Fuzzy Inventory Model for Deteriorating Items with Fluctuating Demand and Using Inventory Parameters as Pentagonal Fuzzy Numbers

ISSN 2319-8133 (Online)

(An International Research Journal), www.compmath-journal.org

Fuzzy Inventory Model for Deteriorating Items with Fluctuating Demand and Using Inventory Parameters as Pentagonal Fuzzy Numbers

Harish Nagar

and

Priyanka Surana

1,2

Department of Mathematics,

Mewar University, Gangrar, Chittorgarh, Raj., INDIA.

[email protected], [email protected] (Received on: February 7, 2015)

ABSTRACT

In this paper, a fuzzy inventory model for deteriorating items with time varying demand and shortages under fully backlogged condition is formulated and solved. Fuzziness is introduced by allowing the cost components (holding cost, shortage cost, etc.), demand rate and the deterioration. In fuzzy environment, all related inventory parameters are assumed to be pentagonal fuzzy numbers. The purpose of this study is to minimize the total cost function in fuzzy environment.

Graded Mean Representation method is used to defuzzify the total cost function and the results obtained by this method are explained by the use of numerical data.

Keywords: Inventory, Deterioration, Fuzzy model, Shortages, Pentagonal Fuzzy Number [PFN], Graded mean representation method.

1. INTRODUCTION

The fuzzy set theory in inventory modeling is the closest possible approach to reality. As reality is not exact and can only be calculated to some extent. Same way, fuzzy theory helps one to incorporate unpredictability in the design of the model, thus bringing it closer to reality

.

The effect of deterioration is very important in many inventory systems.

Deterioration is defined as decay or damage such that the item cannot be used for its original

purpose. Most of the physical goods undergo decay or deterioration over time. Commodities

such as fruits, vegetables, foodstuffs, etc., suffer from depletion by direct spoilage while kept

in store. Highly volatile liquids such as gasoline, alcohol, turpentine, etc., undergo physical

depletion over time through the process of evaporation. In the development of economic

production lot size models, usually researchers consider the deterioration rate, demand rate, unit cost, etc., as fixed, but all of them probably will have some little fluctuations for each cycle in real life situation. So in practical situations, if these quantities are treated as fuzzy variables then it will be more realistic

.

In 2003, Sujit De Kumar, P. K. Kundu and A. Goswami

presented an economic production quantity inventory model involving fuzzy demand rate and fuzzy deterioration rate. In 2007, J. K. Syed and L. A. Aziz

applied signed distance method to Fuzzy inventory model without shortages. In 2011, P. K. De and A. Rawat

proposed a model for fuzzy inventory using triangular fuzzy number without shortages. In 2012, C. K. Jaggi, S. Pareek, A. Sharma and Nidhi

presented a fuzzy inventory model for deteriorating items with time- varying demand and shortages. In 2012, Sumana Saha and Tripti Chakrabarti

proposed a fuzzy EOQ model for time dependent deteriorating items and time dependent demand with shortages. Very recently, D. Dutta and Pavan Kumar published several papers in the area of fuzzy inventory with or without shortages. In 2013, the same authors D. Dutta and Pavan Kumar

proposed an optimal policy for an inventory model without shortages considering fuzziness in demand, holding cost and ordering cost.

In this paper, we first consider a crisp inventory model with constant deteriorating items with constant demand where shortages are allowed with fully backlogged 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. The solution for minimizing the fuzzy cost function has been derived.

2. PRELIMINARIES

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

Definition 2.1: (By Pu and Liu

) A fuzzy set ã on R= (-∞, ∞) is called a fuzzy point if its membership function is

  1,    0,   (1)

Where the point  is called the support of fuzzy set .

Definition 2.2-A fuzzy set 

, 

 where 0    1 and    defined on R, is called a level of a fuzzy interval if its membership function is

  ,      0,  !" (2)

Definition 2.3 A fuzzy number #$  a, b, c  where a   ( and defined on R, is called a triangular fuzzy number

if its membership function is

 *

+,

,

,     

-,+

-,

,     ( 0,  !

. (3)

When     ( , we have fuzzy point (, (, (  (̃. The family of all triangular fu zzy numbers on R is denoted as 0

 2, , (,     (, 3, , ( 4 56

The  -cut of #$  , , ( 4 0

, 0    1 is #  #

, #



Where #

   9  :  and #

  ( : ( :  are the left and right endpoints of #.

Definition 2.4: A trapezoidal fuzzy number #$  , , (, ; is represented

with membership function

as:

 

= >

?

> @A 

,      1 ,     ( 5 

, (    ; 0 ,  ! C > D

> E

(4)

The  -cut of #$  , , (, ;, 0    1 is #  #

, #



Where #

   9  :  and #

  ; : ; : ( are the left and right endpoints of #.

Definition 2.5: A pentagonal fuzzy number(PFN)[9] #$  , , (, ;,  is represented with membership function

as:

 

= >

> ?

> >

@A

 

,      A

 

,     (

1 ,   ( 5

 

, (    ; 5

 

, ;     0 ,  ! C > > D

> >

E

(5)

The  -cut of #$  , , (, ;, , 0    1 is #  #

, #



Where #

   9  :   A



#

   9 ( :   A



and

#

  ; : ; : (  5



#

   :  : ;  5



So

A

  L

α 9 L

α

2   9  :  9  9 ( : 

2

  9  9  :  9 ( : 

2   9  9 ( : 

2 5

  R

α 9 R

α

2  d : d : cα 9  :  : ;

2

 d 9  : ; : ( 9  : ;

2  d 9  :  : (

2

Definition 2.6: If #$  , , (, ;,  is a pentagonal fuzzy number then the graded mean integration representation of # Ois defined as

PQ#$R  S  TA

 9 5



2 U ;

V_W X

S  ;

With 0    Y

and 0  Y

 1

PQ#$R  1 2

S  Z 9  9 ( : 

2 9 d 9  :  : ( 2 [ ;

S  ;



FG

(6)

[a] _ -cut of pentagonal fuzzy number [PFN]:-

[b] Conditions on Pentagonal Fuzzy Number [PFN]:-

A Pentagonal Fuzzy Number PQ#$R 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

.

3.1 Notations

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

2. A is the ordering cost per order.

3. ` is the deterioration rate,0  `  1.

4. T is the length of the Cycle.

5. Q is the ordering Quantity per unit.

6. h is the holding cost per unit per unit time 7. S is the shortage Cost per unit time.

8. C is the unit Cost per unit time.

9. b

, c is the total inventory cost per unit time.

10. dO is the fuzzy demand.

11. `< is the fuzzy deterioration rate.

12. < is the fuzzy holding cost per unit per unit time.

13. e$ is the fuzzy shortage Cost per unit time.

14. f$ is the fuzzy unit Cost per unit time.

15. bO

, c is the total fuzzy inventory cost per unit time.

16. b



, c is the defuzzify value of bO

, c by applying Graded mean representation method.

3.2 Assumptions

1. Demand d  1 9  is assumed to be an increasing function of time i.e. where  and  are positive constants and  h 0,0    1.

2. Replenishment is instantaneous and lead time is zero.

3. Shortages are allowed and fully backlogged.

4. MATHEMATICAL MODEL

Let Q be the total amount of inventory purchased or produced at the beginning of

each period and after fulfilling backorders. Due to reasons of market demand and

deterioration of the items, the inventory level gradually diminishes during the period 0, 



and ultimately falls to zero at   

.The period i

cj is the period of shortages, which are

fully backlogged. Let k be the on-hand inventory level at any time t, which is governed by the following two differential equations:

4.1. Crisp Model

lmn

ln

9 θIt  :Dt, 0  t  t

(4.1) with I(0) =Q, k

  0

lmn

ln

 :Dt, t

 t  T (4.2) with k

  0

The solution of equation (4.1) and (4.2) is given by

k  s

9 v

:

w 

9

:

1 9  (4.3) And

k  

: c 9



: c

 (4.4) By using k

  0,put t= t

in equation (4.3),we get

s  

1 9 

 :

 

: v

:

w (4.5) Now (4.3) becomes

k   _F:  9^t_GF: ^G 9  _FF:  :^uI,u_G ^J9^t_G_FF: ^G:^t_xF: ^]

(4.6) (neglecting higher powers of `).

Total average no. of holding units ( k

) during period [0, T] is given by

k

 S k ; 

 

9



 9  

9



 (4.7) Total no. of deteriorated units ( k

) during period [0, T] is given by

k

 s :Total demand s : S 1 9  ;

k



9



(4.8) Total average no. of shortage units ( k

) during period [0, T] is given by

k

 : S k ; 



: c

:



c :

:



 (4.9)

Total cost of the system per unit time is given by b

, c  1

c # 9 k

9 fk

9 ek



b

, c 

# 9  

9



 9  

9



 9 f 

9



 9 e 



: c

:

v

c :

:



w€ (4.10) 4.2. 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. , <, f$, e$, `<, < may change within some limits.

Let   

, 

, 

, 

, 

, <  

, 

, 

, 

, 

 f$  f

, f

, f

, f

, f

, e$  e

, e

, e

, e

, e

 `<  `

, `

, `

, `

, `

, <  

, 

, 

, 

, 

 be the pentagonal fuzzy numbers.

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

bO

, c 

# 9 < 

9



 9 << 

9



 9 f$ 

9



 9

e$ 



: c

:

v

c :

:



w€ (4.11) We defuzzify the fuzzy total cost bO

, c by graded mean representation method.

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

b



, c  1

12 ib



, c, b



, c, b



, c, b



, c, b



, cj Where

b



, c  1

c „# 9 

… 

2 9

`

6 

‡ 9 



… 

3 9

`

8 

‡ 9 f

`



2 9 



`

3 

‡ 9 e

2 

: c

: 



2 Š

c : c

3 : 2

3 

‹‡Œ

b



, c  1

c „# 9 

… 

2 9 `

6 

‡ 9 



… 

3 9 `

8 

‡ 9 f

`



2 9 



`

3 

‡ 9 e

2 

: c

: 



2 Š

c : c

3 : 2

3 

‹‡Œ

b



, c  1

c „# 9 

… 

2 9

`

6 

‡ 9 



… 

3 9

`

8 

‡ 9 f

`



2 9 



`

3 

‡ 9 e

2 

: c

: 



2 Š

c : c

3 : 2

3 

‹‡Œ

b



, c  1

c „# 9 

… 

2 9 `

6 

‡ 9 



… 

3 9 `

8 

‡ 9 f

`



2 9 



`

3 

‡ 9 e

2 

: c

: 



2 Š

c : c

3 : 2

3 

‹‡Œ

b



, c 

# 9 



9



 9 





9



 9 f



9

_ƒ_ƒt_ƒ

]



 9 e





: c

:

v

c :

:



w€ (4.12) b



, c  1

12 ib



, c 9 3b



, c 9 4b



, c 9 3b



, c 9 b



, cj To minimize total cost function per unit time b



, c , the optimal value of 

and c can be obtained by solving the following equations:

Žb



, c

Ž

 0 and

_‘’u_I,~

~

 0 (4.13) Equation (4.13) is equivalent to

F

FG~_FF_F9^t_G^I_F^G 9 _FF_F_F^G9^t_G^I_F^] 9 f_F2F`<sub>F</sub><sub>F</sub>9 <sub>F</sub><sub>F</sub>`_F_F^G6 9 eF2FF: c : _F_F_Fc : _F^G6 9 3 _GG_F9^t_G^J_F^G 9 _GG_G_F^G9^t_G^J_F^] 9 f_G2_G`<sub>G</sub><sub>F</sub>9 <sub>G</sub><sub>G</sub>`_G_F^G6 9 e_G2GF: c : _G_GFc : _F^G6" 9 4 ]_]_F9^t_G^z_F^G 9 _]]_]_F^G9^t_G^z_F^] 9 f_]2]`<sub>]</sub><sub>F</sub>9 ]]`]FG6 9 e]2]F: c : ]]Fc : FG6" 9 3 ^^F9^t_G^‚FG 9 ^^^FG9^t_G^‚F] 9 f_^2_^`<sub>^</sub><sub>F</sub>9 <sub>^</sub><sub>^</sub>`_^_F^G6 9 e_^2_^_F: c : _^_^_Fc : _F^G6" 9 __F9^t_G^ƒ_F^G 9 ___F^G9

tƒ

GF] 9 f2`F9 `FG6 9 e2F: c : Fc : FG6€

 0 (4.14)

#“; „

e

:



: c :



: c

 9 3e

:



: c :



: c

 9 4e

:



: c :



: c

 9 3e

:



: c :



: c

 9

e

:



: c :



: c

" :

…12# 9 



9



 9 





9

t_I

{



 9 f



9



 9 e





: c

:

v

c :

:



w" 9 3 …



9



 9 





9



 9 f



9



 9 e





: c

:

v

c :

:



w" 9 4 …



9



 9 





9



 9 f



9



 9 e





: c

:

v

c :

:



w" 9 3 …



9

t_‚

x



 9 





9



 9 f



9



 9 e





: c

:

_‚_‚

G

v

c :

:



w" 9 



9



 9 





9



 9 f



9

_ƒ_ƒt_ƒ

]



 9 e





: c

:

v

c :

:



wΠ 0 (4.15) Further, for the total cost function b



, c to be convex, the following conditions must be satisfied

^J_‘’u_I,~

u_I^J

h 0 ,

h 0 (4.16) And

v

IJ

w v

w : v

I~

w h 0 (4.17) The second derivatives of the total cost function b



, c are complicated and it is very difficult to prove the convexity mathematically.

5. NUMERICAL EXAMPLE

Consider an inventory system with following parametric values.

Crisp Model, A=Rs.200/order, C=Rs.20/unit, h=Rs. 5/unit/year, a=100 units/year, b=0.1units/year, `  0.01/year, S=Rs 15 /unit/year.

The solution of crisp model is b

, c = Rs 404.3429, 

=0.7149 year, T = .9639 year.

Fuzzy model,

  60,80,100,120,140, <  0.06,0.08,0.10,0.12,0.14

f$  16,18,20,22,24 , e$  11,13,15,17,19

`<  0.006,0.008,0.010,0.012,0.014, <  1,3,5,7,9

The solution of fuzzy model can be determined by following Graded Mean Representation Method.

1. When ,— <, f,O e$, `<, < all are pentagonal fuzzy numbers.

b



, c  5!. 414.6096 , 

 0.6908˜, c  0.9383˜ 2. When ,— <, f,O e$, `< all are pentagonal fuzzy numbers

b



, c  5!. 406.9852 , 

 0.7135˜, c  0.9560˜ 3. When ,— <, f,O `< all are pentagonal fuzzy numbers.

b



, c  5!. 405.5274 , 

 0.7115˜, c  0.9596˜ 4. When ,— <, `< all are pentagonal fuzzy numbers.

b



, c  5!. 405.2250 , 

 0.7120˜, c  0.9603˜ 5. When  —“; < all are pentagonal fuzzy numbers.

b



, c  5!. 404.8978 , 

 0.7131˜, c  0.9611˜

To show the convexity of cost function b



, c, we plot a 3D graph among 

and c, where values of both 

and c ranging from 

= .65 to 2 with equal interval,

T= .84 to 1 respectively. A three-dimensional graph is shown in the following:

(Figure A) Total fuzzy cost ™š›œ, ž Vs. œ and T.

6. 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..

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distance method”, Applied Mathematics & Information Sciences, Vol. 1(2), pp.203-209

(2007).

14. P. K. De and A. Rawat, “A fuzzy inventory model without shortages using triangular fuzzy number”, Fuzzy Information & Engineering, Vol. 1, pp.59-68 (2011).

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considering fuzziness in demand, holding cost and ordering cost”, International Journal

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