Optimization Of Deteriorating Eoq Model With Shortages And Controllable Lead Time

ISSN 2348 – 7968

Optimization of Deteriorating EOQ Model with Shortages

and Controllable Lead Time

P. Selvaraju S. Kumara Ghuru

Department of Mathematics, RVS College of Engineering and Technology, Coimbatore – 641 402 Department of Mathematics, Chikkanna Government Arts College, Tiruppur -641 602

Abstract:

The main focus of the article is to analyze the optimization inventory system for deteriorating items with shortages are allowed. A proposed model is constructed and obtaining the optimal solutions of lot size optimum level of time which reduced the total cost. A numerical example is illustrated in this model.

Key words:Demand, EOQ, deterioration, Shortages and Inventory cycle.

1. Introduction:

In practical situations, electronic and electrical goods, costlier fashion clothes, road vehicles they fall with its category since they can become absolute overtime and their demand rate will decreases drastically. So Deterioration of items in inventory system has become an interesting feature for its practical importance. Deterioration refers to damages, spoilage, vaporization or obsolescence of the products. Example, deteriorating items include metal parts, which are prove to corrosion and rusting and some perishable food items which are subject to spoilage and decay. In this article an EOQ model is developed for deteriorating items and Shortages. This study is organized as section 2 is concerned with assumption and notations, in section 3 is mathematical model and in section 4 is conclusion.

2. Assumptions and Notations

(i) Assumptions: The following assumptions are used to formulate the problem. a) Asingle period inventory cycle systemof single typeof products is considered.. b) Shortages are permissible.

c)Thelead time is controllable. d)Time horizonis finite.

(ii) Notations: The following notations are used in our analysis. a) R − Demand rate / year

b)C1 −SetupCost/OrderingCost/set c) C2 −HoldingCost/unit/year

d) C3−ShoratgeCost/unit e)C4−Deteriorating Cost/unit

f) L − Lead Time, where Lis controllable

g)Q1 −Maximum Inventory

h) S − Shortages

i) θ−Rate of deteriorative

k) TC −Total Cost

3. Mathematical model

(a) EOQ Model for deteriorating items with Shortages Let I(t) be the inventory level of the system at t(0≤t≤T).

R

Figure: 1 EOQ model for deteriorating items with Shortages The differential equations of I(t) over the period ( 0, T) is given by

1

0 )

( )

(t I t R t T I

dt

d _{+θ =− ≤ ≤}

(1)

T t T R t

I dt

d _{=− ≤ ≤}

1

) (

(2)

ISSN 2348 – 7968 with the boundary conditions are

S T I T

I Q

I(0) = ₁ ( ₁) = 0 ( )=

(3)

dt e R e t I From

T

t t t

∫

− =

1

) ( ), 1

( θ θ

1

T

t t t e

e

R _

     = −

θ

θ θ

[

]

[

(1 ) 1

]

1

− =

− =

− −

t T

t T t

e R

e e e R

θ

θ θ θ

θ θ

(4)

∫

− = = t

T

T t R dt

R t

I From

1

) ( )

( ), 2

( ₁

(5)

( )

1

) 0 ( , 4 )

3

( and I Q

From =

1=

[

1 −1

]

T

e R

Q θ

θ

   

 

+ =

   

 

− +

+ =

2

1 2 1

2 1 1

2 1 2

1

T R T R

T T R

θ θ θ θ

(6)

Total cost (T_c): Total cost consists of ordering cost, holding cost, deteriorating cost and Shortage cost

T

C Cost Ordering

i 1

)

( =

(7)

∫

= 1

0

2 _{( )}

) (

T

dt t I T C Cost Holding ii

[

]

dt e

R T

C T _{T t}

1 ) ( 0 2 1 1 − = −

∫

θ θ 1 1 0 ) ( 2 T t T t e T C R       − − − = − θ θ θ

[

₁

]

1

0 )

( 2

2 T t T

t e T C R θ θ θ + − = −

[

1

]

1 2

2 ₁ T

e T T C R _θ θ θ + − − =       − − − + − = 2 1 1 2 1 2 1 1 2 2 T T T T C

R _{θ θ} θ

θ       − − = 2 2 1 2 2 2 T T C R θ θ T T C R 2 2 1 2 = (8)

∫

= 1 0

4 _{( )}

) ( T dt t I T C Cost ing Deteriorat iii θ

[

]

dt e R T

C T t

T 1 ) ( 0 4 1 1 − = −

∫

θ θ θ 1 1 0 ) ( 4 T t T t e T C R       − − − = − θ θ θ θ

[

₁

]

1

0 )

(

4 T t T

t e T C R _θ θ θ ₊ − = −

[

1

]

1 4

1 T e T T

C

R _θ θ

θ + − − =       − − − + − = 2 1 1 2 1 2 1 1 4 T T T T C

R _{θ θ} θ

ISSN 2348 – 7968 dt T t R L T C T T ) ( 1 1 3

∫

− = T T t T t L T C R 1 1 2 3

2 _

    − =      + − − = 2 1 2 1 1 2 3 2 2 T T T T T L T C R _     − + = 2 2 1 12 2

3 T TT T

L T C R

[

]

2 1 3

2TL T T C R − = (10)

Total Cost = Ordering Cost + Holding Cost + Shortage Cost + Deteriorating Cost

T T C R T T L T C R T T C R T C T_C 2 ) ( 2 2 2 1 4 2 1 3 2 1 2

1 _{+ + − +} θ

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

2 TL T T

C R T C C T R T C

TC = + +θ + −

(11)

( )

11 Partially withrespect toT1 ating Differenti

( )

0 0 ) 1 ( ) ( 2 2 ) ( 2 2 ) ( 2 1 2 1 3 1 4 2 1 〉 ∂ ∂ = − − + + = ∂ ∂ C C T T and T T L T C R T C C T R T T θ ) ( )

( 3 ₁

1 4

2 T T

L C T

C

C +θ = −

T L C T L C C C 3 1 3 4 2 ) ( +θ + = L C C C T L C T 3 4 2 3 1 + + = θ (12)

( )

Partially with respect toT ating Differenti 11 0 ) ( ) ( 2 2 2 ) ( ) ( ₂ 2 1 1 3 2 2 1 4 2 2

1 _ =

2 ( ) (2 2 2 ) 0 2 1 1 2 1 2 3 2 1 4 2

1 − + + − − + − =

− T TT T TT T

L C R T C C R C θ 0 ) ( ) (

2 2 3 2 ₁2

1 4 2

1 − + + − =

− T T

L C R T C C R C θ 2 1 3 4 2 1 2

3 _{2 T}

L C C C R C T L C R       _{+ +} + = θ             + + + + + = 2 3 4 2 2 2 3 3 3 4 2 3 1 2 ) ( ) ( 2 L C C C T L C L C L C C C L C R C T θ θ L C R C L C C C L C T 3 0 3 4 2 3 2 2 1 =             + + − θ L C R C L C C C C C T 3 0 3 4 2 4 2

2 ₌ 2

          + + + θ θ             + + + = ) ( ) ( 2 4 2 3 3 4 2 1 2 C C L C R L C C C C T θ θ and C C L C R L C C C C T           + + + = ) ( ) ( 2 4 2 3 3 4 2 1 θ θ _           + + + = ) ( ) ( 2 4 2 3 3 4 2 1 C C L C L C C C C R Q θ θ Numerical Example 2 01 . 0 , 150 10 10 100

5000 ₁= ₂= ₃= ₄= = =

= C C C C and L

R

Let θ

Optimal Solution:

ISSN 2348 – 7968 T = 0.1071, T₁ =0.0325 Ordering cost = 933.71, Holding cost = 246.56, Deteriorating Cost = 36.98, Shortage Cost = 649.53, and Total cost = 1866.78

2 3

3 2

1( )

2 0

:

C L C R

L C C C T

then when

Note

+ =

=

θ

2 3

3 2

1( )

2

C L C

L C C C R Q

and

+ =

which are standard Inventory Models.

(b) EOQ model for deteriorating Items

Let I(t) be the inventory level of the system at t(0≤t≤T).

The differential equations of I (t) over the period (0, T) is given by

0 )

( )

(t I t R t T

I dt

d

≤ ≤ −

= +θ

with the boundary conditions are

) 14 ( 0

) ( )

( =Qand I T = I θ

From (14);

I(t)= e−θt

∫

T

t

R t

eθ dt

= e−θt _θR

( )

T t t

eθ

= _θ

R _t

e−θ

(

eθT −eθt

)

I(t) = _θR {eθ(T−t) −1} (15)

Total cost (TC): Total cost consists of setup cost, deteriorating cost and holding cost.

(i) Setup Cost =

T C1

(16)

(ii) Holding Cost =

∫

T

dt t I T C

0

2 _{( )}

=

∫

T

T C

0 2

θ

R

} 1 {eθ(T−t) − dt

=

T C R

θ 2

∫

T

0

} 1

{eθ(T−t) − dt

=

T C R

θ 2

T t T

t e

0 ) (

   

 

− −

−

θ

θ

=

[

]

T t

T

t e

T C R

0 )

( 2

2 θ

θ

θ ₊

− −

=

[

e T T T e T

]

T C

R θ _θ θ

θ + −

− ( − )

2 2

=

[

T e T

]

T C

R _θ θ

θ + −

−

1

2 2

= 2

[

1

]

2 − −

T e

T C

R T θ

θ

θ

= 

  

 

− − +

+ 1

2 1

2 2

2

2 _T T _T

T C

R _θ θ _θ

θ

ISSN 2348 – 7968 = _

    

2

2 2

2

2 T

T C

R θ

θ

=

2

2T C R

(17)

(iii) Deteriorating Cost =

∫

T

dt t I T

C

0

4 _{( )}

θ

=

∫

T

T C

0 4

θ

θ

R

} 1 {eθ(T−t) − dt

=

T C R

θ θ ₄

∫

T

0

} 1

{eθ(T−t) − dt

=

T C

R 4

T t T

t e

0 ) (

   

 

− −

−

θ

θ

=

[

]

T t

T

t e

T C R

0 )

(

4 θ

θ

θ ₊

− −

=

[

e T T T e T

]

T C

R θ _θ θ

θ + −

− 4 ( − )

=

[

T e T

]

T C

R _θ θ

θ + −

− 1

4

= 4

[

e − T−1

]

T C

R T θ

θ

θ

= 

  

 

− − +

+ 1

2 1

2 2

4 _T T _T

T C

R _θ θ _θ

θ

= _

    

2

2 2

4 T

T C

R θ

θ

=

2

4T C Rθ

(18)

(iv) Total Cost = Setup Cost + Holding Cost + Deteriorating Cost

T_C =

2 2

4 2

1 RC T R C T

T

C θ

+ +

(19)

Differentiating the total Cost (19) with respect to T,

( )

T_C = dT

d

_{( )}

0 2 0

2

2 3

1 2

2 4

2 2

1 + + = = 〉

−

T C T

dt d and C

R C R T C

C

θ

[

]

2

4 2 2

1 R C C

T

C +θ

=

[

2 4

]

1

2 2

C C R

C T

θ + =

[

2 4

]

1

2

C C

R C T

θ

+ =

(20)

Numerical Example

01 . 0 150

10 100

5000 ₁= ₂= ₄= =

= C C C θ

R Let

Optimal Solution:

T =0.059, Ordering Cost = 1694.92, Holding Cost = 1475 Deteriorating Cost = 221.25 and Total Cost = 3391.17

Which is the stationary inventory model.

4 2

1

2

1 2

2 0

:

C C

C R Q

and C

R C T

then If

Note

θ θ

+ =

= =

ISSN 2348 – 7968

4. Conclusion:

This proposed model is developed the effect of deterioration items in the EOQ inventory model by two sections, one is the inventory model with shortages when lead time is controllable and another one describes the EOQ model for deteriorating items. We conclude that the lead time is inevitable even when shortages are fulfilled immediately. Infact, the lead time is controllable sense illustrated by suitable numerical examples. The main focus of this mathematical model to optimization of lot size and the time which minimizes the total cost in an inventory cycle.

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