View of A FUZZY INVENTORY MODEL FOR DETERIORATING ITEMS UNDER INFLATION WITH DISPLAYED STOCK DEPENDENT DEMAND USING GENETIC ALGORITHM

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Vol. 04, Issue 03,March 2019 Available Online: www.ajeee.co.in/index.php/AJEEE

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A FUZZY INVENTORY MODEL FOR DETERIORATING ITEMS UNDER INFLATION WITH DISPLAYED STOCK DEPENDENT DEMAND USING GENETIC ALGORITHM

R B Singh

Department of Statistics, D. N. College, Meerut, Uttar Pradesh Sanjay Kumar

Department of Mathematics, SRM University, Delhi-NCR, Sonepat, Haryana

Abstract- This study develops a single-item, inventory model for deteriorating items with stock dependent demand rate in the fuzzy environment. In practice there are many commodities such as food, medication, fashionable items exist. Here the researcher presents the search timeliness difficulty by various operational constraints in manufacturing plants over and above business strategic policy, profit points, and variety of refinements. In addition, we discussed a crisp and a fuzzy inventory model, which considers scarcity and partially backlog constraints even when looking at stock-dependent demand rates for erratically deteriorating commodities. The model is designed to optimize the possibility of ambiguous ambitions of intended tasks. In this model we are using fuzzy logic and genetic algorithms for get better the effectiveness and efficient optimal inventory models. To calculate the optimal cost in defuzzify model, Graded mean representation, Signed distance and Centroid methods are used. Accordingly, we are comparing the optimal cost through these methods with the help of numerical illustration; along with sensitively table is also presented. A genetic algorithm (GA) is used to solve the model, which is illustrated on a numerical example.

Keywords: Fuzzy, Inflation, Stock-Dependent Demand, Genetic Algorithm, Webull‟s distribution, partial backlogging.

1 INTRODUCTION

Fuzzy-genetic algorithm is a valuable tool for global optimization; Fuzzy logic is a controlled tool for dealing with impurities and uncertainty. However each of these devices has its inherent limitations.

Combined techniques are developed at the same time to remove the limitations of component components and utilize their strengths. A large number of joint techniques have been developed to solve a wide variety of problems. These techniques include fuzzy-genetic algorithms, genetic-fuzzy systems, and fuzzy-genetic algorithms already use fuzzy logic techniques to improve the performance of genetic algorithms.

GA is found as an efficient tool for global optimization but its local search capability is poorly observed. On the other hand an FLC is a major tool for narrowband search.

In real life, the demand rate for a particular type of item depends on many factors, such as time, stock level in the showroom, quality of the item and so on.GA is an algorithmic tool used to reduce the total cost of inventory problem that is obtained by the fuzzify model.

Fuzzification is a way of converting crush set values into fuzzy set values, we are using different types of membership functions to obtain these

values. Fuzzification is a method for converting a crush set value to a fuzzy set value, for these deficiencies, a decipherment means changing the fuzzy model to a crisp model. Zadeh [1965]

describein relation to fuzzy sets, membership functions means characteristics of the set. Park et al.

[1987] introduced the idea for fuzzy set to EOQ formula by indicating the inventory carrying cost with a fuzzy number. Chang et al. [1998] calculated the backorder inventory issues with fuzzy backorder such that the backorder quantity is a triangular fuzzy number. Chang et al.

[1999] developed the fuzzy production;

inventory models for fuzzify the product quantity as the triangular fuzzy number.

Yao and Chiang [2003] examined the total cost of inventory without backorder, fuzzified the total demand and method used for defuzzify by the centroid and the signed distance methods. Dutta et al.

[2005] developedan inventory model in presence of fuzzy random variable demand and the optimum is obtained with the help of a graded mean integration representation method. Gani and Maheswari[2010] developed an EOQ model with inferior characteristics items with shortages and defective rate, demand, holding cost, ordering cost and

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2 shortage cost getting from triangular fuzzy numbers and get the total profit with the help of the Graded mean integration method of defuzzify. Singh et al. [2011]

introduced a limited storage size fuzzy model. Uthayakumar and Valliathl [2011]

developed a commercial production model for Weibull deteriorating thing over an infinite horizon under fuzzy environment and deal with some cost component as triangular fuzzy numbers and find the cost function with the signed distance method of defuzzify the cost function.

Kumar et al. [2013] determined the inventory model of linear demand rate and backlogging. They used signed distance method to defuzzify average profit and order quantity. Kumar and Rajput [2015] developed a fuzzy model of inventory for deteriorating thing where demand rate is time dependent with demand rate and partially backlog is based on triangular fuzzy number. They used the signed distance and the centroid method of defuzzify the total profit as per unit time. Kumar and Kumar [2015]

introduced an inventory model for deteriorating items with inventory dependent demand rate and partial backlogging with capability constraints.

Kumar and Kumar [2016] developed an inventory modelfor Non-Instantaneous deterating items with stock dependant demand and partially backlogging. Kumar and Kumar [2016] further explain a model of inventory of deteriorating items for which GA is used for permissible postponed in funds. Kumar at.el. [2016]

discuss the model of deteriorating items of stock dependent demand rate where inflation, shortage and partially backorder are allowed and GA is used to decrease the total cost of inventory. Kumar and Kumar‟s [2017] also present a model of fuzzy inventory of deteriorating and partially backlog of linear demand for the effective and satisfactory process. Sarkar et al (2018) A multi-retailer supply chain model with back order and variable production cost. Saha and Sen (2018) introduced an inventory model for deteriorating items with time and price dependent demand and shortages under the effect of inflation. Sarkar et al (2019) An application of time dependent holding costs and system reliability in a multi- item sustainable economic energy efficient reliable manufacturing system.

In this paper, we have developed an inventory model for deteriorating items with stock-dependent demand rate with consideration the impact of inflation and under fuzzy environment. In this article we have also used the Genetic Algorithm for determined the optimum results.

Numerical example and sensitivity analysis is carried out to study the various parameters on decision variable and objective function by using Matlab for the feasibility and applicability of our model.

2. DEFINITIONS OF FUZZY CONTROL In order to treat fuzzy inventory model by using Graded Mean representation, Signed Distance and Centroid to defuzzify, we need the following definitions.

Definition 2.1 (By Pu and Liu [25, Definition 2.1]). A fuzzy set ãon

 ^, 

R   

is called a fuzzy point if its membership function is𝜇_𝑎(𝑥). Where the point a is called the support of fuzzy set ã.

( ) 1,

a 0,

x a

x x a

 _{ }^ ^ ^_

  



Definition 2.2. A fuzzy set [a∝, b∝] is called a level of a fuzzy interval if its membership function is𝜇_a∝,b∝(𝑥). Where 0≤∝≤1 and a<b defined for R.

,

[ ]

( ) ,

a b

0,

x x a

otherwise

 b



_ _

_{ } ^ ^{ } ^ _

 

Definition 2.3. A fuzzy number

A 

_{= (a, b,}

c) is called a triangular fuzzy number if its membership function is𝜇_𝐴 𝑥 .where a< b <

c and defined for R.

,

( ) ,

0,

A

x a

a x b

b a c x

x b x c

x b

Otherwise



    

  

 

    

 

 

When a=b=c, we have a fuzzy point (c, c, c) = c.

The family of all triangular fuzzy numbers

on R is denoted as

{( , , ) | , , }

FN  a b c a  b c a b cR

Definition 2.4. If A= (a, b, c) is atriangular fuzzy number then the graded mean integration representation of

A 

_is

defined as

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3

1 1

0

( ) ( )

( ) 2

A

A w

w

L h R h

h dh

P A

hdh

 

  

 

 





 With 0< h≤ wA and

0< wA ≤ 1

1

0

1

0

[ ( ) ( )]

( ) 1 / 2 4

6 h a h b a c h c a dh

a b c P A

hdh

    

 











Definition 2.4. If

A 

= (a, b, c) is a triangular fuzzy number then the signed distance of

A 

is defined as

1

0

( , 0) ( ( ) , ( ) ), 0 2

L R 4

a b c P A^{ } 



d A



_ A



_ ^ ^ ^

Definition 2.4. The Centroid method on the triangular fuzzy number

A 

= (a, b, c) is a defined as

( ) 3

a b c

C A  

 

Figure 1: ∝- Cut Of a Triangular Fuzzy Number

3. NOTATIONS AND ASSUMPTIONS 3.1 Notations

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

 A is the ordering cost per order.

 T is the length of the Cycle.

 Q is the ordering Quantity per unit.

 h is the holding cost per unit per unit time.

 S is the shortage Cost per unit time.

 C is the unit Cost per unit time.

 TC t T1, is the total inventory cost per unit time.

 D is the fuzzy demand.

 h^~ is the fuzzy holding cost per unit per unit time.

 _S^~ is the fuzzy shortage Cost per unit time.

 C^~ is the total fuzzy inventory cost per unit time.

 TC_{d G}t T1, is the defuzzify value of



1



~

,

TCd G t T by applying Graded mean integration method.

 TC_{d S}



t T1,



is the defuzzify value of



1



~

,

TCd S t T by applying Signed distance method.



TC

_{d C}

 t T

1

, 

is the defuzzify value of



1



~

,

TC

d c

t T

by applying Centroid method.

3.2 Assumptions

 The demand rate D(t) at time t is

1 1

b q(t), 0 (t) o,

a t t

D D t t T

  

   

where

a

and b are non negative constraints.

 A fraction z(t) of the on hand inventory deteriorates per unit time where z(t) =t^^¹, 0 < α < 1, t > 0, β

>1.

 Inflation considered.

 Replenishment is instantaneous and lead-time is zero.

 Shortages are allowed and partially backlogged. Unsatisfied demand is backlogged, and the fraction of shortages backordered is

1/(1    ( T t ))

^{, where}



^{is a}

positive constant.

4 FORMULATION OF INVENTORY MODEL

It is considered that the q(t) be the on- hand inventory at time t with initial inventory Q. The inventory level gradually diminishes during the time period [0, t1] because of market demand, satisfaction of the customers and deterioration of the items, and after that during the time interval [t1, T] shortages period are partially backlogged. At any instant of time, the inventory level q(t) is governed by the differential equations.

4.1 Crisp Model and Fuzzy Model 4.1.1 Crisp Model

Figure 1 Graphical representation Inventory model

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4 Let q(t) be the inventory level at any time, which is governed by the following two differential equations

  ¹ ^(t)  ^{( )}

d q t

t q a bq t

dt

 ^

    0 t t1(1) With the boundary condition q(0) = Q and q(t1) = 0, the solution of equation (1) is

 

1 1 2 2

1 1 1

(t) ( ) (t t ) 1

1 2

q Q a t  t^ t^ b t^ bt

 ^ ^

  

          

(2) By putting q(t1) = 0, we have

1 2

1 1

1 1 2

t bt Q a t

 



  

     

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Now, equation (2) becomes

 

1 1 ¹ ¹ 1² ²

(t) 1 ( ) ( ) (t t )

1 2

q a t^ bt t t  t^ t^ b



 

 

          

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Again in the second time interval [t1, T]

the instantaneous inventory will satisfy.

Thus, the differential equation below represents the inventory status

   

 

1 d q t Do

dt ^{ }  Tt ¹

t   t T

(5)

With the condition q (t1) = 0, we get the solution of equation (5) is



1

 

1



q( ) D^olog 1 D^olog 1

t  T t  T t

 ^  ^

        

(6) Total average no. of h holding units (IH) during period[0, T] is given by

1

0

q(t) dt

t Rt

IH



e^

     

2

3 4 2 5 2

1

1 1 1 1

3 2 1 ( 2)

2 6 8 15 ( 1)( 2)

a R b bR ab R a b a R

at t t ab Rt    t^

 

       

      

 

3

 

4

1 1

2 ( 3) 1

2( 1)( 3) ( 1)( 2)( 4)

a R ab R R

t^ t^

     

    

  

    

       

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Total no. of deteriorated units (ID) during period[0, T] is given by

 

1

0

, (t) dt

t

D D

I  Q Total Demand I  Q



a bq

 

2 2 4 5

2 3 2 2

1 1 1

1 1

3 6

3 3 7 6 6 5

2 6 30 24 60

at b b R abt abRt

at b b R abt b bR b

       

             

2 2 3 2 4 2 5

1 1 1 1 1

2 1

2 2 ( 3)( 1) 3 2 2 6( 4) 2( 3)

ab t ^ abt  ab t ^ b R abRt^ ab R t ^

      

         

                (8)

Total average no. of shortage units (IS) during period [0, T] is given by

 

1

1 2 1

q( ) dt 1 log 1

T

o s

t

D T t

I t  T t

  

    

         

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Total cost of the system per unit time is given by

1   

, 1 _H _D _S

TC t T A h I C I S I

T   

1  1 1² 1³ ²

1 3 4

, (3 b) CR

2 6 6 30

h R b bR b b R

TC t T A Cat at Cb at h C C

T

           

              

 

   

4 2

2 5 2

1 1

3 2 3 6 7 1 6 5

4 2 6 60 15 ( 1)( 2)

abt h C C h a t

R a b b bR abRt b b



 

    

             

 

   



^h ^^^R ^^² ^^Cb ²^^²



^^{ab C t}² ₂^¹^^²^₍_^₃₎₍^_¹₁₎^^₂₍_₃₎₍^{a t}^_¹^^₂₎₍³ _₁₎



²

 

²



³

 

²



¹



²



²



¹



¹ ⁴ ¹

4 ( 1)( 2) 6

ab t R Cb

h R Cb RCb h

  

     

  

      

             

        

 

2 5

1 0 1

1 2

1log 1 2( 3)

SD

Cab R t T t

T t

 

   

   

       

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4.2 Fuzzy Model

Throughout the development of EOQ models, previous authors have assumed that the deterioration rate is constant. In the above developed crisp model, it was assumed that all

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5

the parameters were fixed or could be predicted with certainty; but in real life situations, due to uncertainly in the environment it is not easy to define all the parameters specifically, Accordingly we assume some of these parameters namelya^~, b^~,^C^~ , S^~ , _^~_, _^~ ,h^~may change within some limits.

Let 1 2 3 1 2 3 2

~ ~

1

~

(a , a , a ), (C , C , C ), (S ,S ,S ),3

a C S ^~(  ₁, ₂, ₃), ^~(  ₁, ₂, ₃), h^~(h , h , h ),₁ ₂ ₃ b^~(b , b , b ),₁ ₂ ₃ are as triangular fuzzy numbers.

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

1  1 1² 1³ ²

1 3 4

, (3 b) CR

2 6 6 30

h R b bR b b R

TC t T A Cat at Cb at h C C

T

           

              

 

        



    

  

      

     

4

2 5 2

1

3 2 3 6 7 1 6 5

4 2 6 60 15

abt h C C h

R a b b bR abRt b b

   

           

   

    

       

 

   

 

2 2 2

1 1 1

2 2 2

( 1)( 2) 2 ( 3)( 1)

a t ab C t

h R Cb

 

     

   

    

           

  

 

      

   

          

 

3

1 2 2 3 2 1 2 2 1

2( 3)( 2)( 1)

a t h R Cb RCb

      

  

         

  



         

  

 

4 2 5

0

1 1 1

2 1

1 1

log 1 6

4 ( 1)( 2) 2( 3)

SD

ab t R Cb Cab R t T t

h T t

 

    

  

   

          

              

      

  

   

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We used the Graded mean representation; Signed distance and Centroid method for defuzzify the fuzzy total cost.

(1) By Graded Mean Representation Method, Total Cost is given by



1



₁



1



₂



1



₃



1



, 1 , , , , ,

dG 6 dG dG dG

TC t T  TC t T TC t T TC t T  Where

 

1

2

2 1 3 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

3 4

, 1 (3 b ) R

2 6 6 30

dG

h R b b R b b R

TC t T A C a t a t C b C a t h C C

T

           

              

     

4

2 5 2

1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1

3 2 3 6 7 6 5

4 2 6 60 15

a b t h C C h

R a b b b R a b Rt b b

   

           

   

 

 

2 2 2

1 1 1 1 1 1 1 1 1

1 1 1 1 1 1

1 1 1 1

2 2 2 1

( 1)( 2) 2 ( 3)( 1)

a t a b C t

h R C b

 

     

   

    

           

        

 

3 1 1 1 2

1 1 1 1 1 1 1 1 1 1 1

1 1 1

2 2 3 1 2 2 1

2( 3)( 2)( 1) a t

h R C b RC b

       

  

         

  

 

4 2 5

1 0

1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 2 1

1 1 1 1

1 1

log 1

4 ( 1)( 2) 6 2( 3)

S D

a b t R C b C a b R t T t

h T t

 

    

      

          

              

(12)

 

2

2 2 3 2 2 2 2

1 2 2 1 2 1 2 2 2 2 2 1 2 2 2

3 4

, 1 (3 ) R

2 6 6 30

dG

h R b b R b b R

TC t T A C a t a t C b b C a t h C C

T

           

              

     

4

2 5 2

2 2 1 2 2 2 2

2 2 2 2 2 2 1 2 2

3 2 3 6 7 6 5

4 2 6 60 15

a b t h C C h

R a b b b R a b Rt b b

   

           

   

 

   

 

2 2 2

2 2 1 2 2 2 1 2

2 2 2 2 2 2

2 2 2 2

2 2 2 1

( 1)( 2) 2 ( 3)( 1)

a t a b C t

h R C b

 

    

   

    

           

 



 



    

 

3 2 2 1 2

2 2 2 2 2 2 2 2 2 2 2

2 2 2

2 2 3 1 2 2 1

2( 3)( 2)( 1)

a t

h R C b RC b

       

  

         

  

 

4 2 5

2 0

2 2 2 1 2 2 2 2 2 2 2 2 1 1

2 2 1

2 2 2 2

1 1

log 1

4 ( 1)( 2) 6 2( 3)

S D

a b t R C b a C b R t T t

h T t

 

    

      

          

               (13)

 

3

2 3 3 3 3

1 3 3 1 3 1 3 3 3 3 3 1 3

, 1 (3 ) R

2 6

dG

h R b b R

TC t T A C a t a t C b b C a t h

T

       

           

View of A FUZZY INVENTORY MODEL FOR DETERIORATING ITEMS UNDER INFLATION WITH DISPLAYED STOCK DEPENDENT DEMAND USING GENETIC ALGORITHM

 , 

R   

( ) ,

0,

x x a

otherwise

 b



      

 

A 

A 









A 

A 







A 













TC

 t T

, 





,

TC

t T

a

1/(1    ( T t ))



 

 

t   t T



 





     

 

 



 

   







 



 











     

 

   

 

          

 

 

















     

 ^, 

_{ } ^ ^{ } ^ _