Cmjv03i04p0440 by Journal of Computer and Mathematical Sciences

J. Comp. & Math. Sci. Vol.3 (4), 440-445 (2012)

An Inventory Model for Weibull Deteriorating Items with Linear Demand Pattern L. K. RAJU1, U. K. MISRA2, SRICHANDAN MISHRA3 and G. MISRA4 1

Department of Mathematics NIST, Paluru Hills, Berhampur, Odisha, INDIA. 2 Department of Mathematics Berhampur University, Berhampur, Odisha, INDIA. 3 Department of Mathematics Govt. Science College, Malkangiri. Odisha, INDIA. 4 Department of Statistics, Utkala University, Bhubaneswara Odisha, INDIA. (Received on: June 30, 2012) ABSTRACT The objective of this model is to investigate the inventory system for perishable items with linear demand pattern where two parameter Weibull distribution for deterioration is considered. The Economic order quantity is determined for minimizing the average total cost per unit time. As the rate of deterioration increases, the optimal time of the inventory decreases and the required number of items for the fixed length of each ordering or the production cycle, the minimum total operational cost of the inventory and the required items for the fulfillment of backorder increases. The application is illustrated with suitable examples. Keywords: Inventory system, Weibull distribution. AMS Classification No:

Linear demand, Deterioration,

90B05 .

1. INTRODUCTION The deterioration of many items during storage period is a real fact. In many inventory models, it is assumed that the items can be stored indefinitely without any

risk of deterioration. However, certain types of items undergo changes while in storage so that, with time, they become partially or entirely unfit for use. Deterioration refers to damage, spoilage, vaporization, or obsolescence of the products. There are

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L. K. Raju, et al., J. Comp. & Math. Sci. Vol.3 (4), 440-445 (2012)

several types of items that will deteriorate if stored for extended periods of time. Examples of deteriorating items include metal parts, which are prone to corrosion and rusting, and food items, which are subject to spoilage and decay. Electronic components and fashion clothing also fall into this category, because they can become obsolete over time and their demand will typically decrease drastically. Inventory control for deteriorating items has been a well-studied problem. Numerous optimal and heuristic approaches have been developed for modeling and solving different variations of this problem. C. K. Tripathy, L. M. Pradhan and U.K. Mishra8 have developed a model with linear deterioration rate where the demand rate is considered to be constant. Sarbjit and S.S. Raj4 have developed a model whose demand as well as perishability rate increases with time. Ghare and Schrader2 use the concept of deterioration. Covert and Philip1 also formulated a model with variable rate of deterioration with two parameter Weibull distributions which was further extended by Shah6. Sarbjit and Raj5 also developed the model for items having linear demand and variable Deterioration rate with trade credit. Tomba et. al.7 developed a model with linear demand pattern and deterioration with shortages. Sanjay Jain and Kumar3 established the model having power demand pattern, Weibull distribution deterioration and shortages. In this paper an attempt has been made to develop an inventory model for perishable items with two-parameter Weibull density function for deterioration and the linear demand pattern is used over a finite planning horizon. Nature of the model

is also discussed for shortage state. Optimal solution for the proposed model is derived and the applications are investigated with the help of some numerical examples. 2. ASSUMPTIONS AND NOTATIONS Following assumptions are made for the proposal model: i. Single inventory will be used. ii. Lead time is zero. iii. The model is studied when shortages are allowed. iv. Demand follows the linear demand pattern v. Shortages are allowed and are completely backlogged. vi. Replenishment rate is infinite but size is finite. vii. Time horizon is finite. viii. There is no repair of deteriorated items occurring during the cycle. ix. The second and higher powers of α are neglected in this analysis of the model hereafter. Following notations are made for the given model: I (t ) = On hand inventory level at any time t ,t ≥ 0 . R(t ) = a + bt is the demand rate at time t .

θ : θ = αβ t β −1 , The two-parameter Weibull distribution deterioration rate (unit/unit time). Where 0 < α << 1 is called the scale parameter, β > 0 is the shape parameter.

Q = Total amount of replenishment in the beginning of each cycle. V = Inventory at time t = 0

Journal of Computer and Mathematical Sciences Vol. 3, Issue 4, 31 August, 2012 Pages (422-497)

L. K. Raju, et al. al., J. Comp. & Math. Sci. Vol.3 (4), 440-445 (2012)

442

T = Duration of a cycle. cd = The deterioration cost per unit item.

V be the level of initial inventory. In the period (0, t1 ) the inventory level gradually

ch = The holding cost per unit item.

decreases due to market demand and deterioration. At t1 , the level of inventory reaches zero and after that the shortages are allowed to occur during the interval [t1 , T ] , which are fully backlogged. Only the backlogging items are replaced by the next replenishment. The behavior vior of inventory during the period (0, T ) is depicted in the following inventory-time time diagram.

cb = The shortage cost per unit. U =The total average cost of the system. 3. FORMULATION The objective of the model is to determine the optimal order quantity in order to keep the total relevant cost as low as possible. The optimality is determined for shortage of items. Taking Q be the total amount of replenishment in the beginning of each cycle, and after fulfilling backorders let

Here we have taken the total duration T as fixed constant.. The objective here is to determine the optimal order quantity in order to keep the total relevant cost as low as possible.

If I (t ) be the on-hand hand inventory at time t ≥ 0 , then at time t + ∆t , the on-hand on inventory in the interval [0, t1 ] will be

I (t + ∆ t ) = I (t ) − θ (t ) I (t ) ∆ t − R (t ) ∆ t

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L. K. Raju, et al., J. Comp. & Math. Sci. Vol.3 (4), 440-445 (2012)

Dividing by ∆t and then taking as ∆t → 0 we get dI ( t ) + αβ t β −1 I (t ) = − ( a + bt ) ; dt

0 ≤ t ≤ t1

aα β +1 β +1 b 2 2 bα β +2 β +2  (t1 −t ) + (t1 −t ) + (t1 −t ); a(t1 −t) + β + 1 2 β +2  

α tβ 

−

I (t) = e

0 ≤ t ≤ t1

(3.3)

(3.1) For the next interval [t1 , T ] , where the shortages are allowed we have

On solving equation (3.2) with boundary condition we have

I (t + ∆t ) = I (t ) − R (t ) ⋅ ∆t .

I (t ) = a (t 1 − t ) +

Dividing by ∆t and then taking as ∆t → 0 we get

t1 ≤ t ≤ T

dI ( t ) = − ( a + bt ); dt

I (t1 ) = 0 . On solving equation (3.1) with boundary condition we have t1

;

V = at1 +

aα β +1 b 2 bα β + 2 t1 + t1 + t1 β +1 2 β +2

(3.5)

The total amount of deteriorated units in 0 ≤ t ≤ t1 is given by

β t β −1 I ( t ) dt

∫α

(3.4)

Using I (0) = V in equation (3.3) we have

t1 ≤ t ≤ T (3.2)

The boundary conditions are I (0) = V and

b 2 2 (t 1 − t ) 2

(3.6)

V β α V 2 β a b aα β bα β β +1 β +2 2 β +1 2β +2  t1 − t1 − t1 + t1 t1 = αβ  t1 − +  2β β +1 2( β + 2) ( 2 β + 1)( β + 1) 4( β + 1)( β + 2) β 

The total cost function consists of the following elements:

(i ) Holding cost per cycle t1

C h ∫ I ( t ) dt

(3.7)

 a t 12 b t 13 aα β bα β α V β +1 β +2 β +3  = C h V t 1 − − − t1 + t1 + t1  2 3 β +1 ( β + 1)( β + 2 ) 2 ( β + 2 )( β + 3 )  

(ii ) Deterioration cost per cycle t1

C d ∫ αβ t β −1 I ( t ) dt

(3.8)

V β α V 2 β aα β bα β a b β +1 β +2 2 β +1 2β +2  = αβ C d  t1 − t1 − t1 − t1 + t1 + t1  β 2 β β + 1 2 ( β + 2 ) ( 2 β + 1 )( β + 1 ) 4 ( β + 1 )( β + 2 )   Journal of Computer and Mathematical Sciences Vol. 3, Issue 4, 31 August, 2012 Pages (422-497)

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L. K. Raju, et al., J. Comp. & Math. Sci. Vol.3 (4), 440-445 (2012)

(iii ) Shortage cost per cycle T

∫ I ( t ) dt t1

 aT = C b  a t1 T − 2 

b t 12 T bT − 2 6

−

a t 12 b t 13  −  2 3 

(3.9)

Taking the relevant costs mentioned above, the total average cost per unit time of the system is given by U (V , t 1 ) =

1 {Holding cost + Deterioration cost - Shortage cost} T

(3.10)

 C h 

 a t 12 b t 13 α V β +1 aα β bα β β +2 β +3  − − − + + V t t1 t1 t1  1  2 3 β +1 ( β + 1)( β + 2 ) 2 ( β + 2 )( β + 3 )   V β α V 2 β aα β bα β a b β +1 β +2 2 β +1 2β +2  + αβ C d  t1 − t1 − t1 − t1 + t1 + t1  β 2 β β 1 2 ( β 2 ) ( 2 β 1 )( β 1 ) 4 ( β 1 )( β 2 ) + + + + + +   =

1 T

 a T 2 b t12 T b T 3 a t12 b t13   − C b  a t1 T − + − − −  2 2 6 2 3    Eliminating V from equation (3.10) we have

U (t1 ) = 1 α β C d  1  a t 1β + 1 + aα t 12 β + 1 + b t 1β + 2 + b α t 12 β + 2  T  β +1 2 β +2  β  aα 3 β +1 b 2 β + 2 bα 3β + 2  α  2 β +1 − + t1 + t1 + t1  a t1  2β  β +1 2 β +2  −

β +1

t 1β + 1 −

(3.11)

 b aα β bα β t 1β + 2 − t 12 β + 1 + t 12 β + 2  2(β + 2) ( 2 β + 1 )( β + 1 ) 4 ( β + 1 )( β + 2 ) 

 a α β +2 t1 + C h  a t 12 + + β +1  a t 12 b t 13 − − + 2 3 (β

 b 3 bα a α 2β +2 b β +3 bα α  β +2 t1 + t 1β + 3 − t1 t 12 β + 3  + + t1 +  a t1 2 β +2 β +1 β +1 2 β +2   aα β b α β t 1β + 2 + t 1β + 3  + 1 )( β + 2 ) 2 ( β + 2 )( β + 3 ) 

 aT − C b  a t1 T − 2 

b t 12 T bT − 2 6

Now equation (3.11) can be minimized but as it is difficult to solve the problem by deriving a closed equation of the solution of equation (3.11), Matlab Software has been * used to determine optimal t1 and hence the

−

a t 12 b t 13   −  2 3   *

optimal cost U (t1 ) can be evaluated. Also ∗

level of initial inventory level V and the total amount of replenishment Q * in the beginning of each cycle can be determined.

Journal of Computer and Mathematical Sciences Vol. 3, Issue 4, 31 August, 2012 Pages (422-497)

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L. K. Raju, et al., J. Comp. & Math. Sci. Vol.3 (4), 440-445 (2012)

4.0. EXAMPLES

of β increases, the optimal time increases ∗

Example 1: The values of the parameters are considered as follows:

whereas Q * , V decrease. REFERENCES

ch = $40 / unit / year, cd = $70 / unit, cb = $100 / unit

α = 0.001, β = 1, a = 400, b = 30, T = 1 year. Now using equation (3.11) which can be minimized to determine optimal * t1 = 0 .7191 Years and hence the average optimal cost U (t1* ) = $5894.0 / unit . Also level of initial inventory level V ∗ = 295.5037 units. Total amount of replenishment

Q * = 415 .1071 units . Example 2: The values of the parameters are considered as follows: ch = $40 / unit / year, cd = $70 / unit, cb = $100 / unit

α = 0.001, β = 2, a = 400, b = 30, T = 1 year. Now using equation (3.13) which can be minimized to determine optimal * t 1 = 0 . 7192 Years and hence the optimal cost U ( t 1 * ) = $ 5889 . 9 / unit . Also level of initial inventory level V ∗ = 295 .4903 units .

Total amount of replenishment

Q * = 415 .0516 units. 5. CONCLUSION Here an EOQ model is derived for perishable items with linear demand pattern. Two-parameter Weibull distribution for deterioration is used. The model is studied for minimization of total average cost .Numerical examples are used to illustrate the result where we saw that when the value

1. Covert, R.P. and Philip, G.C. An EOQ Model for Items with Weibull Distribution Deterioration: IIE Trans., Vol. 5, pp. 323-326 (1973). 2. Ghar, P.M. and Schrader, G.F. A Model for exponentially Decaying Inventories: Ind. Engg., Vol. 14, pp. 238-243 (1963). 3. Jain, S. and Kumar, M. An Inventory Model with power demand pattern, Weibull Distribution deterioration and Shortages: Journal of Ind. Acad. of Mathematics; Vol. 30,No.1,pp.55-61 (2008). 4. Sarbjit, S. and Raj, S. S. Lot Sizing decisions under trade credit with Variable demand Rate under Inflation: Ind. J. Math. Science, Vol. 3, pp. 29-38 (2007). 5. Sarbjit, S. and Raj, S. S. An Optimal Inventory Policy for Items having linear demand and variable Deterioration Rate with trade credit: J. of Mathematics and Statistics, Vol. 5(4): pp. 330-333 (2009). 6. Shah, Y. K. An order-level lot-size inventory model for Deteriorating Items: IIE Trans., Vol. 9, pp. 108-112 (1977). 7. Tomba, I. and Brojendro, Kh. An Inventory model with Linear demand pattern and Deterioration with Shortages: J. of Ind. Acad. Math., Vol. 33, No. 2, pp. 607-612 (2011). 8. Tripathy, C. K., Pradhan, L. M. and Mishra, U. An EPQ Model for linear Deteriorating Item with variable holding cost: Int. Journal Comp. and Applied Math. Vol. 5, No 2, pp. 209-215 (2010).

Journal of Computer and Mathematical Sciences Vol. 3, Issue 4, 31 August, 2012 Pages (422-497)