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International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 7, Issue 12, December 2017)

80

Vendor-Buyer Integrated Optimal Policy in an Intuitionistic

Fuzzy Environment

Mahuya Deb

1

, Prabjot Kaur

2

1Department of Commerce, Gauhati University, Assam 781014, India

2Department of Mathematics, Birla Insitute of Technology, Mesra, Jharkhand -835215, India Abstract: In this paper a vendor buyer integrated

inventory model without shortages is taken with the primary aim to reduce the total cost for the mutual benefit of the buyer and the vendor. The importance of studying this model is heightened since quite a few number of organisations hope to settle down with JIT policy which ensures purchase and manufacture of items available at a time when it is required for consumption. For the successful implementation of JIT it is presumed that a new bond of support or mutual aid exist between the vendor and buyer. An example is taken to explain the proposed model.

Keywords: vendor-buyer, fuzzy, intuitionistic fuzzy

I. INTRODUCTION

Inventory represents a substantial investment capital for many firms. The way inventory accumulates and depletes over time have long been recognized as a major factor that contributes to the fluctuations in business activity. Among the various costs that a manufacturing sector bears, the inventory cost seems to be the dominant one and therefore it is essential to formulate strategies which could ensure the smooth transition of products and services to the customers at a minimum cost. The pertinent queries of when to order and in what quantities remain critical for both the manufacturing and the service industries. Inventories assume the role of current assets in an organisation and therefore a slight reduction in inventories lowers assets relative to liabilities that affect the current ratio, generally taken as a measure of liquidity. It is also evident that a small change in the inventory level may disturb revenues and operating expenses which may lead to an alarming situation in the operating profits of an organisation and in turn affect its return on investment. Against this scenario business units nowadays are allocating a considerable proportion of their resources to keep a check on their inventory level else it may affect their revenue status.

Normally, problems of inventory are treated separately for the vendor and the buyer. But combining the duo works wonder and this cooperation evident in this approach leads to the successful operation of supply chain administration through minimising the joint inventory cost. JIT as a system of inventory is used by many industries these days to reduce the inventory cost. However to support such a system it is essential to ensure that the production system available with the vendor is a reliability one.

Therefore the model discussed in this chapter includes the JIT (just in time) concept that can significantly lead to a reduction in the joint inventory cost .

An extensive study of literature reveals that the economic order quantity (EOQ) models and economic production quantity (EPQ) models are treated independent of each other from the perspective of the buyer or the vendor. Quite often the result which is optimal for one player becomes non-optimal to the other player. A number of unique entities constitute the supply chain who plays a major role in the transformation of the finished product and in the process make those commodities available to the final consumers at a least possible cost. However the prominent players (eg. vendors, retailers, distributors, etc.) who belong to different corporate entities in the supply chain may be more interested in achieving their self interest ie in terms of minimising their cost than that of the chain as a whole. But with the growing impact and research on management of supply chain , firms apprehend that inventories that pile up across the supply chain can be more proficiently handled with better synchronization among its members thereby leading to reduction in lead time and cost without compromising on customer service. Therefore with proper coordination between the vendor and buyer inventory cost and reaction time of both the vendor and buyer system can be efficiently reduced and is therefore considered a prerequisite for successful supply chain management which ensures joint inventory cost reduction and response time of the vendor buyer system .

Banerjee [1] in his paper concentrated on a combined economic lot size model for a single vendor where the production rate is assumed to be finite. He was one of the first few researchers who dealt with the integrated vendor buyer problem. His assumption was that whatever a vendor manufactures is at a finite rate and each buyer shipment is produced as a separate batch. Goyal [2]

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International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 7, Issue 12, December 2017)

81

Yang et al [7] brought modification to Goyal’s model wherein he tried to bring cost reduction through his approach. They also considered the JIT implementation in a vendor buyer system with deterministic demand wherein the demand is assumed to rely on the displayed items. Nagoorgani [9] presented a paper on an integrated inventory model wherein the parameters work on a fuzzy environment. He assumed all costs in the model as triangular fuzzy numbers.

Taking a novel approach this paper proposes to construct the vendor buyer inventory model taking the parameters as triangular intuitionistic fuzzy numbers. The signed distance function defined by [4] is used for defuzzification. The organisation of the paper is as follows: The second section deals on the preliminaries of fuzzy and Intuitionistic fuzzy sets. Section 3 deals in establishment of the Inventory Model in the crisp and in the Intuitionistic fuzzy environment. Section 4 deals with a numerical to illustrate the result. The ending Section 5 provides the conclusion for this study.

II. PRELIMINARIES ON FUZZY AND INTUITIONISTIC FUZZY SET

Definition 2.1: Fuzzy Set

A fuzzy set is defined by

]}

1

,

0

[

)

(

,

/

)

(

,

{(

x

x

x

A

x

A

A

A

In the pair

(

x

,

A

(

x

))

the first element x belong to the classical set A, the second element

A

(x

)

belong to the interval [0, 1] is called membership function or grade of membership. The membership function is also a

degree of compatibility or a degree of truth of x in

A

.

Definition 2.2: Fuzzy Numbers

The notion of fuzzy numbers was introduced by

Dubois and Prade [ 52]

A fuzzy subset

A

of the real line R with membership function

A

(

x

)

:

R

[

0

,

1

]

is called a fuzzy number if

i.

A

is normal, (i.e.) there exist an element

x

0 such that

1

)

(

x

0

A

ii .

A

is fuzzy convex,

ie

A

[

x

1

(

1

)

x

2

]

A

(

x

1

)

A

(

x

2

),

x

1

,

x

2

R

,

[

0

,

1

]

iii .

A

(

x

)

is upper continuous, and iv. supp

A

is bounded, where supp

}

0

)

(

:

{

x

R

x

A

A

Definition 2.3: Triangular Fuzzy Number:

A fuzzy number

A

of the universe of discourse U may be characterized by a triangular distribution function

parameterized by a triplet

(

a

1

,

a

2

,

a

3

)

.

Fig 2.1 Membership function of TFN

The membership function of the fuzzy number

A

is

defined as

3 3 2

2 3

2 1

1 2

1

1

,

0

,

,

,

0

)

(

a

x

a

x

a

a

x

x

a

a

x

a

a

a

a

x

a

x

x

A

Definition2.4: Trapezoidal Fuzzy Number:

The fuzzy set

A

(

a

1

,

a

2

,

a

3

,

a

4

)

where

4 3 2

1

a

a

a

a

and defined on R, is called the

trapezoidal fuzzy number if membership function of

A

is given by

Otherwise

a

x

a

for

a

a

x

a

a

x

a

for

a

x

a

for

a

a

a

x

x

x

A

0

1

)

(

3 4

3 4

4

3 2

2 1

1 2

1

[image:2.595.338.524.554.764.2]

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International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 7, Issue 12, December 2017)

82

Definition 2.5: Arithmetic operations of fuzzy numbers

Let

A

and

B

be two fuzzy numbers (TFN)

parameterized by the triplet say

(

a

1

,

a

2

,

a

3

)

and

)

,

,

(

b

1

b

2

b

3 respectively

Suppose A=(a1,a2,a3) and B=(b1,b2,b3) are two

triangular fuzzy numbers

Suppose

A

(

a

1

,

a

2

,

a

3

,

a

4

)

and

)

,

,

,

(

b

1

b

2

b

3

b

4

B

are two trapezoidal fuzzy numbers, then

(i) The addition of

A

and

B

is

)

,

,

,

(

a

1

b

1

a

2

b

2

a

3

b

3

a

4

b

4

B

A

where a1,a2,a3,a4,b1,b2,b3,b4 are real numbers

(ii) The product of

A

and

B

is

)

,

,

,

(

a

1

b

1

a

2

b

2

a

3

b

3

a

4

b

4

B

A

(iii)

B

(

b

4

,

b

3

,

b

2

,

b

1

)

then the subtraction of

B

from

A

is

)

,

,

,

(

a

1

b

4

a

2

b

3

a

3

b

2

a

4

b

1

B

A

(iv)

) , , , (

) 1 , 1 , 1 , 1 ( 1

1 4

2 3

3 2

4 1

1 2 3 4 1

b a

b a

b a

b a

B A then

b b b b B B

 

 

(v)

0 ) , , , (

0 ) , , , ( ,

1 2 3 4

4 3 2 1

 

 

    

      

a a a a

a a a a A then R Let

Defuzzification Using Signed Distance Method

Fuzziness helps in evaluating rules but the final output of a fuzzy system needs to be a crisp number. Therefore defuzzification is a process of transforming fuzzy values to crisp values. It is such a inverse transformation which maps the output from the fuzzy domain back into the crisp domain.

Defuzzification methods have been widely studied for some years and were applied to fuzzy systems. The major idea behind these methods was to obtain a typical value from a given set according to some specified characters. Defuzzification method provides a correspondence from the set of all fuzzy sets into the set of all real numbers.

Signed Distance Method

The signed distance has some similar properties to the properties induced by the signed distance in real numbers. For any

a

and

0

R

, define the signed

distance from a to 0 as

D

0

(

A

,

0

)

a

If a > 0, the distance from a to 0 is a =

D

0

(

a

,

0

)

If a < 0, the distance from a to 0 is -a =

D

0

(

a

,

0

)

From the definition of signed distance, the signed distance of two end

points of the α cut

B

(

)

[

B

L

(

),

B

U

(

)]

of

B

to the origin 0 is

ly

respective

B

B

D

and

B

B

D

U U

L L

)

(

)

0

),

(

(

)

(

)

0

),

(

(

0 0

.

Their average

2

)

(

)

(

U

L

B

B

is taken as the

signed distance of

[

B

L

(

),

B

U

(

)]

to0.

That is signed distance of interval

[

B

L

(

),

B

U

(

)]

to 0 is defined as

)]

(

)

(

[

2

1

)]

0

),

(

(

(

)

0

),

(

(

[(

2

1

]

0

),

(

),

(

[

0 0

U L

U L

U L

o

B

B

B

D

B

D

B

B

D

The signed distance of

B

to 0 is definedas

1

0

0

[

(

)

(

)]

2

1

)

0

,

(

B

B

B

d

D

L U

For the triangular numbe

A

(

a

1

,

a

2

,

a

3

,

a

4

)

the α

cut of

A

is

A

(

)

[

A

L

(

),

A

U

(

)],

[

0

,

1

]

where

A

L

(

)

a

1

(

a

2

a

1

)

And

A

U

(

)

a

3

(

a

3

a

2

)

The signed distance of

A

to 0 is

}

2

[

4

1

)

0

,

(

1 2 3

0

A

a

a

a

D

For the trapezoidal number

B

(

b

1

,

b

2

,

b

3

,

b

4

)

the α

cut of

B

is

]

1

,

0

[

)],

(

),

(

[

)

(

B

L

B

U

B

where

)

(

)

(

b

1

b

2

b

1

B

L

and

)

(

)

(

b

4

b

4

b

3

B

U

The signed distance of

B

to 0 is

]

[

4

1

)

0

,

(

1 2 3 4

0

B

b

b

b

b

(4)

International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 7, Issue 12, December 2017)

83

Definition 2.6: Intuitionistic Fuzzy Set: Let a set X be

fixed. An IFS

A

~

in X is an object having the form

}

:

)

(

),

(

,

{

~

~

~

x

x

x

X

x

A

A

A

, where the

]

1

,

0

[

:

)

(

~

x

X

A

and ~

(

x

)

:

X

[

0

,

1

]

A

define

the degree of membership and degree of non membership

respectively of the elememt

x

X

to the set

A

~

, which is a subset of the set X , for every element of

1

)

(

)

(

0

,

~

~

X

x

x

x

A A

Definition 2.7:: Intuitionistic Fuzzy Number

Intuitionistic Fuzzy Number: An IFN

A

~

is defined as follows:

(i) an intuitionistic fuzzy subset of the real line

ii) normal, i.e., there is any

x

0

R

such

that ~

(

x

)

1

(

so

~

(

x

)

0

)

A

A

(iii) a convex set for the membership function

)

(

~

x

A

i.e.

]

1

,

0

[

,

,

))

(

),

(

min(

)

)

1

(

(

1 2 ~ 1 ~ 2 1 2

~

A

x

x

A

x

A

x

x

x

R

(iv) a concave set for the non-membership function

)

(

~

x

A

ie

]

1

,

0

[

,

,

))

(

),

(

max(

)

)

1

(

(

1 2 ~ 1 ~ 2 1 2

~

A

x

x

A

x

A

x

x

x

R

Definition2.8::A triangular intuitionistic fuzzy number

)

,

;

,

,

(

~

~ ~ 3 2

1

a

a

w

a

u

a

a

A

is a subset of intuitionistic fuzzy set set on the set of real number R whose membership and non membership are defined as follows:

                                               otherwise a x a a a a x a x a a a x a x A and otherwise a x a a a x a a x a a a a x x A 1 2 3 , 2 3 2 2 1 , 2 1 2 ) ( ~ , 0 3 2 , 2 3 ) 3 ( 2 1 , 1 2 ) 1 ( ) ( ~  

Definition 2.9 Arithmetic Operations of Triangular Intuitionistic Fuzzy Number

If

A

~

(

a

1

,

a

2

,

a

3

)(

a

1

,

a

2

,

a

3

)

and

)

,

,

)(

,

,

(

~

3 2 1 3 2

1

b

b

b

b

b

b

B

are two TIFNs, then

.Addition of two TIFN

) 3 3 , 2 2 , 1 1 ( ) 3 3 , 2 2 , 1 1 ( ~ ~ b a b a b a b a b a b a B A            

is also TIFN

Subtraction of two TIFN

)

,

,

(

)

,

,

(

~

~

1 3 2 2 3 1 1 3 2 2 3 1

b

a

b

a

b

a

b

a

b

a

b

a

B

A

is also TIFN

Multiplication of two TIFN

)

,

,

)(

,

,

(

~

~

3 3 2 2 1 1 3 3 2 2 1

1

b

a

b

a

b

a

b

a

b

a

b

a

B

A

is also a TIFN

If TIFN

A

~

(

a

1

,

a

2

,

a

3

)(

a

1

,

a

2

,

a

3

)

and y=ka (with

k>0) then

~

y

k

A

~

Division of two TIFN is

)

,

,

)(

,

,

(

~

~

1 3 2 2 3 1 1 3 2 2 3 1

b

a

b

a

b

a

b

a

b

a

b

a

B

A

is also a TIFN

Definition 2.10:: Trapezoidal Intuitionistic Fuzzy Number: An Intuitionistic fuzzy number

)

,

,

,

)(

,

,

,

(

~

4 3 2 1 4 3 2 1

a

a

a

a

a

a

a

a

A

is said to be a

trapezoidal intuitionistic fuzzy number if its membership function and non-membership function are given by

                                                     4 4 3 2 1 1 1 4 3 3 4 3 3 2 0 2 1 1 2 2 ) ( ~ 0 4 3 3 4 4 3 2 1 1 1 2 1 ) ( ~ a a a a a a where otherwise a x fora a a a x a x a for a x a for a a x a x A otherwise a x a for a a x a a x a for a x a for a a a x x A  

Definition2. 11: Arithmetic operations of TrFIN

Various arithmetic operations are carried out on intuitionistic fuzzy sets.

If

A

~

(

a

1

,

a

2

,

a

3

,

a

4

;

a

1

,

a

2

,

a

3

,

a

4

)

and

)

,

,

,

;

,

,

,

(

~

4 3 2 1 4 3 2 1

b

b

b

b

b

b

b

b

B

are two trapezoidal

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International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 7, Issue 12, December 2017)

84

1. Addition of two TrIFN is

) , , , ; , , , ( ~ ~

4 4 3 3 2 2 1 1 4 4 3 3 2 2 1 1

            

B a b a b a b a b a b a b a b a b

A

2. Subtraction of two TrIFN is

) , , , ; , , , ( ~ ~

1 4 2 3 3 2 4 1 1 4 2 3 3 2 4 1

            

B a b a b a b a b a b a b a b a b A

3. Multiplication of two TrIFN

)

4

4 , 3 3 , 2 2 , 1 1 (

) 4 4 , 3 3 , 2 2 , 1 1 ( ~ ˆ

 

  

b a b a b a b a

b a b a b a b a B A

If

A

~

(

a

1

,

a

2

,

a

3

,

a

4

;

a

1

,

a

2

,

a

3

,

a

4

)

and y=ka (with

k>0)

Then

)

,

,

,

;

,

,

,

(

~

~

4 3 2 1 4 3 2 1

k

A

ka

ka

ka

ka

ka

ka

ka

ka

Y

is a TrIFN

4. Division of TrIFN

If

(

,

,

,

;

,

,

,

)

~

4 3 2 1 4 3 2 1

a

a

a

a

a

a

a

a

A

and

)

,

,

,

;

,

,

,

(

~

4 3 2 1 4 3 2 1

b

b

b

b

b

b

b

b

B

are two TrIFN then

)

,

,

,

(

)

,

,

,

(

~

~

1 4

2 3

3 2

4 1

1 4

2 3

3 2

4 1

b

a

b

a

b

a

b

a

b

a

b

a

b

a

b

a

B

A

III. FORMULATION OF THE MODEL IN CRISP ENVIRONMENT

The growing focus on supply chain management has made it imperative for firms to more efficiently manage their inventories for better coordination between the buyers and sellers. Nagoorgani [8] considered such a scenario and formulated a mathematical model wherein the various cost parameters of the model are considered in the fuzzy environment. Here the researcher extended the model to incorporate intuitionistic fuzzy numbers in the cost parameters as they include non membership along with membership in the representation of fuzziness.

3.1 Assumptions

a) In the model is considered an integrated system with one-vendor and one-buyer .

b) The rate at which production is carried on is assumed to be fixed.

c) The rate of meeting demand does not vary. d) The vendor and buyer is aware of each other . e) There is no scope for shortage.

3.2 Notations

Q – Amount of inventory available to the buyer in a single delivery

n - Number of deliveries from the vendor to the buyer per vendor’s replenishment interval

Sv –Cost incurred per set up of the vendor

Ab –Cost incurred for ordering per unit by the buyer

Cv- Cost incurred for producing per unit by the vendor

Cb- Cost incurred for purchasing per unit by the buyer

i – Carrying cost incurred per year P - Production rate per year , P>D D – Demand available per year

TC (Q, n) - Total integrated cost of the vendor and the buyer when they collaborate

Q

~

- Intuitionistic fuzzy Amount of inventory available to the buyer in a single delivery

n

~

- Intuitionistic fuzzy number of deliveries from the vendor to the buyer

v

S

~

- Intuitionistic fuzzy Cost incurred per set up of the vendor

b

A

~

- Intuitionistic fuzzy Cost incurred for ordering per unit by the buyer

v

C

~

- Intuitionistic fuzzy Cost incurred for producing per unit by the vendor

b

C

~

- Intuitionistic fuzzy Cost incurred for purchasing per unit by the buyer

i

~

- Intuitionistic fuzzy inventory carrying cost per year

)

,

(

~

n

Q

C

T

- Intuitionistic fuzzy integrated total cost of the buyer and the vendor

3.3 The Framework for Calculation

The interval for refilling used by the buyer is

D Q

The interval for refilling used by the vendor’s is

D Q n

The normal level of inventory for the buyer isQ/2

The normal level of inventory II of the vendor is

(6)

International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 7, Issue 12, December 2017)

85

) 1 ( ]

) 1 )( 1 [( 2

)] 1 1 ( 2

) 1 ( 2 [

! !

). 1 -1 )( 1 ( ... ) 1 -1 ( 2 ) 1 -1 ( 2

' -'

2 2

2 2

2

P D P D n

Q

P D Q n n P nQ nQ

D

D nQ

P D n P

D Q P D Q P nQ

inventory ent

replenishm s

vendor

ventory weightedin time

s Vendor II

 

 

  

 

[image:6.595.53.288.138.423.2]

Figure 2.3: Inventory level of the buyer and time-weighted inventory of the vendor

The implication in (1) results when the buyer receives the good till it is enough to make up the batch size, thereby bringing a reduction in the cost of the inventory during the production period. This is different from the model which Goyal [3] had considered due to a different modelling strategy wherein buyer received goods as soon as the production period is terminated.

3.4 Intuitionistic Fuzzy Mathematical Model

Intuitionistic fuzzy Cost incurred for ordering per unit

by the buyer=

Q

A

D

~

b

Intuitionistic fuzzy cost incurred for carrying

inventory of the Buyer =

2

~

~

b

C

Q

i

Then, intuitionistic fuzzy annual cost of the buyer =

Q

A

D

~

b

+

2

~

~

b

C

Q

i

Intuitionistic fuzzy set up cost incurred by the

vendor =

nQ

S

D

v

~

Intuitionistic fuzzy cost incurred for carrying inventory of the vendor

=

[(

1

)(

1

)

]

2

~

~

P

D

P

D

n

C

Q

i

v

Then, Vendor’s intuitionistic fuzzy annual cost =

nQ

S

D

~

v

+

[(

1

)(

1

)

]

2

~

~

P

D

P

D

n

C

Q

i

v

From equation (1), the intuitionistc fuzzy integrated total cost incurred for the vendor and the buyer per year is

)

,

(

~

n

Q

C

T

=

Q

A

D

~

b

+

2

~

~

b

C

Q

i

+

nQ

S

D

~

v

+

]

)

1

)(

1

[(

2

~

~

P

D

P

D

n

C

Q

i

v

(2)

In the right-side of (2), the first two terms are the intuitionistic fuzzy ordering cost and carrying cost incurred for the buyer respectively, and the end terms are the intuitionistic fuzzy setup cost and intuitionistic fuzzy carrying cost incurred for the vendor respectively.

Now taking the first derivative of (2) with regard to Q and equating it to zero and thereby solving the equation, the economic lot size of the buyer is given by,

After substituting the value of (3) into (2), the optimal integrated total cost is given by

(7)

International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 7, Issue 12, December 2017)

86

Since the value of n is positive integer, the optimal value of n, denoted by n*, is

)

6

(

)

1

(

)

(

)

-1

(

~

~

)]

2

-1

(

~

~

[

~

* *

*

n

TC

n

whenTC

P

D

C

A

P

D

C

C

S

n

v b

v b v

Or

)

7

(

)

1

(

)

(

)

1

(

~

~

)]

2

-1

(

~

~

[

~

* * *

*

n

TC

n

whenTC

P

D

C

A

P

D

C

C

S

n

v b

v b v

Where [x] is used t the o denote the largest integer less than or equal to x , and TC(n) represents that the TC is a function of n

From equation (3) the economic order quantity (ie when n=1) is

The result shown in equation (8) is the same as that obtained by Banerjee’s model (1986) [3]

From equation (3) it is seen that when the rate of production is very large , the economic lot size becomes

IV. NUMERICAL EXPOSURE

The example illustrated here is taken from Goyal [3] and Yang et. al [7].

Taking the cost parameters as triangular intuitionistic fuzzy numbers

Intuitionistic fuzzy s cost incurred per setup of the vendor

S

~

v= (350, 400, 450)(300,400,500)

Intuitionistic fuzzy cost incurred per order of the buyer

A

~

b=( 20,25,30)(15,25,35)

Intuitionistic fuzzy production cost incurred for the vendor

C

~

v = (19, 20,21)( 17,20,23)

Intuitionistic fuzzy purchase costincurred by the buyer

b

C

~

= (24, 25, 26)(22,25,28)

Intuitionistic fuzzy Annual inventory carrying cost

~

i

= (0.1, 0.2, 0.3)(0.0, 0.2,0.5)

Production rate per year P=3200 units Demand available per year D= 1000units

Therefore the number of deliveries is given by

Using the signed distance for defuzzification we have

Integrated Total Cost with intuitionistic fuzzy parameters

Intuitionistic Fuzzy ordering cost incurred for the

buyer=

Q

A

D

~

b

= 197.36

Intuitionistic Fuzzy Annual Carrying Cost of the

buyer =

2

~

~

b

C

Q

i

(8)

International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 7, Issue 12, December 2017)

87

Therefore the buyer’s intuitionistic Fuzzy Annual Cost

=197.36+250.8

=448.16

Intuitionistic fuzzy annual set up cost of the vendor

=

nQ

S

D

~

v

= 710.22

Intuitionistic fuzzy cost the vendor for carrying the inventory

=

2

[(

1

)(

1

)

]

~

~

P

D

P

D

n

C

Q

i

v

= 504.824

Therefore the vendor’s total cost = 710.22+504.824

= 1215.044

[image:8.595.42.288.486.579.2]

Now the total cost of the vendor and the buyer taken in collaboration = 1663.204

Table 1.1

Sensitivity Analysis with TIFN

Q TC(Q,n)

n=1 386.3231 2458.283

n=2 226.274 2139.882

n=3 163.245 2048.86

n=4 129.200 2021.28`

Table 1.2

Comparing the given problem in fuzzy as used by Nagoorgani [9]

Q(fuzz y )

Q(intuitionisti c fuzzy)

TC(fuzzy) TC(IF)

n= 1

386 Units

357.77 Units Rs2465.8 2

Rs 2375.82 2 n=

2

234 Units

216.93 Units Rs. 2155.83

Rs 2074.39 8 n=

3

172 Units

159.26 Units Rs. 2067.49

Rs 1988.31 2 n=

4

137 Units

127.38 Units Rs. 2043.35

Rs 1962.54

Now Using Trapezoidal Intuitionistic Fuzzy Number as:

Intuitionistic fuzzy cost for set up of the Vendor

S

~

v

= (350, 400, 450,475)(300,400,450,500)

Intuitionistic fuzzy cost for ordering on part of the Vendor

A

~

b=( 20,25,30,45)(15,25,30,50)

Intuitionistic fuzzy cost for production for the Vendor

v

C

~

= ( 19, 20,21, 22)( 18,20,21,23)

Intuitionistic fuzzy cost for purchase used by the buyer

b

C

~

= (24, 25, 26, 27)(23,25,26,28)

Annual inventory carrying cost

~

i

= (0.1 , 0.2, 0.3,0.4)(0.0, 0.2,0.3,0.5)

Production rate per year P=3200 units Demand available per year D= 1000units

Using Signed Distance Method for defuzzification the various parameters of the model are obtained as

n=4.2

Q=115.7 =116(approx)

Intuitionistic Fuzzy Ordering Cost incurred by the buyer

for a year=

Q

A

D

~

b

= 258.620

Intuitionistic Fuzzy Carrying Cost incurred by the buyer

for a year =

2

~

~

b

C

Q

i

= 369.75

Therefore the intuitionistic Fuzzy Annual Cost of the buyer

=628.37

Intuitionistic fuzzy cost incurred for set up of the

vendor =

nQ

S

D

~

v

= 853.08

Intuitionistic fuzzy carrying cost of the vendor=

]

)

1

)(

1

[(

2

~

~

P

D

P

D

n

C

Q

i

v

[image:8.595.40.287.596.753.2]
(9)

International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 7, Issue 12, December 2017)

88

Therefore the vendor’s total cost =1599.92

[image:9.595.42.288.188.418.2]

Therefore the integrated total cost of the vendor and the buyer= 2228.299

Table 1.3 Sensitivity with TrIFN

Q TC

n=1 373Units Rs 2691

n=2 219.1176 Units Rs 2420

n=3 158.54 Units Rs 2261

n=4 125 Units Rs 2237.76

Table 1.4

Comparison of total cost between TIFN and TrIFN

n Q Buyer

s total cost

Vendor s total cost

TC

Triangula r

4. 5

117.4 1

543.14 3

1475.29 2

2018.43 5

Trapezoid al

4. 2

116 628.37 1599.92 2228.29 9

V. CONCLUSION

This paper is an attempt to consider a system where there is a collaboration between the vendor and buyer to determine the economic lot size policy using triangular and trapezoidal intuitionistic fuzzy numbers. The results obtained showed that cost can be considerably reduced with triangular intuitionistic fuzzy numbers when compared with fuzzy model as used by Nagoorgani [9]. The example is also worked for trapezoidal intuitionistic fuzzy numbers with sensitivity analysis.

Calculation here involves finding the order size of the buyer, lot size of the vendor and total cost of the vendor and the buyer who work in collaboration . A simple procedure is explained which helped in obtaining an approximate solution to the proposed model. The solution explained through this example presents a dynamic business situation which sets to explore the uncertainty associated with vagueness in the various parameters of the total expected cost.

REFERENCES

[1] Banerjee, A joint economic lot size model for purchaser and vendor, Decis. Sci., 17 (1986), pp 292– 311

[2] Goyal, S.K (1988), A joint economic lot size model for purchaser and vendor: A comment, Decis. Sci., 19 pp 236–241

[3] Goyal ,S.K and Gupta ,Y.P, (1989) Integrated inventory model: the buyer vendor coordination, Euro. J.Oper.Res.41pp 261-269. [4] Chang H.C,Ho,C.H, Ouyang L.Yand Su C.H(2009),The optimal

pricing and ordering policy for an integrated inventory model when trade credit linked to order quantity, Applied Mathematical Modeling,Vol.33, No.7,pp.2978-2991

[5] Ha, D and Kim, S.L(1997), Implementation of JIT purchasing: an integrated approach, Prod. Plan. Control, 8 pp 152–157

[6] Thomas, D.J. and .Griffin, P.M Coordinated supply chain management, European J.Oper. Res., 94 (1996), 1–15

[7] Yang, P.C., Wee H.M. and Yang, H.J. Global optimal policy for vendor-buyer integrated inventory system within just in time environment, Journal of Glob. Optim.37 (2007), 505-511.

[8] Nagoorgani,A and Ponnalagu.K A new approach on solving intuitionistic fuzzy linear programming problem, Applied Mathematical Sciences, 6(70) (2012), 3467-3474

[9] Nagoor Gani . A and Sabarinathan..G, Optimal Policy for Vendor-Buyer Integrated InventoryModel within Just In Time in Fuzzy Environment, Intern. J. Fuzzy Mathematical Archive, Vol. 2, 2013, 26-35 ISSN: 2320 –3242 (P), 2320 –3250 (online) [10] Atanassov, K (1986) ‘Intuitionistic fuzzy sets’, Fuzzy Sets and

Figure

Fig 2.2 Membership function of TrFN
Figure  2.3: Inventory level of the buyer and  time-weighted inventory of the vendor
Table 1.1 Sensitivity Analysis with TIFN
Table 1.3 Sensitivity with TrIFN

References

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