Cmjv01i05p0572 by Journal of Computer and Mathematical Sciences

J. Comp. & Math. Sci. Vol. 1(5), 572-585 (2010)

An Optimal Ordering Policy for Weibull Deteriorating Items with Progressive Payment Scheme Under DCF Approach U.K.MISRA1, BHASKAR BHAULA2 and SRICHANDAN MISHRA3 1 Department of Mathematics Berhampur University, Berhampur, Orissa e-mail: umakanta_misra@yahoo.com. 2 Department of Mathematics Sanjay Memorial Institute of Technology (Degree Engineering College)Orissa e-mail:bhaskarbholo@gmail.com. 3 Department of Mathematics Dhaneswar Rath Institute of Engineering & Management Studies (DRIEMS), Tangi, Cuttack-754022, Orissa. e-mail: srichandan.mishra@gmail.com.

ABSTRACTS This paper develops an optimal inventory replenishment policy for Weibull deteriorating items with a progressive payment scheme under discounted cash flow approach over infinite planning horizon. The progressive trade credit offered by the supplier to the retailers is defined as follows. If the retailer pays the outstanding amount by the time

T1 ,

the supplier does not charge any interest. If the retailer pays after the time

T1 but before T2 T2  T1  , supplier charges interest at the rate of

I c1 . If the amount is paid after time T2 , the retailer has been charged





the interest at a rate of I c 2 I c 2  I c1 . All three possible cases for flow of the present value of a cycle to all infinite number of future cycles have been studied. The model also studied through numerical examples and sensitivity analysis. Key Words: Inventory, Weibull deterioration, Progressive payment scheme, DCF approach. Journal of Computer and Mathematical Sciences Vol. 1, Issue 5, 31 August, 2010 Pages (528-635)

U. K. Mishra et al., J. Comp. & Math. Sci. Vol. 1(5), 572-585 (2010)

INTRODUCTION The mathematical modeling of real world inventory problems needs the simplification of assumption to make the mathematics malleable. One of the most important problems faced in inventory management is how to regulate and maintain the inventories of deteriorating items. The effect of deterioration of physical goods cannot be neglected in any inventory system because almost all the physical goods deteriorate over time. Food products, vegetables, f ruits, pharmaceuticals, chemicals, blood, photographic films, drugs, electronic goods, radioactive substances and volatile liquids are examples of some items in which sufficient amount of deterioration takes place during the normal storage period and should be taken into consideration in inventory system. During the last few years many researchers have focused their study of inventory model for deteriorating systems. To get an up to date research activity over the model we may refer articles of Datta and Pal6, Chang3, Raafat13 and Hardik Soni9. On the other hand the trade credit period offered by the supplier to the retailer not only encourages the retailers but also attracts new customers. It also functions as a role of price discounting approach because it does not force competitors to reduce their prices. Goyal7 developed an EOQ model when the supplier offers the retailer a permissible delay in payment.

573

Shah 14 extended Goyal's model by incorporating a constant rate of deterioration. The model enriched by Jamal11 allowing shortages in addition to deterioration. Hwang and Shinn10 derived optimal selling price and lot size for the retailer under conditions of permissible delay in payments. Shah 14 developed probabilistic inventory model under tradecredit policy. Lio et al.12 developed an inventory model for stock dependent demand rate when delay in payment is permissible. Teng15 advised that buyers should order in smaller quantity and take opportunity for the regularity of trade-credit. Chang 2 extended Teng's model by incorporating constant rate of deterioration. Discounted-Cash-Flow (DCF) approach is a very significant matter used in decisions to know the time value of money which is completely neglected in classical inventory system. The DCF approach permits a proper recognition of the functional implication of the opportunity cost and out of pocket costs in inventory analysis. In addition to the above, it also permits an explicit recognition of the timings of cash-flow associated with an inventory model. Hadley8 compared the optimal ordering quantities obtained by two different functions such as average cost approach and NPV approach (Net Present Value). He concluded that when discount rate is excessive, the optimal reorder interval has significant difference for these

Journal of Computer and Mathematical Sciences Vol. 1, Issue 5, 31 August, 2010 Pages (528-635)

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U. K. Mishra et al., J. Comp. & Math. Sci. Vol. 1(5), 572-585 (2010)

two models. Chao5 developed an EOQ model and NPV with deterministic and stochastic demand under DCF approach. Aggarwal and Jaggi1 extended Shah's14 model using DCF approach. Chang 4 developed an inventory model f or deteriorating items with shortages under DCF approach. Various inventory models taking constant rate of deterioration under DCF approach have been developed by researchers. It has been empirically observed that failure and life expectancy of many items can be expressed in terms of Weibull distribution function. Hence the consideration of Weibull distribution function for rate of deterioration in our model is more justified. The present model is discussed and solved analytically and conditions for concavity of the optimal function is established and studied through numerical examples and sensitivity analysis. NOTATIONS AND ASSUMPTIONS The present model of inventory replenishment policy is developed under the following notations: A = Replenishment cost per order. R = Demand rate. C = Purchase cost per unit. P = Selling price per unit (P > C). h = Inventory holding cost excluding interest charges per unit.  (t ) = Time proportional deterioration. r = Discount rate per unit time.

T1 = 1st permissible trade credit in settling account without any extra charges. T2 = Second permissible delay period in settling the account with an interest charged I c1 T2  T1  . I c1 = Interest charged in stock per year by the supplier when the retailer pays after T1 but before T2. I c 2 = Interest charged in stock per year by the supplier when the retailer pays after time T2 , I c 2  I c1 . I e = Interest earned per year.. T = Length of the replenishment cycle. Q = Optimum order quantity. I (t ) = Instantaneous inventory level at time t 0  t  T  . PV1 (T ) = Present value of cash-out-flows for T  T1 . PV2 (T ) =Present value of cash-out-flows for T1  T  T2 . PV3 (T ) = Present value of cash-out-flow for T  T2 . APV (T ) = Present value of all future cashout-flows. In addition to the above notations, we have the following assumption: i. The inventory system consists of single item only. ii. Replenishment rate is infinite. iii. The demand rate R is constant during the entire cycle time T. iv. Lead time is zero. v. Shortages are not allowed. vi. The items deteriorate at a time varying rate    t  1 , where 0    1,   1, 0    1 .

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U. K. Mishra et al., J. Comp. & Math. Sci. Vol. 1(5), 572-585 (2010)

vii. If the retailer pays by time T1 , then supplier does not charge retailer and the deposits the surplus in an account earning the interest at a rate of I e per unit per year. If the retailer pays after T1 and before T2 , the supplier charges the retailer at an interest rate I c1 per year for the unpaid balance at the end of T2 . If the retailer pays after T2 , the supplier charges interest at the rate I c 2 /year where I c 2  I c1 . viii. There is no repair or replenishment of deteriorated units during the entire cycle. MATHEMATICAL FORMULATION

obtained from equation (2) by putting t  0

I (0)  I 0  Q 

R  1 T  T  1  1





(3)

COMPONENTS OF THE SYSTEM The elements consisting of total inventory cost of the model per cycle are as follows i. Ordering cost = OC = A (4) ii. Purchasing cost = PC = CQ =

RC   1 T  T  1  1





(5)

iii. Inventory holding cost = IHC = T

The depletion of inventory happens due to combined effect of the demand and deterioration during the interval 0, T  . Hence the instantaneous inventory level is governed by the differential equation.

Rh  6

dI (t )   t  1 I (t )   R , 0  t  T , (1) dt

INTEREST CHARGED AND INTEREST EARNED

h  I (t ) e  rt dt 0

 2 6T   2  3 3T  rT    1  2 

(6)

I (0)  I 0  Q and I (T )  0 .

The following three possibilities arise, during the computation of interest charged and interest earned depending

Solving the differential equation (1) under

upon the length of the cycle time:

the given boundary conditions, we have

Case-I: T  T1 , Case-II: T 1  T  T2 ,

with boundary conditions



I(t) 

Ret  1 T t   T 1  t 1 , (2)  1







after neglecting higher power of  , as   1 . The ordering quantity can be

Case-III: T  T2 . Case-I: According to the agreement of the supplier with retailer, the interest charged

Journal of Computer and Mathematical Sciences Vol. 1, Issue 5, 31 August, 2010 Pages (528-635)

U. K. Mishra et al., J. Comp. & Math. Sci. Vol. 1(5), 572-585 (2010)

576

by the supplier is zero. Hence



I c1  0

(7) during the interval [0, T1 ] as T  T1 . During the interval 0, T1  , the retailer sells the product, deposits the revenue into interest earning account at the rate I e per year and pay the amount CQ to the supplier but during the interval T ,T1  , he deposits the total amount only to an account earning interest at the rate of I e . Hence the total interest earned during the

APV1 (T ) 



PV1 (T ) e  nrt 

n 0

 1 1 rT      PV1 (T ) (10) 4   rT 2 The necessary condition f or optimality can be obtained after differentiating equation (10) with respect to T partially and equating zero. Hence

interval 0, T1  is

T  IE1  RPI e   te rt dt  T T1  T  e r T1 T   0 



PRIe 1  1 rT  erT  r 2T T1  T  er T1 T  2 r





PV1 (T ) 1  e  rT

r T 4 PV (T) 4T 2rT 2 2

r2T3

 PVT(T)  0 1

That is

(8) The present value of total inventory cost per cycle is the sum of ordering cost, purchasing cost, inventory holding cost (excluding interest charged), interest charged and minus interest earned from selling revenue during the interval 0, T1  . Hence present value of cash-out flow per

r T 4 PRI 11rT e

cycle is



2 2

 r

 A

rT

r2TT1 TerT1T



 RC  1 T T  1  Rh  3T 2  rT 3  1 6 





6T 2  2 2 3 rT   4T  2rT  r T  PRIe Te  1  2



PV1 T   OC  PC  IHC  IC1  IE1 as IC1  0 ,









 T1  2T  rT1T  rT 2 er T1T   RC 1 T 



 PV1 T   OC  PC  IHC  IE1 (9) All future cash-out-flows taking the present value is given by

Rh  2

 2 T  1    2T  rT 2     0 (11)   1   

Journal of Computer and Mathematical Sciences Vol. 1, Issue 5, 31 August, 2010 Pages (528-635)

U. K. Mishra et al., J. Comp. & Math. Sci. Vol. 1(5), 572-585 (2010)

The optimum cycle length can be obtained after solving equation (11) for T. Differentiating equation (10) with respect to T twice we have,

2 APV1(T) 2PV1(T)  1 1 rT  2PV1(T)      2 T 2 rT3  Tr 2 4  T (12) 2

 OC  PC  0,   RCT  1  0 2 T T 2



 2 IE1   PRI e 1  2r (4T  T1 )  0 T 2 Hence

 2 APV1 (T )  0 for all T . T 2

period 0, T1  and also he earned the interest IE2 during the same period 0, T1  . Hence there are two sub-cases are possible depending upon the procurement cost and total revenue earned. Case-2.1: PRT1 e  rT1  IE 2  CQ Case-2.2: PRT1 e  rT1  IE 2  CQ Case-2.1:

 2 IHC  Rh 1  rT   T   0 T 2



577

(13)

Equation (13) represents the sufficient condition for optimality (min). Case-2:

Here the retailer has enough money in his account to pay off the total procurement cost at time T1 . Hence the interest charged I c 2.1  0 . The present value of cash-out-flows per cycle during this sub-case is PV2.1 (T )  OC  PC  IHC  IE 2 (15) The present value of all future-cash-flows is

 1 1 rT  APV 2 (T1 )      PV2.1 (T ) (16)  rT 2 4  Differentiating equation (16), we have

Since T  T1 , the interest earned during the interval 0, T1  is T1

IE2  PRIe  te rt dt  0

PRIe 1  1  rT1  e rT1 2 r





(14) The retailer has to pay the procurement cost CQ purchasing Q units at time T  T1 and earned the revenue of PRT1 e  rT1 selling RT1 units during the

APV 2.1 (T )  r 1     2  PV2.1 (T ) T  4 rT   1 1 rT  PV 2.1 (V )     T  rT 2 4 

(17)

We have

APV2.1 (T ) OC PC IHC IE2     T T T T T

Journal of Computer and Mathematical Sciences Vol. 1, Issue 5, 31 August, 2010 Pages (528-635)

U. K. Mishra et al., J. Comp. & Math. Sci. Vol. 1(5), 572-585 (2010)

578



PC IHC  T T

PV2.1 (T )  0,

 2 PV2.1 (T )  2 IE 2  0 ,  0. T 2 T 2

For the necessary condition of optimality, we have

Equation (19) represents the sufficient condition of optimality.

APV2.1 (T ) 0 T

Case-2.2:

Then we have

r T 2



 PRI  4  2 e 1  1  rT1  e rT1  A  r







RC   1 T  T  1  Rh    1 6

3T



 2

 61T  2 

V1  CQ  PRT1 e  rT1  IE 2

 rT 3 

 



In this case the retailer does not have enough money in his account to pay at permissible credit time T1 . Hence the supplier charges the retailer for the unpaid balance



 4T  2rT 2  r 2 T 3 RC 1  T  

Rh 2

2T  rT

(18)

Equation (18) represents the necessary condition of optimality. The optimum value of the cycle of length T can be obtained after solving equation (18) for T.

 2 APV2.1 (T ) 1   2 PV2.1 (T )   2 T T  2  1 1 rT   PV2.1 (T )    0  T 2  rT 2 4 

IE2  PRI e  t e rt dt  0

PRI e 1  1  rt e rT1 2 r





(21) The interest charged during T1 , T  is

I c 2..2



(19)

(20)

at the interest I c1 after time T1 . The interest earned during 0, T1  is T1

2 T  1    0    1 



V12 I c1  PR

 I (t ) e

 rt

V12 I c1h    1 T 2 r   1T 3    P(  1)  2 6

 T   2  2

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U. K. Mishra et al., J. Comp. & Math. Sci. Vol. 1(5), 572-585 (2010)

  TT12 T13  T12     1 T1T    r  1    2  3   2

2  1 T  T1  .  0 As T  T1 ,

 T  2     T1T  1  1      1   2 

 2 I c 2.2 0 , T 2

 TT1 1 T   2       1   2  

IE 2  2 IE 2  0 T T 2

 KV12 B, where k  I c1  h /( P (   1))

I c 2.2 V B    k 2V1 B 1  V12 T T T  

2 B  V1  T 2 

V1  RC 1  T   0 T





 2V1   RCT  1  0 T 2

(29)

(23)

Differentiating equation (29) with respect to T, we have

PV2.2 (T ) OC PC IHC    T T T T +

(24)

I c 2.2 IE2  T T

 (25)

PV2.2 (T ) PC IHC I c 2.2    T T T T (30) 2

(26)

B   1T  T1   2  r (T  T1 )  T  T 2





 PV2.2 (T )  PC  IHC   + T 2 T 2 T 2  2 I c 2.2 T 2

(31)

The present values of all future cash-flowsout is given by

(  1) T  T1 



PV 2.2 (T )  OC  PC  IHC  I c 2.2  IE 2

(22)

  V1  2  2 I c 2.2  2V1  k 2 B  2 B V     1 T 2 T 2   T  V B 4V1 1 T T

(28)



   T  1  T1 1  0 as T  T1

(27)

1 rT   1 APV2.2 (T )      PV2.2 (T ) 2 4   rT

2B  (   1) 1  rT      1 T  1 2 T Journal of Computer and Mathematical Sciences Vol. 1, Issue 5, 31 August, 2010 Pages (528-635)

(32)

U. K. Mishra et al., J. Comp. & Math. Sci. Vol. 1(5), 572-585 (2010)

580

APV2.2 (T )  r 1      2  PV22 (T )   T  4 rT   1 rT  PV2.2 (T )  1     4  T  rT 2









RC    1 T  T  1   Rh  1 6

Rh  2 6 T   2  3 3 T  rT     1(  2)     KV12 B   4T  2rT 2  r 2T 3 



 



Rc 1  T 



Rh  2



V B    k  2V1 B 1  V12   0 T T  

 1 1 rT  APV3.1 (T )      PV2.1 (T ) (35) 4   rT 2 (33)

Equation (33) represents the necessary condition of optimality. So optimum value of T can be obtained after solving equation (33) for T.

T 2



PV2.2 (T ) T2



Case-3: T  T2 The procurement cost of the retailer is CQ the total revenue available in his account at time T1 and T2 are PRT1e  rT1  IE 2 and PRT2 e  rT2  IE 2 respectively. Hence the following three sub-cases are possible depending upon the procurement cost and available revenue in his account. Case-3.1: PRT1e  rT1  IE 2  CQ This is the same as sub-case 2.1. All future cash-out-flows taking the present value is given by

 2 T  1   2T  rT 2     1  

 2 APV2.2 (T )

(34)

Equation (34) represents the sufficient condition of optimality (min).

 PRI  r 2T 2  4  2 e 1  1  rT1  e rT1  A  r



2  1 1 rT   PV2.2 (T )   0   T 2  rT 2 4 



APV3.1 (T )  r 1     2  PV 2.1 (T ) T  4 rT 

 1 1 rT  PV2.1 (T )    0 .  4  T  rT 2

(36)

Equation (36) represents necessary condition for optimality. Hence the optimum value of T can be obtained after solving equation (36) for T.

 2 APV3.1 (T ) PV3.1 (T )   T 2 T2

Journal of Computer and Mathematical Sciences Vol. 1, Issue 5, 31 August, 2010 Pages (528-635)

U. K. Mishra et al., J. Comp. & Math. Sci. Vol. 1(5), 572-585 (2010)

2  1 1 rT   pV2.1 (T )   0   4  T 2  rT 2

(37)

Inequation (37) represents sufficient condition of optimality (min).

PR (T2  T1 ) e

 PRI e

 te

 rt

Equation (39) represents the necessary condition for optimality.

 2 APV3.2 (T ) 1  2 PV2.2 (T )  2 T T 1 rT   2 PV3.2 (T )  1   0   4  T 2  rT 2

Case-3.2: PRT1 e  rT1  IE 2  CQ But  r ( T2 T1 )

581

(40)

Inequation (40) represents the sufficient condition of optimality (min).





 CQ  PRT1 e  rT1  IE 2 . This is the case when retailer does not have sufficient money at T1 to pay the supplier completely but he can pay the balance amount on or before T2 . Hence the supplier charges the interest to the retailer for the unpaid balance



V1  CQ  PRT1 e  rT1  IE 2

But

PR (T2  T1 ) e  r (T2 T1 )  PRI e

 te

 rt

 PRI e T22   .  CQ   PRT2  2  



at the rate of I c1 . This is similar to Case 2.2. Hence all future cash-out-flows taking the present value is given by

1 rT   1 APV3.2 (T )      PV2.2 (T ) 4   rT 2 (38)



PRI eT22  CQ Case-3.3: PRT2  2

APV3.2 (T )  r 1     2  PV2.2 (T )  T  4 rT 

 1 1 rT  PV2.2 (T )   0   T  rT 2 4 

(39)

In this case the retailer does not have sufficient money to pay off the total procurement cost at T2 . Hence the retailer has to pay the supplier of amount

PRT1 e  rT1  IE 2 with the interest charged on the unpaid balance





V1  CQ  PRT1 e  rT1  IE 2 at the rate of I c1 at T1 and he also makes the payment off

PR T2  T1  e  r T2 T1   PRI e

 te

 rt

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U. K. Mishra et al., J. Comp. & Math. Sci. Vol. 1(5), 572-585 (2010)

582

with interest charged on the unpaid

PRI e 1  1  rT1  e  rT1 r2



balance

IE3.3 

T2   V2  V1  PRT2 T1  er(T2 T1 )  PRIe  tert dt   T1

IE3.3  2 IE3.3  0 T T 2

 V1 I c1

V22 I c 2 PR

 I (t) e

 rt

r   1T 3  T   2   6  2    TT 2 T 3     1  T2T  T22   r   1  2  2  3   2 2  

 TT  T      1   2  

 2 I c 3.3  2V1    I c1 T2  T1  K1 T 2 T 2    V1  2  2 V1   2V 2 B1  B1  T 2    T 

V1 B1  2 B2   V22  T T T 2 

T1   1 T  T2 

 V1 I c1 T2  T1   K 1V 22 B

The interest earned is

V1 V2  , T T

B1   1T T 2   2  r T  T2   T 2

 2

hI c 2 where K 1  P (   1)

(43)

 4V2

 T  2     T  1 T2  2      1   2   1 2

V1  2 B1   2V2 B1 T  V 2 T   

V22 hI c2    1T 2 T2  T1    P (   1)  2

(42)

I c 3.3 V  I c1 T2  T1  1  K 1 T T

at the rate of I c 2 at time T2 . Therefore the charged interest I c 3.3 is

I c3.3  V1 I c1 T2  T1  







  T  1  T2 1  0 as T  T2

(41)

 2 B1    1 1  rT      1 T  1 T 2

2  1 T  T2  > 0.

Journal of Computer and Mathematical Sciences Vol. 1, Issue 5, 31 August, 2010 Pages (528-635)

(44)

U. K. Mishra et al., J. Comp. & Math. Sci. Vol. 1(5), 572-585 (2010)

Therefore

 2 I c 3.3  0. T 2

583

1 rT   2 PV3.3 (T )  1     0 (47)  4  T 2  rT 2

Since

Inequation (47) represents sufficient

PV3.3 (T )  OC  PC  IHC  I c3.3  IE3.3

condition of optimality (min).



PV3.3 (T ) OC PC IHC     T T T T

NUMERICAL EXAMPLES Let R = 1000, h = 4, A = 100, P = 35, C = 20, I c1  0.1, I c 2  0.12 ,

I c3.3 IE3.3 PC IHC IC3.3     T T T T T 

I e  0.08, T1 

 2 PV3.3 (T )  2 PC  2 IHC    T 2 T 2 T 2 365

 IC 3.3 0. T 2 value is given by

1 rT   1 APV3.3 (T )      PV3.3 (T ) 2 4   rT (45)

APV3.3 (T )  r 1     2  PV3.3 (T ) T  4 rT 

 1 1 rT  PV3.3 (T )    0  T  rT 2 4 

(46)

Equation (46) represents necessary condition for optimality. Differentiating equation (45) with respect to T partially twice, we have

 2 APV3.3 (T ) 1  2 PV3.3 (T ) T 2 T

 0.16438,   0.06,   2.1, r  0.06 optimal solution for different values of

All future cash-out-flows taking the present



45 60  0.123287, T2  365 365

 and  . SENSITIVITY ANALYSIS The effect of optimality due to change of values of different parameters associated in this model is given in the tables. Important points of the tables 1. T, Q and APV are moderately sensitive to the change of Weibull parameters  and  under constant discount rate as increase in  causes decrease in both T and Q, but increase in APV. On the other hand  has reverse effect that of  for the same. 2. T and Q are moderately sensitive to the change of discount rate r and APV is very much sensitive to change of r. Under constant deterioration rate, increase in r causes decrease in T, Q and APV.

Journal of Computer and Mathematical Sciences Vol. 1, Issue 5, 31 August, 2010 Pages (528-635)

U. K. Mishra et al., J. Comp. & Math. Sci. Vol. 1(5), 572-585 (2010)

584

Table - 1

ď Ą 0. 03

0. 04

0. 05

0. 06

0. 07

ď ˘

1.9

2. 0

2.1

2.2

2.3

T Q A PV T Q A PV T Q A PV T Q A PV T

0.1710241 171.15384 348601.364 0.170241 170.322236 348642.13 0.16941 169.510114 348682.66 0.168597 168.715473 348722.74 0.167794

0.1715372 171.587675 348578 0.17074 170.806366 348611.27 0.170021 170.102914 348644.26 0.169316 169.413079 348676.95 0.16863

0.1718205 171.861662 348559. 09 0.171175 171.229245 348576. 43 0.170558 170.625052 348613. 13 0.169947 170.026572 348639. 84 0.16935

0. 1720807 172. 114298 348 543.76 0. 17155 171. 594357 34 856 5.9 0. 17102 17 1. 0749 348 587.88 0. 170492 170. 557231 348 609.71 0. 16997

0.17234 172.367459 348531.32 0.1 71 866 171.902281 348549.4 0.1 71 416 171.46 096 348567.36 0.1 70 966 171.019486 348585.21 0.1 70 518

Q A PV

167.930318 348762.45

168.741887 348709.4

169.441827 348666. 35

170. 04536 348 631.39

170.579862 348602.95

Table - 2 Optimal Solution for Different Values of r

r 0.03 0.04 0.05 0.07 0.08

T3.1 0.18109 0.17745 0.17401 0.166912 0.16403

Q 181.18689 177.54098 174.09562 166.98725 164.10

APV 695,486.39 522,071.81 418,016.14 299,081.49 261,908.86

Table - 3 Optimal Solution for Different Values of A

A 90 80 70 60 50

T 0.15909 = T3.1 0.14737 = T3.1 0.134555 = T3.1 0.12092 = T1 0.11058 = T1

3. T, Q and APV are more sensitive to any change in A. Under constant deterioration rate and constant discount rate, decrease in A causes decrease in T, Q and APV. CONCLUSION This model is incorporated with some kinds of inventory happening day to day in the business world. Deterioration over time is a natural phenomenon for any

Q 159.15485 147.42115 134.59358 120.9477 110.601

APV 347,621.92 346,529.2 345,341.62 344,028.17 342,583.84

kinds of goods. The inclusion of progressive trade credit policy in this model is special attraction for new customers. In any inventory analysis discounted cashflows approach creates a proper recognition of financial implication of the opportunity cost. Consideration of Weibull distribution deteriorating function in our model is appropriate regulator to measure

Journal of Computer and Mathematical Sciences Vol. 1, Issue 5, 31 August, 2010 Pages (528-635)

U. K. Mishra et al., J. Comp. & Math. Sci. Vol. 1(5), 572-585 (2010)

reasonable amount of deterioration of perishable items and failure of any kind of items. Taking above realities into account, we have developed this model and also studied the effect of various parameters through sensitivity analysis. This model is very useful in business community. This can be widely used for fashionable clothes, electronic components, perishable products, domestic goods and other products. This model can further be enriched by allowing shortages. REFERENCES 1. Aggarwal, S.P and Jaggi, C.K. Ordering policies of deteriorating items under permissible delay in payment, Journal of the OR Society, 46(5), 658-662 (1995). 2. Chang, C.T, Ouyang, L.Y and Teng, J.T. An EOQ mode for deteriorating items under supplier credits linked to ordering quantity, Applied Mathematical Modeling, 27: 983-996 (2003). 3. Chang, K.J. A theorem on the determination of economic order quantity under condition of permissible delay in payments, Computers and OR, 25, 49-52 (1998). 4. Chang, Liang, Tsss-Pang. Inventory and Pricing strategies for deteriorating items with a discounted cash flows approach, 52, 29-40 (2007). 5. Chao, H. The EOQ model with stochastic demand and discounting, EJOR, 59, 434-443 (1992). 6. Datta, T.K. and Pal, A.K. Effects of inflation and time value of money on an inventory model with linear time dependent rate and shortages, EJOR, 52, 326-333 (1991). 7. Goyal, S.K. Economic order quantity

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Journal of Computer and Mathematical Sciences Vol. 1, Issue 5, 31 August, 2010 Pages (528-635)