View of FUZZY INVENTORY MODEL FOR DETERIORATING ITEMS WITH FLUCTUATING DEMAND AND SHORTAGE UNDER FULLY BACKLOGGED AND USING INVENTORY PARAMETERS AS PFN

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Vol.04,Special Issue 04, ²^nd Conference (ICIRSTM) April 2019, Available Online: www.ajeee.co.in/index.php/AJEEE

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FUZZY INVENTORY MODEL FOR DETERIORATING ITEMS WITH FLUCTUATING DEMAND AND SHORTAGE UNDER FULLY BACKLOGGED AND USING INVENTORY

PARAMETERS AS PFN Priyanka Surana Department of Mathematics

Modi Institute of management & Technology, Kota

Abstract:- In this paper, we first consider a crisp inventory model with constant deteriorating items with constant demand where shortages are allowed with fully back logged condition. Thereafter we developed the corresponding fuzzy inventory model for fuzzy deteriorating items with fuzzy demand rate under full backlogging. The average total inventory cost in fuzzy sense is derived. All inventory parameters including deterioration rate are fuzzified as the pentagonal fuzzy numbers. The fuzzy model is defuzzified by using the Graded mean representation method and Signed distance method. The solution for minimizing the fuzzy cost function has been derived. The results obtained by this method are explained by the use of numerical data.

Keywords:- Inventory, Deterioration, Fuzzy model, Shortages, Pentagonal Fuzzy Number [PFN], Signed distance method.

1. INTRODUCTION

You have to crawl before you can fly, so we’re going to ease into the Fuzzy World Tour with some very elementary fuzzy definitions. The real world is up and down, constantly moving and changing, and full of surprises. In other word, Fuzzy. Fuzzy techniques let you successfully handle real world situations. In every day content most of the problems involve imprecise concept. To handle the imprecise concept, the conventional method of set theory and numbers are insufficient and need to be extended to some other concepts. Fuzzy concept is one of the concepts for this purpose.

1.1 Fuzzy Concept:-

A fuzzy concept is a concept of which the boundaries of application can vary considerably according to context or conditions, instead of being fixed once and for all[1]. This means the concept is vague in some way, lacking a fixed, precise meaning, without however being unclear or meaningless altogether [2]. It has a definite meaning, which can become more precise only through further elaboration and specification, including a closer definition of the context in which the concept is used. A fuzzy concept is understood by scientists as a concept.

1.2 Example -What’s The Process Of Parallel Parking A Car?

First you line up your car next to the one in front of your space. Then you angle the car back into the space, turning the steering wheel slightly to adjust your angle as you get closer to the curb. Now turn the wheel to back up straight and—nothing. Your rear tire’s wedged against the curb. OK. Go forward slowly, steering toward the curb until the rear tire straightens out. Fine except, you’re too far from the curb. Drive back and forth again, uses shallower angles. Now straight forward. Good, but a little too closes to the car ahead. Back up a few inches. Think! Oops, that’s the bumper of the car in back. Forward just a few inches. Stop! Perfect!! Congratulations.

You’ve just parallel-parked your car. And you’ve just performed a series of fuzzy operations. Not fuzzy in the sense of being confused. But fuzzy in the real-world sense, like

“going forward slowly” or “a bit hungry” or “partly cloudy”—the distinctions that people use in decision-making all the time, but that computers and other advanced technology haven’t been able to handle. What kind of problems? For one, waiting for an elevator at lunch hour.

How do you program elevators so that they pick up the most people in the least amount of time? Or how do you program elevators to minimize the waiting time for the most people?

Suppose you’re operating an automated subway system.

How do you program a train to start up and slow down at stations so smoothly that the passengers hardly notice? For that matter, how can you program a brake system on an automobile so that it works efficiently, taking road and tire conditions into account? Which is “to an extent applicable” in a situation, and it therefore implies gradations of meaning.

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The best known example of a fuzzy concept around the world is an amber traffic light, and indeed fuzzy concepts are nowadays widely used in traffic control systems.

 Fuzzy number:- A fuzzy number is a convex, normalized fuzzy set ̃ whose membership function is at least piecewise continuous and has the functional value ( ) at precisely one element. So fuzzy number (fuzzy set) represents a real number interval whose boundary is fuzzy.

 Crisp and fuzzy realtion:- One of the most fundamental notions in pure and applied sciences is the concept of a relation. Science has been described as the discovery of relations between objects, states and events. Fuzzy relations generalize the concept of relations in the same manner as fuzzy sets generalize the fundamental idea of sets.

 Crisp Relation:-Crisp relation is defined on the Cartesian product of two universal sets determined as

*( )| + (1) The crisp relation R is defined by its membership function

( ) { ( )

( ) } (2)

Here “1” implies complete truth degree for the pair to be in relation and “0” implies no relation. When the sets are finite the relation is represented by a matrix R called a relation matrix If a crisp relation R represents that of from sets A to B, for B, its membership function ( ) is,

( ) { ( )

( ) } (3) This membership function maps to set * +

* + (4)

We know that the relation R is considered as a set. Recalling the previous fuzzy concept, we can define ambiguous relation.

 Fuzzy relation:-The fuzzy relation equation is an equation of the form A · R = B, where A and B are fuzzy sets, R is a fuzzy relation, and A · R stands for the composition of A with R. A relation is a mathematical description of a situation where certain elements of sets are related to one another in some way. Fuzzy relations are significant concepts in fuzzy theory and have been widely used in many fields such as fuzzy clustering, fuzzy control and uncertainty reasoning. They also play an important role in fuzzy diagnosis and fuzzy modeling. When fuzzy relations are used in practice, how to estimate and compare them is a significant problem.

Uncertainty measurements of fuzzy relations have been done by some researchers.

Similarity measurement of uncertainty was introduced by Yager who also discussed its application.

2. DEFINITIONS AND PRELIMINARIES

In order to treat fuzzy inventory model by using graded mean representation method to defuzzify, we need the following definitions

Definition: A pentagonal fuzzy number (PFN) [9] ̃ ( ) is represented with membership function ̃ as:

̃( )

{

( )

}

( )

The -cut of ̃ ( ) is ( ) , ( ) ( )-

Where ( ) ( ) ( ), ( ) ( ) ( ) and

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( ) ( ) ( ), ( ) ( ) ( ) So

( )

( ) ( ) ( ) ( ) ( ) ( )

( )

( ) ( ) ( ) ( ) ( ) ( )

Fig (1): Graphical representation of Pentagonal Fuzzy Number (PFN)

Definition : If ̃ ( ) is a pentagonal fuzzy number then the signed distance method of ̃is defined as

( ̃ ̃) ∫ (, ( ) ( ) - ̃) ( ) ( )

2.1 Conditions On Pentagonal Fuzzy Number [PFN]:-

A Pentagonal Fuzzy Number ( ̃) should satisfy the following conditions;

1. ̃( ) is a continuous function in the interval [0,1].

2. ̃( ) is strictly increasing and continuous function on [a, b] and [b, c].

3. ̃( ) is strictly decreasing and continuous function on [c, d] and [d, e].

3. NOTATIONS AND ASSUMPTIONS:

The mathematical model in this paper is developed on the basis of the following assumptions and notations.

Notations:

D (t) is the demand rate at any time t per unit time.

1. A is the ordering cost per order.

2. is the deterioration rate, 3. T is the length of the Cycle.

4. Q is the ordering Quantity per unit.

5. h is the holding cost per unit per unit time 6. S is the shortage Cost per unit time.

7. C is the unit Cost per unit time.

8. ( ) is the total inventory cost per unit time.

9. ̃ is the fuzzy demand.

10. ̃ is the fuzzy deterioration rate.

11. ̃ is the fuzzy holding cost per unit per unit time.

12. ̃ is the fuzzy shortage Cost per unit time.

13. ̃ is the fuzzy unit Cost per unit time.

14. ̃( ) is the total fuzzy inventory cost per unit time.

15. ( ) is the defuzzify value of ̃( ) by applying Signed Distance Method.

0 0.2 0.4 0.6 0.8 1 1.2

a b c d e

y→

x→

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Assumptions:

1. Demand ( ) ( ) is assumed to be an increasing function of time i.e. where and are positive constants and

2. Replenishment is instantaneous and lead time is zero.

3. Shortages are allowed and fully backlogged.

4. FUZZY MODEL

Due to uncertainly in the environment it is not easy to define all the parameters precisely, accordingly we assume some of these parameters viz. ̃ ̃ ̃ ̃ ̃ ̃ may change within some limits.

Let

̃ ( ) ̃ ( )

̃ ( ), ̃ ( ) ̃ ( ), ̃ ( )

be the pentagonal fuzzy numbers.

Total cost of the system per unit time in fuzzy sense is given by

̃( ) * ̃ ̃ , ̃

- ̃ ̃ ̃ , ̃

- ̃ , ̃ ̃ ̃ ̃ ̃ - ̃ , ̃

( ) ̃ ̃

( )-+ ( )

We defuzzify the fuzzy total cost ̃( ) by graded mean representation method and signed distance methods.

By Graded Mean Representation Method, Total Cost is given by.

( )

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

Where

( ) * , - , - , -

, ( ) ( )-+

( ) * , - , - , -

, ( ) ( )-+

( ) * , - , - , -

, ( ) ( )-+

( ) * , - , - , -

, ( ) ( )-+

( ) * , - , - , -

, ( ) ( )-+ ( )

( )

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

To minimize total cost function per unit time ( ) , the optimal value of and can be obtained by solving the following equations:

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5 ( )

and

( )

( ) Equation (4.13) is equivalent to

[ { } { } * +

* ( ) ( )+

{ { } { } * +

* ( ) ( )+}

{ { } { } * +

* ( ) ( )+}

{ { } { } * +

* ( ) ( )+} { } { }

* + * ( ) ( )+]

( )

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6 [

{ { ( ) ( )} { ( ) ( )}

{ ( ) ( )} { ( ) ( )}

{ ( ) ( )}}

{ , - , - , -

, ( ) ( )-}

{ , - , - , -

, ( ) ( )-}

{ , - , - , -

, ( ) ( )-}

{ , - , - , -

, ( ) ( )-} , -

, - , -

, ( ) ( )-] ( )

Further, for the total cost function ( ) to be convex, the following conditions must be satisfied

( )

( ) And

( ( )

) ( ( )

) ( )

The second derivatives of the total cost function ( ) are complicated and it is very difficult to prove the convexity mathematically .However, with the help of graph, we can easily demonstrate convexity of total fuzzy cost function.

(i) We defuzzify the fuzzy total cost ̃( ) by signed distance method.

By signed distance method, Total Cost is given by.

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

Where

( ) * , - , - , -

, ( ) ( )-+

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( ) * , - , - , -

, ( ) ( )-+

( ) * , - , - , -

, ( ) ( )-+

( ) * , - , - , -

, ( ) ( )-+

( ) * , - , - , -

, ( ) ( )-+ ( )

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

To minimize total cost function per unit time ( ) , the optimal value of and can be obtained by solving the following equations:

( )

and

( )

( ) Equation (4.13) is equivalent to

[ { } { } * + * ( ) ( )+

{ { } { } * +

* ( ) ( )+}

{ { } { } * +

* ( ) ( )+}

{ { } { } * +

* ( ) ( )+} { } { }

* + * ( ) ( )+]

( )

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8 [

{ { ( ) ( )} { ( ) ( )}

{ ( ) ( )} { ( ) ( )}

{ ( ) ( )}}

{ , - , - , -

, ( ) ( )-}

{ , - , - , -

, ( ) ( )-}

{ , - , - , -

, ( ) ( )-}

{ , - , - , -

, ( ) ( )-} , -

, - , -

, ( ) ( )-] ( )

Further, for the total cost function ( ) to be convex, the following conditions must be satisfied

( )

( ) And

( ( )

) ( ( )

) ( )

The second derivatives of the total cost function ( ) are complicated and it is very difficult to prove the convexity mathematically. Thus, the convexity of total cost function has been established graphically. (Figure 1 and 2)

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[1]Total fuzzy cost ( ) Vs. and T. [2] Total fuzzy cost ( ) Vs. and T.

5. CONCLUSIONS

This paper presents a fuzzy inventory model for deteriorating items with shortages under fully backlogged condition in which demand is an increasing function of time. Shortages and deterioration are natural in any inventory control system. The proposed model is developed in both the crisp and fuzzy environments. In fuzzy environment, all related inventory parameters are assumed to be pentagonal fuzzy numbers. For defuzzification, graded mean method is employed to evaluate the optimal time period of positive stock and total cycle length T which minimizes the total cost. By given numerical example it has been tested that graded mean representation method gives minimum cost.

6. FUTURE RESEARCH DIRECTION

1. The other uncertainties such as supply uncertainty, lead time uncertainty and costs uncertainty can be present in inventory control.

2. Comparing the result of this study with the result of the other studies with different Fuzzy system.

REFERENCES

1. Bansal. A., (2010), Some non linear arithmetic operations on triangular fuzzy number (m,α,β), Advances in FuzzyMathematics, 5,147-156.

2. Bansal. Abhinav., (2011), Trapezoidal Fuzzy Numbers (a,b,c,d); Arithmetic Behavior, International Journal of Physical and Mathematical Sciences, ISSN: 2010-1791.

3. Dinagar. D. Stephen and Latha. K., (2013), Some types of Type-2 Triangular FuzzyMatrices, International Journal of Pure and Applied Mathematics, Vol-82, No.1, 21-32.

4. Dubois. D and Prade. H., Operations on Fuzzy Numbers, International Journal of Systems Science, Vol-9, No. 6., pp.613-626.

5. Dutta. Palash and Ali. Tazid., (2011), Fuzzy Arithmetic with and without α-cut

6. Method; A Comparative Study, International Journal of latest trends in Computing, E-ISSN: 2045-5364, Vol-3.

7. Hass. Michael., (2009), Applied Fuzzy Arithmetic, Springer International Edition, 8. ISBN 978-81-8489-300.

9. Klir. G.J and Bo Yuan., (2005), Fuzzy Sets and Fuzzy logic, Prentice Hall of India Private Limited.