three commonly used

domains of steganography.

Most of the techniques that can be used on

images, can also be applied on audio files.

Compressing an audio file with lossy compression

will result in loss of the hidden message as it will

change the whole structure of a file. Also, several

lossy compression schemes use the limits of the

human ear to their advantage by removing all

frequencies that cannot be heard. This will also

remove any frequencies that are used by a

steganographic system which hides information in

that part of the spectrum.

Another possible way of removing steganograms

is lowering the bitrate of the audio file. In that

case, there will be less available space to store

hidden data and therefore, at least parts of it will

get lost.

For video, once more again, the same methods as

for images and audio files can be applied to

remove hidden information. To defeat the use of

signals or gestures however, human insight is still

necessary, as computer systems are not yet

capable of detecting this with a reasonable rate of

success.

REFERENCES

[1] Ahmed Ibrahim, Steganalysis in Computer

Forensics, Security Research Centre

Conferences, Australian Digital Forensics

Conference, Edith Cowan University Year

2007.

[2] H. Faheem Ahmed and U. Rizwan, An

Alternative Technique in Data Embedding,

Advanced Materials in Physics, 233-242,

2012.

[3] Greg Goth, Steganalysis Gets Past the Hype,

IEEE, Distributed Systems Online 1541-4922

© 2005 Published by the IEEE Computer

Society Vol. 6, No. 4; April 2005.

[4] Guillermito, Steganography: A few tools to

discover hidden data. Retrieved September

29, 2007,

http://www.guillermito2.net/stegano/tools/ind

ex.html.

[5] J. Kelley, Terrorist instructions hidden online.

Retrieved September 14, 2007,

http://www.usatoday.com /tech/news/2001-

02-05-binladen-side.html

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k-Stage Fuzzy Transportation Problem Based On Interval Valued Fuzzy Numbers

S. Elizabeth and L. Sujatha

Abstract: The transportation problem is a typical problem where a product is to be transported from ‘m’ sources to ‘n’ destinations. In general, crisp transportation problems are solved with the assumption that the supplies, demands and cost parameters are specified in a precise way, but this is not possible in real life situations due to uncertainty in judgments, lack of evidence etc., which gives rise to fuzzy environment and hence fuzzy decision making method is needed here. In this paper, a procedure is proposed for k-stage fuzzy transportation problem. Illustrative example is also included to demonstrate the proposed approach. Index Terms - Fuzzy transportation problem, Interval valued fuzzy numbers, Transportation Network, Decision Maker.

—————————— ——————————

1. INTRODUCTION

HE basic transportation problem was

originally developed by Hitchcock [4]. It can

be modeled as a standard linear programming

problem. An Initial basic feasible solution (IBFS)

for the transportation problem can be obtained by

using the North-west corner rule, Row minima,

Column minima, Matrix minima or the Vogel‘s

approximation method (VAM). The Modified

distribution method (MDM) is useful for finding

the optimal solution for the transportation

problem. It is not often possible to get relevant

precise data for supplies, demands and cost

parameters hence fuzzy numbers namely interval

valued fuzzy numbers, triangular fuzzy numbers,

trapezoidal fuzzy numbers may represent this

data. Zimmerman [12] showed that solutions

obtained by fuzzy linear programming are always

efficient. Subsequently, Zimmermann‘s fuzzy

linear programming has developed into several

fuzzy optimization methods for solving the

transportation problems. Chanas et.al. [1]

presented a fuzzy linear programming model for

solving transportation problems with crisp cost

coefficients and fuzzy supply and demand values.

Chanas and Kuchta [2] proposed the concept of

the optimal solution for the transportation problem

————————————————

S. Elizabeth is serving in the Department of Mathematics, Auxilium

College, Vellore, India. E-mail: elizafma@gmail.com

L. Sujatha is pursuing Ph.D. degree in Mathematics, Auxilium College, Vellore, India. E-mail: sujathajayasankar@yahoo.co.in

with fuzzy coefficients expressed as fuzzy

numbers, and developed an algorithm for

obtaining the optimal solution. Saad and Abbas

[10] discussed the solution algorithm for solving

the transportation problem in fuzzy environment.

Liu and Kao [7] described a method for solving

Fuzzy transportation problem (FTP) based on

extension principle. Lin [6] introduced a genetic

algorithm to solve a transportation problem with

fuzzy objective functions. Dinagar and Palanivel

[11] investigated FTP, with the aid of trapezoidal

fuzzy numbers. Fuzzy modified distribution

method is proposed to find the optimal solution in

terms of fuzzy method is proposed to find the

optimal solution in terms of fuzzy numbers.

Pandian and Natarajan [9] proposed a new

algorithm namely, fuzzy zero point method for

finding a fuzzy optimal solution for a FTP, where

the transportation cost, supply and demand are

represented by trapezoidal fuzzy numbers. Thus

numerous papers have been published in fuzzy

transportation problem.

The rest of the paper is organized as follows: In

section 2, basic definitions on Interval valued

fuzzy numbers are reviewed and some new

definitions are coined for the same. In section 3, a

procedure is proposed for k-stage transportation

problem in fuzzy environment. An example is also

illustrated for the proposed approach. Section 4,

concludes the paper.

T

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2. PRE-REQUISITES

Definition 1. [5] When interval is defined on real

number . This interval is said to be a subset of

. The interval valued fuzzy number is denoted

by 313131

,,],,[ aaaaaaA and its membership

function is given as follows:

3

31

1

,0

,1

,0

)(

ax

axa

ax

xA

Fig. 1. Interval valued fuzzy number ],[31

aaA

If a1 = a3, this interval indicates a point

],[ 11aaA =

1a

Definition 2. Operations on Interval Valued fuzzy numbers [5] .

Let ],[31

aaA and ],[31

bbB , 31

,31

,, bbaa be

two interval valued fuzzy numbers then :

Addition Operation )(

33113131 ,,)(,)( bababbaaBA

Subtraction Operation )(

13313131 ,,)(,)( bababbaaBA

Multiplication Operation )( :

3313311133133111

3131

....,....

,)(,)(

babababababababa

bbaaBA

The following definitions are introduced in this

paper.

Definition 3. The Centroid Measure for α- cut interval number is given in [3]. The same

procedure is followed here for Interval Valued

fuzzy number. Let ],[31

aaA be an interval valued

fuzzy number.

Then Centroid2

.1

.131

3

1

3

1)()(

aa

dx

xdx

a

a

a

aACA

Definition 4. Let ],[31

aaA and ],[31

bbB be

two interval valued fuzzy numbers. If )()( BCAC

then A is called fuzzy maximum then B and if

)()( BCAC then A is called fuzzy minimum then

B.

Definition 5. Fuzzy zero for interval valued fuzzy number is denoted by IFNO and it is taken as

[0,0] (or) ],[ kk where k and δ are positive

scalars.

Definition 6. Let ],[31

aaA and ],[31

bbB ,

31

,31

,, bbaa be two interval valued fuzzy

numbers. The multiplication Operation on Interval valued fuzzy numbers based on Centroid measure is defined as

Case (i) If 0)(,0)( BCAC and if

0)(,0)( BCAC ,

)(

2),(

2)( 31

331

1bb

abb

aBA

Case (ii) If 0)(,0)( BCAC and if

0)(,0)( BCAC ,

)(

2),(

2)( 3131

13bb

abb

aBA

2.1 Degeneracy in Fuzzy Transportation Problem

Fuzzy Initial basic feasible solution is said to be

degenerate, if the number of allotted cells is less

than m + n – 1, where m is the number of fuzzy

origins and n is the number of fuzzy destinations.

In such cases, we allocate small quantity namely

],[ satisfying the following conditions to one or

more empty cells so that the total number of

allotted cells is equal to m + n – 1 independent

cells.

(i) 0],[

],[],[)(]k,-k[],[)(],[

and ],[],[)(]0,0[]0,0[)(],[ (ii)

kkkk

where )]k,[-k(C and)],[( C

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(iii) ],[)(],[)3()1(

ijijxx

ijijijijijijxxxxxx ,],[],[

)3()1()3()1(

(iv)

(1) (3) (1) (3)

(1) (3)

[ , ]( )[ , ] [ , ]

[ , ] ,

ij ij ij ij

ij ij ij ij

x x x x

x x x x

where ],[ )3()1(

ijijxxxij is a quantity transported

from ith

fuzzy origin to jth

fuzzy destination. Thus

we obtain the fuzzy initial basic feasible solution

as non-degenerate solution.

3. PROCEDURE FOR k-STAGE FUZZY

TRANSPORTATION PROBLEM

The k-stage fuzzy transportation problem is

framed with an assumption that the destinations

are unable to receive the quantity in excess due to

storage capacity. In such situation after consuming

part of the initial product, they are prepared to

receive the excess quantity in the successive

stages whereas the cost parameters remain the

same in all the k-stages. In k-stages the

transportation of the product from sources to the

destination is done in parallel. Nagoor Gani and

Abdul Razak [8] presented two stage cost

minimizing fuzzy transportation problem in which

supplies and demands are taken as trapezoidal

fuzzy numbers with crisp cost. In this section a

procedure is presented for k-stage fuzzy

transportation problem where the supplies,

demands and cost parameters are taken as Interval

Valued Fuzzy Numbers. A parametric approach is

followed to obtain a fuzzy solution which aim to

minimize the sum of the fuzzy transportation cost

in k-stages.

The linear programming model representing the k-

stage fuzzy transportation problem is given by

Minimize Z =

m n

ijiji j

xc CC1 1

))( ( ( I )

subject to the constraints

)(

1)( i

n

ij aCCj

x

for i= 1,2,…..,m (Row sum)

)(

1)( j

m

ij bCCi

x

for j= 1,2,…..,n (Column sum)

jiC ijx ,0( )

The following steps are followed for k-stage fuzzy

transportation problem.

Step 1. Construct a fuzzy transportation network

with m fuzzy origins (rows) and n fuzzy

destinations (columns). Let fuzzy supply ai = [ai(1)

,

ai(3)

] be the quantity of commodity available at

fuzzy origin i. Let fuzzy demand bj = [bj(1)

, bj(3)

]

be the quantity of commodity needed at fuzzy

destination j and let fuzzy cost cij = [cij(1)

, cij(3)

] be

the cost for transporting one unit of the product

from ith

fuzzy origin to jth

fuzzy destination, where

i = 1,2,….,m (Number of rows), j = 1,2,…..,n

(Number of columns).

Table 3.1. Fuzzy Transportation Problem

FO\

FD

FD1 FD2 ... FDn ai

FO1 [c11(1)

,

c11(3)

]

[c12(1)

,

c12(3)

]

... [c1n(1)

,

c1n(3)

]

a1=[a1(1)

,

a1(3)

]

FO2 [c21(1)

,

c 21(3)

]

[c22(1)

,

c22(3)

]

... [c2n(1)

,c2n(3)

] a2=[a2(1)

,

a2(3)

]

FOm [cm1(1)

,

cm1(3)

]

[cm2(1)

,

cm2(3)

]

... [cmn(1)

,cmn(3)

] am=[am(1)

,

am(3)

]

bj b1=[b1(1)

,

b1(3)

]

b2=[b2(1)

,

b2(3)

]

... bn=[bn(1)

,bn(3)

]

The given fuzzy transportation problem is said to

be balanced if

n

j

m

ijjii

bbaa11

)3()1()3()1(,,

and unbalanced if

n

j

m

ijjii

bbaa11

)3()1()3()1(,, .

The unbalanced fuzzy transportation problem is

converted into balanced fuzzy transportation

problem as follows:

If

n

j

m

ijjii

bbaa11

)3()1()3()1(,,

then the dummy column with cost parameters

[0,0] or ],[ kk , where k and δ are positive

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scalars, is introduced which will provide for the

excess demand.

If

n

j

m

ijjii

bbaa11

)3()1()3()1(,,

then the dummy row with cost parameters [0,0] or

],[ kk where k and δ are positive scalars, is

introduced which will provide for the excess

supply. First we have to check whether the given

Fuzzy Transportation Problem is balanced. If so,

divide it into k-stages as follows:

1- stage

Minimize 11

11 c)(z

1

n

jijij

m

i

xcC

subject to the constraints

n

ijj

x1

1 Ai1 for mi ..,,.........2,1 (Row sum)

m

iijx

11

B1j for nj ..,,.........2,1 (Column sum)

jix ,0ij1

2- stage

Minimize 22

12 c)(z

1

n

jijij

m

i

xcC

subject to the constraints

n

ijj

x1

2 Ai2 for mi ..,,.........2,1 (Row sum)

m

iijx

12

B2j for nj ..,,.........2,1 (Column sum)

jix ,0ij2

k- stage

Minimize k

1k c)(z

1

n

jijkij

m

i

xcC

subject to the constraints

n

ijkj

x1

Aik for mi ..,,.........2,1 (Row sum)

m

iijkx

1

Bkj for nj ..,,.........2,1 (Column sum)

jix ,0ijk

(1) (3)

1

(1) (3)

2

,

... ([ , ])

( )2

ij ij ij

ij ij

ij

ij ijk C

C X i jij

x x x x x

x xx

where ],[ )3()1(

ijijxxxij is a quantity transported

from ith

fuzzy origin to jth

fuzzy destination.

Here the optimal parameters are obtained using

centroid measure as follows:

If ai = [ai(1)

, ai(3)

] , bj = [bj(1)

, bj(3)

] , cij = [cij(1)

,

cij(3)

] then

number integer an is and value

decimal a is if

integer an is if

*

*

2)(

)3()1(

iA

iAia iAiA

iAiA

Cii

aa

Similarly,

number integer an is and value

decimal a is if

integer an is if

*

*

2)(

)3()1(

ijC

ijCijCijC

ijCijC

ijij

ij

cccC

mtoiiAoriAiaC 1,)(*

and ntojjBorjBjbC 1,)(*

Now *iAoriA are divided into k- parts namely

iki

AAAi .....,,,21 (or) **

2

*

1,......,,

ikiiAAA ,

where

mmkmmkAAAAAAAA .......,.......,.....

21111211

(or) ***

2

*

1

*

1

*

1

*

12

*

11......,......,......

mmkmmkAAAAAAAA

Similar procedure follows for *

jj BB or . Also

n

jkj

m

iik

n

jj

m

ii BABA

1111

11 ,....., .

If not, that is if

n

jkj

m

iik

n

jj

m

ii BABA

1111

11 ,....., ,

then dummy row with cost parameter zero is

introduced which will provide for the excess

supply.

number integer an is and value

decimal a is if

integer an is if

*

*

2)(

)3()1(

jB

jBjbC jBjB

jBjB

jjbb

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If

n

jkj

m

iik

n

jj

m

ii BABA

1111

11 ,.....,

then dummy column with cost parameter zero is

introduced which will provide for the excess

demand. But note, most probably the divided

values are chosen such that the FTP in all the k-

stages are balanced.

Step 2. Calculate Fuzzy initial basic feasible

solution using VAM and the fuzzy optimal

solution using MDM for k-stages.

Step 3. Calculate minimum fuzzy transportation

cost kzzz ,......,, 21 from fuzzy optimal solution in

k-stages and it is taken as kccc ,......,, 21

respectively.

Step 4. Declare 1 2 ...... kc c c c as the fuzzy

optimal value of the fuzzy transportation problem

(I) .

4. ILLUSTRATIVE EXAMPLE

Construct a fuzzy transportation network where

nodes denote the fuzzy origin (FO) and fuzzy

destinations (FD), edges denote the fuzzy cost.

Fig. 2. Fuzzy Transportation Network

Table 3.2. Fuzzy Initial Table

FO\

FD

FD1 FD2 FD3 FD4 ai

FO1 [-2,6] [-2,6] [-2,6] [-1,3] [-3,9]

FO2 [4,16] [3,13] [2,8] [1,7] [-3,17]

FO3 [3,11] [2,10] [2,10] [3,13] [-20,30]

bj [-11,19] [-3,9] [-13,21] [1,7] [-26,56]

Table 3.3. Centroid Measure for

FO \

FD

FD1 FD2 FD3 FD4 C (ai)

FO1 2 2 2 1 A1=3

FO2 10 8 5 4 A2=7

FO3 7 6 6 8 A3=5

C (bj) B1=4 B2=3 B3=4 B4=4 15

Here

4

1

3

1

)3()1()3()1(,,

jijjii

bbaa =[-26,56]

15)(b C)(a C

3 4

ji1 1

i j

.

Hence it is a balanced FTP. Let k = 3

1-Stage.

Table 3.4. Fuzzy IBFS using VAM

FO\

FD

FD1 FD2 FD3 FD4 Ai1

FO1 2

1

2

-

2

-

1

-

A11=1

FO2 10

-

8

-

5

1

4

1

A21=2

FO3 7

-

6

1

6

-

8

-

A31=1

B1j B11=1 B12=1 B13=1 B14=1 4

Table 3.5. Fuzzy Optimal Solution

FO\

FD

FD1 FD2 FD3 FD4 FO\ FD

FO1 2

1

2

2

-

1

FO1

FO2 10

-

8

-

5

1

4

1

FO2

FO3 7

-

6

1

6

-

8

-

FO3

Minimize 1z C11 x111+ C12 x121+ C14 x141+

C23x231+ C24x241+ C32 x321

= 1614151212

= 2+5+4+6+3 117317 c

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2-Stage.

Table 3.6. Fuzzy IBFS using VAM

FO\

FD

FD1 FD2 FD3 FD4 Ai2

FO1 2

1

2

-

2

-

1

-

A12=1

FO2 10

-

8

-

5

1

4

1

A22=2

FO3 7

-

6

1

6

-

8

-

A32=1

B2j B21=1 B22=1 B23=1 B24=1 4

Table 3.7. Fuzzy Optimal Solution

FO\

FD

FD1 FD2 FD3 FD4 FO\ FD

FO1 2

1

2

2

-

1

FO1

FO2 10

-

8

-

5

1

4

1

FO2

FO3 7

-

6

1

6

-

8

-

FO3

Minimize 2z C11 x112+ C12 x122+ C14 x142

+ C23 x232+ C24 x242+ C32 x322

= 2 1 2 1 5 1 4 1 6 1

= 2+5+4+6+3 217 3 17 c

3- Stage.

Table 3.8. Fuzzy IBFS using VAM

FO\ FD FD1 FD2 FD3 FD4 Ai3

FO1 2

1

2

-

2

-

1

-

A13=1

FO2 10

-

8

-

5

1

4

2

A23=3

FO3 7

1

6

1

6

1

8

-

A33=3

B3j B31=2 B32=1 B33= 2 B34=2 7

Here Fuzzy IBFS given in table 3.8 is an optimal

solution

Minimize 3z C11 x113+ C23x233+ C24x243

+ C31 x313+ C32 x323+ C33x333

3

2 1 5 1 4 2 7 1 6 1 6 1

34 c

Therefore the optimal value of the objective

function of the fuzzy minimum cost transportation

problem is

Minimize Z = 321 ccc

= 17 + 17 + 34

= 68 = c (1)

For the sake of verification, calculating

ijkijij xxx ......21 = Xij

from all k-stages for i=1,2,….,m; j=1,2,…..,n

which is given in the below table 3.9

Table 3.9. Table for verification

FO\

FD

FD1 FD2 FD3 FD4 C (ai)

FO1 2

3

2

2

2

-

1

2

A1=3

FO2 10

-

8

-

5

3

4

4

A2=7

FO3 7

1

6

3

6

1

8

-

A3=5

C (bj) B1= 4 B2= 3 B3= 4 B4=

4

15

Minimize Z = C11 X11+ C12 X12 + C14 X14

+ C23 X23 + C24 X24+ C31X31

+ C32 X32 + C33 X33

2 3 2 2 1 2 5 3 4 4

7 1 6 3 6 1

68

Results and Discussions.

The fuzzy optimal solution using fuzzy VAM and

fuzzy optimality test for Table 3.2 is given in

below Table 3.10

The general linear programming model

representing the fuzzy transportation problem for

table 3.1 is given by

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Minimize Z =

subject to the constraints,

],,[ )3()1(

1

)3()1([] ii

n

ijij aaxxj

for mi ..,,.........2,1

],,[ )3()1(

1

)3()1([] jj

m

ijij bxx bi

for nj ..,,.........2,1

Table 3. 10. Fuzzy Optimal Solution

FO\

FD

FD1 FD2 FD3 FD4 ai

FO1 [-2,6]

[-3,9]

[-2,6]

-

[-2,6]

-

[-1,3]

-

[-3,9]

FO2 [4,16]

-

[3,13]

-

[2,8]

[-4,10]

[1,7]

[1,7]

[-3,17]

FO3 [3,11]

[-8,10]

[2,10]

[-3,9]

[2,10]

[-9,11]

[3,13]

-

[-20,30]

bj

[-11,19]

[-3,9]

[-13,21]

[1,7]

[-26,56]

The general linear programming model

representing the fuzzy transportation problem for

table 3.1 is given by

Minimize Z =

subject to the constraints,

],,[ )3()1(

1

)3()1([] ii

n

ijij aaxxj

for mi ..,,.........2,1

],,[ )3()1(

1

)3()1([] jj

m

ijij bxx bi

for nj ..,,.........2,1

For the table 3.10, the fuzzy minimum

transportation cost is

Minimize Z

2, 6 ( ) 3,9 ( ) 2,8 ( ) 4,10 ( ) 1, 7 ( ) 1, 7

( ) 3,11 ( ) 8,10 ( ) 2,10 ( ) 3,9 ( ) 2,10 ( ) 9,11

=

10,2)(30,6)(11,3)(28,4)(24,6)(18,6

=[15,121]

068 ZC (2)

Hence as given in Step 1, most probably the

divided values are chosen such that the FTP in all

the k – stages are balanced so that the solution

obtained for k-stage FTP (equation (1)) coincides

with the solution obtained using fuzzy VAM and

fuzzy optimality test (equation (2)). If not, they

are approximately equal.

5. CONCLUSION

In Today‘s highly competitive market, the

organization or agency should plan their work in

such a way that they should create and deliver the

products to customers in a cost – effective

manner. Fuzzy Transportation models play a vital

role in solving these types of problems. In this

paper, we developed a method for finding the

fuzzy optimal solution for k-stage fuzzy

transportation problem. We conclude that the

proposed method is very easy to understand and

provide a powerful framework to meet the

challenging problems in real life situations.

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INTERNATIONAL JOURNAL OF SCIENTIFIC & ENGINEERING RESEARCH, VOLUME 3, ISSUE 11, NOVEMBER -2012 162 ISSN 2229-5518

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