An Inventory Model for Deteriorating Items with Permissible Delay in Payment and Inflation Under Price Dependent Demand

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On the Occasion of his 75th Birthday Anniversary PJSOR, Vol. 8, No. 3, pages 583-592, July 2012

An Inventory Model for Deteriorating Items with Permissible Delay

in Payment and Inflation Under Price Dependent Demand

Manisha Pal

Department of Statistics University of Calcutta, India [email protected]

Hare Krishna Maity Department of Statistics University of Calcutta, India [email protected]

Abstract

The paper studies an inventory model for deteriorating items when demand for the item is dependent on the selling price. Shortages are allowed and backlogged, and price inflation is taken into consideration. Further, it is assumed that the supplier permits the inventory manager to settle his accounts within a given specified time period. Numerical examples are cited to illustrate the model and to study the sensitivity of the model to change in model parameters.

Keywords and phrases: Inventory; Delay in payment; Deteriorating items; Backlogging of shortage; Selling price dependent demand; Inflation.

1. Introduction

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The above models were developed under the assumption that inflation does not play a significant role on the inventory policy. However, from financial point of view, one may consider an inventory to be a capital investment, and, as such, it should compete with other assets for an organization’s limited capital fund. It is, therefore, important to investigate how time-value of money influences various inventory policies. The first study in this direction has been reported by Buzacott (1975), who considered EOQ model with inflation, subject to different types of pricing policies. Misra (1979) developed a discounted-cost model and included internal (company) and external (general economy) inflation rates for various costs associated with an inventory system. Sarker and Pan (1994) surveyed the effects of inflation and the time value of money on order quantity with finite replenishment rate. Some studies were also conducted with variable demand, see, for example, Uthayakumar and Geetha (2009), Maity (2010), Vrat and Padmanabhan (1990), Datta and Pal (1991), Hariga (1995), Hariga and Ben-Daya (1996) and Chung (2003).

In this paper, we consider a dynamic inventory model for deteriorating items allowing shortages and under inflation, when demand is price dependent and the inventory manager enjoys a fixed permissible delay in payment. The paper is organized as follows. In section 2, we analyze the model. In section 3, examples are cited and a sensitivity analysis of the model is carried out. Finally, in section 4, a discussion on the model is given.

2. The Mathematical Model and Its Analysis The following notations have been used in the study:

H : The finite planning horizon

r : The constant inflation rate, 0r1

p(t) : The selling price at timet,p(0) =p

R(t) : The price dependent demand rate at timet  : The constant deterioration rate

M : The permissible delay in payment

Ir : The interest charged per unit of money per annum by the supplier

Ie : The interest earned per unit of money per annum

T : Length of a replenishment cycle

T1 :Time to exhaust stock within a replenishment cycle, 0T1<T

A : The ordering cost per order at timet= 0

c : Purchase cost per unit at timet= 0

I : Fraction of the purchase cost per unit defining the holding cost per unit per annum

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The demand rate at timetis given by

 

_t _a _bpert

R   , _

     

bp a r

t 1log

= 0, for 1log _,

     

bp a r

t

wherea,b0..

We take the length H of the planning horizon to be such that the demand rate at the end of the planning horizon remains non-negative, that is, 1log _.

     

bp a r

H This may also be

ensured by taking ab.

We assume that the planning horizon is divided into n reorder intervals of length T, so that we have H nT. Further, the costs and selling price during a reorder cycle is assumed to remain the same as that at the beginning of the cycle. Thus, the price in the_sth cycle is given by _per s1T_{, and hence the demand rate is}

  _,₍ ₁₎ _,

)

(_t _a _bpe 1 _s _T _t _sT

R _ _ rs T _ _ _ ₁__s__n_.

The inventory policy is to place an order at the beginning of each reorder interval, and the order quantity is just sufficient to meet the backorders in the previous period and the demand during the firstT1units of time in the current period.

Let,Is(t) denote the inventory level at time point tin thesthcycle, 1sn. Since depletion

of stock occurs owing to demand and deterioration, the following differential equations define transitions in inventory:

 

_{ }

,

1T s r s

s _I _t _a _bpe

dt t

dI __ __ _ 

1

0tT 1sn

 

_{ }

,

1

1 rs T

s _a _bpe

dt t

dI  __ _  _T __t__T

1 1sn

whereI_s

 

T₁ 0, for1sn.

Solving the differential equations, we get

 



 







   

    

 

, 1,

,

1, 0

, 1

1 1

1

n s T t T T

t D

n s T t e

D t I

s

t T s

s

 

where r s 1T.

s a bpe

D __ _ 

The different costs incurred during the planning horizon are as follows: (i) Total ordering cost:

 n T r rT

H T T A Ae Ae

A 1

1)

,

( _ _ _ _ 



1 1 rH

rT

e A

e  

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(ii) Total purchasing cost:

 

  



 



 



1 1

1 0 1 2

( , ) 0 rT 0 T r n T 0 T r n T

H n

T T

C T T cI ce I a bp dt ce  I a bpe  dt



   

  _   _  _   _





  













_

  

 

  

    

 

  

  

1 1 1

1 1

1

1 ₂ 1 2 ₂ 2

2

1

rT rT rH rT

rT rH rT

rH rT

rH T

e e bp e

e a T T c e

e bp e

e a e

c 



(iii) Total holding cost:

 

  _

 

    

 1 1  1 _

0 1

1

0 0 0

1)

,

( T T rn TT _n

t rT

H T T Ic I t dt Ice I t dt Ice I t dt

K 





_

  

 

  

 

1 1 1

1

1 2₂

1

2 1 rT

rH

rT rH T

e e bp e

e a T e

Ic _



(iv) Total deterioration cost:

 

  _

 

    

 1 1  1 _

0 1

1

0 0 0

1)

,

( rn TT n

T T

t rT

H T T c I t dt ce I t dt ce I t dt

D    





_

  

 

  

 

1 1 1

1

1 2₂

1

rT rH

rT rH T

e e bp e

e a T e

c _



(v) Total shortage cost:

 



I T e I T e I T



s T T

S rT rn T _n

H( , 1) 0  1  1 1





_

  

 

  



1 1 1

1

2 2

1 rT

rH

rT rH

e e bp e

e a T T s

(v) Total selling price:









 

_

  

 

  

  

 1 1

0 0

0 1)

,

( T rT T rT

H T T p a bp dt pe a bpe dt I T

P

 



 



 

   

 

  



  1  _

0 2

1

1 T

n T

n r T

n

r _a _bpe _dt _I _T

pe







_

  

 

  

    

 

  

 

  

1 1 1

1 1

1

2 2 2 1

2 2

1 rT

rT rH

rT rT rH rT

rT rH

e e bp e

e a T T pe e

e bp e

e a pT

(vi) Total interest earned:

Since the inventory manager can earn revenue by selling his goods whenM T1, we have

IEH(T, T1) = total interest earned during (0,H)















      

 1 1

0 1 0 1

T T

rT rT

e

e a bp T t dt pI e a bpe T t dt

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 



 







  



  1 

0 1

1 1

... rn TT r n T

ee a bpe T t dt

pI , when M T1

= 0, when M T₁

            1 1 1 1 2 2 2 2 1 rT rH rT rH e e e bp e e a T

pI , when M T1

= 0, when M T1

(vii) Total interest payable:

 

     

 1 1  1 _

1 1 1 0 1) , ( T M n rT n r T M rT r T M r

H T T cI I t dt cI e I t dt cI e I t dt

IP   









, 1 1 1 1

1 2₂

1 1                 rT rH rT rH M T r e e bp e e a M T e cI _

  whenM T1













_              1 1 0 1 T M T

r a bp T t dt a bp M t dt

cI

















_              1 1 0 1 T M T rT rT rT

re a bpe T t dt a bpe M t dt

cI  



 









 







             

  1  

1 0 1 1 1 1

... T M

T T n r T n r T n r

re a bpe T t dt a bpe M t dt

cI





_, 2 2 1 1 1

1 ₁2 ₁ 2

2 2         _            

 T M T

e e bp e e a

cI_r rH_rT rH_rT whenM T1.

Hence, the total profit made in the interval[0, ]is:











,



,for , for , , , 1 1 2 1 1 1 1 T M T T C T M T T C T T C M M M     where







₁





 





₁



2 [

, 12 ₁ ₂ ₁

1 1

1 1  1   

     _    

 _pT _pI T _e _T Ic c cI _e  _T _M

T T

C T r T M

e M _      

















1 1 1 1 1 1 1 1 1 1 ] 1 2 2 2 1 2 2 1 1                                   rT rH rT rT rH rT rT rH rT rT rH rT rH T e e A e e bp e e a pe c T T e e bp e e a e c T T s  













      _            _      

 _T Ic c

e T M T cI pT T T

CM r T 1 ₂

2 1 2 1 1 1

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 

























1 1 1 1 1 1 1 1 1 1 ] 1 1 2 2 2 1 2 2 1 1 1 1                                        rT rH rT rT rH rT rT rH rT rT rH rT rH T M T r e e A e e bp e e a pe c T T e e bp e e a e c T T s M T e

cI  

 



Lemma 1: Both C1M



T,T1



and C2M



T,T1



are decreasing function ofIands, for fixed

T1andT.

Proof: We have,









0 1 1 1 1 1 , 2 2 1 2 1

1 1 

                 rT rH rT rH T M e e bp e e a T e c I T T C _  









0 1 1 1 1 1 , 2 2 1 2 1

2 1 

                 rT rH rT rH T M e e bp e e a T e c I T T C _   and,









₀

1 1 1 1 , 2 2 1 1 1 _                 rT rH rT rH M e e bp e e a T T s T T C









0 1 1 1 1 , 2 2 1 1 2 _                 rT rH rT rH M e e bp e e a T T s T T C

Thus, the total profit decreases with increase in the inventory holding cost and the shortage cost.

Lemma2: C₁M



T,T₁



_and





1 2 T,T

CM _{are both concave in}_T

1, for givenT.

Proof: We have









0 ) 2 ( 1 1 1 1

, 2 ( )

2 2 2 1 1 1 2

1  

                  e M T r T rT rH rT rH M pI e cI I ce H T e e bp e e a T T T

C  

, since 0 1 1 1 1 2 2             rT rH rT rH e e bp e e

a and



( 2) (1 )



0

e M T r

T _I _cI _e _pI

ce 

.

Hence CM



T_,T



1

1 is a concave function ofT1, for givenT.

Similarly it can be shown thatCM



T_,T



1

2 is concave inT1, for givenT.

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It may be noted that the optimal values of (T₁,T)maximizing CM



T _,T



1 1 satisfy 0 ) , ( 1 1 1 _   T T T CM

and 1 ( , 1) _0

  T T T CM

, which reduce to the following equations:

                                 1 1 1 1 1 1 1 1 ) ( ) ( ) 2 ( 2 2 2 2 2 1 1 rT rH rT rH rT rT rH rT rT rH rT r e T M r e e bp e e a e e bp e e a pe c s p cI T pI e I e I

c  _ 

(2.1)











 









 



                

    ₂

2 2 2 2 2 2 1 1 1 1 2 1 1 1 rT rT rT rH rT rT rH rT rT rT rH rT rT rH rT e e e e e rbp e re e e re a pe c T T

-

₁





 





₁



2

[ 12 ₁ ₂ ₁

1 pI T  e 1  T  _ Ic  c_cI e 1  T M 

pT T r T M

e   _ _ _   s



T T1





1 _1



]

 c _eT











_           _rT rT rH rT rT rH re e e bp re e e a 2 2 2 2 2 2 1 1 1

1 _



_c_ _perT



_

             1 1 1 1 2 2 2 rT rT rH rT rT rH e e bp e e

a -





_

             1 1 1 1 2 2 2 1 _rT rT rH rT rT rH rT e e bp e e a T T pre



rT



rT rH re e e A ₂ 1 1    _] 1 2 2 1 1 1 1             rT rH rT rH e e bp e e

a =s. (2.2)

Similarly, the optimal values of (T₁,T)maximizing CM



T_,T



1

2 satisfy ( , ) 0

1 1 1 _   T T T CM

and 1 ( , 1) _0

  T T T CM

, which reduce to the following equations:

      _ _       

 M

r T

r M I cT I ce I I e

I c s

p  

 

] ( ) 2

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









  

    

  _ 

      

 

 _

 1 2 ₁ ₂ ₁

2

1 ₁

2 2

[ _cI T M T _e T1 _T Ic c _pT

r   _ _ 



e 1 



T1M



1



cIr T M _





















  

  

  

  

   



 rT

rT rH rT

rT rH

T _re

e e bp re

e e a e

c T T

s 2

2 2

1 2

1 1 1

1 1

1



 



cperT









_

  

 

  

    

 

  



   



1 1 1

1 {

1 1 1

1

2 2 2 1

2 2 2

rT rT rH rT

rT rH rT rT

rT rH rT

rT rH

e e bp e

e a pre T T e

e bp e

e a







 





 



}

1

2 1 1

2 1

1 1

2 2

2 

  

  



 



    

rT

rT rT

rH rT

rT rH

rT

rT rT

rH rT

rT rH rT

e

re e

e re bp e

re e

e re a pe c



rT



rT rH

re e

e

A ₂

1 1

 

 _] s

e e bp e

e

a rH_rT rH_rT _ 

  

 

  



 1

2 2

1 1 1

1 _. _(2.4)

4. Numerical Examples

Example 1:Suppose A= Rs. 250, c=Rs. 20, p=Rs. 24, s=Re 0.1, I= 0.1,Ie=.12,Ir=.15,

r=.02,=.02,M=0.1 year,H=4 years,a=2000,b= 0.1.

Using the software MATLAB, we get the following output:

For M T1,

T1opt= 0.226, Topt= 0.433 and maxC1M



T1,T



28879.11

For M T1,

T1opt= 0.098, Topt=0.443 and maxC2M



T1,T



28060.3.

Hence, the optimal solution isT1= 0.226, T= 0.433, and CM



T1,T



28879.11.

Example 2: Suppose A=250, c=Rs. 20, p=Rs. 25, s =Re 0.1, I= 0.1, Ie=.12, Ir=.15, r =

0.15,= 0.4,M=0.1 year,H=2 years,a=2000,b= 0.1.

MATLAB output is:

For M T1,T1opt= 0.01, Topt= 0.395 and maxC1M



T1,T



21067.82;

For M T1,T1opt= 0.002, Topt= 0.393 and maxC2M



T1,T



21073.55.

Hence, the optimal solution isT1= 0.002, T= 0.393, and CM



T1,T



21073.55.

Example 3:The following tables show the change in the optimal values ofT1andTwith

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Table1: p= Rs.25,r=.02,θ=.02.

H M=0.01 M=0.05 M=0.1 M=0.3 M=0.6

0.8 1.0 2.0 4.0 6.0

T1 T Profit T1 T Profit T1 T Profit T1 T Profit T1 T Profit

0.294 0.294 0.271 0.165 0.124

0.294 0.294 0.299 0.394 0.513

8413.3 8430.2 8522.5 9002.0 9460.2

0.298 0.298 0.298 0.206 0.165

0.298 0.298 0.298 0.388 0.505

8630.3 8647.6 8735.0 9100.9 9517.2

0.311 0.311 0.311 0.256 0.217

0.311 0.311 0.311 0.379 0.495

8857.7 8875.5 8965.2 9227.4 9590.8

0.423 0.423 0.423 0.423 0.420

0.423 0.423 0.423 0.423 0.434

9405.8 9424.7 9519.9 9714.2 9914.2

0.675 0.675 0.675 0.675 0.675

9730.6 9750.2 9848.7 10050 10256

Table2: p= Rs. 25,H=2,M= 0.01

r θ=0.02 θ=0.08 θ=0.15 θ=0.2 θ=0.4

0.02 0.05 0.08 0.10 0.12 0.15 0.20

0.271 0.217 0.163 0.125 0.085 0.019 0.010

0.299 0.309 0.327 0.344 0.364 0.396 0.463

8522.5 8816.1 9201.4 9516.1 9882.4 10539 11850

0.132 0.109 0.084 0.065 0.045 0.010 0.004

0.265 0.286 0.312 0.334 0.357 0.395 0.460

8210.5 8615.9 9091.2 9452.7 9854.1 10538 11866

0.083 0.069 0.054 0.042 0.029 0.010 0.004

0.255 0.278 0.307 0.330 0.355 0.395 0.46

8090 8537.3 9047.4 9427.4 9842.7 10537 11866

0.065 0.055 0.043 0.033 0.023 0.003 0.004

0.251 0.275 0.305 0.329 0.354 0.394 0.460

8046 8508.5 9031.3 9418.1 9838.6 10537 11866

0.036 0.030 0.024 0.018 0.013 0.002 0.004

0.246 0.271 0.302 0.326 0.353 0.393 0.460

7968.8 8457.7 9002.8 9401.6 9831.1 10537 11866

Table 3: r=.02,H=4,θ=.02

M P=22 P=24 P=26 P=28 P=30

0.01 0.05 0.10 0.30 0.50

0.077 0.114 0.161 0.350 0.539

0.721 0.717 0.713 0.706 0.716

3359.0 3381.5 3408.8 3506.0 3583.4

0.137 0.177 0.226 0.416 0.576

0.446 0.441 0.433 0.416 0.576

7058.1 7129.5 7219.8 7591.6 7831.7

0.194 0.236 0.287 0.430 0.596

0.361 0.355 0.345 0.430 0.596

10985 11114 11280 11837 12111

0.260 0.305 0.328 0.446 0.618

0.326 0.319 0.328 0.446 0.618

15051 15246 15483 16085 16393

0.322 0.327 0.341 0.464 0.642

19241 19468 19709 20336 20680

From the above tables we have the following observations: (i) As M increasesT1increases.

(ii) AsHincreasesT1decreases.

(iii) AsrincreasesT1decreases, butTincreases.

(iv) Asincreases, bothT1andTdecrease.

(v) Aspincreases, bothT1andTincrease

Conclusion

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References

1. Aggarwal, S.P. and Jaggi, C.K.: Ordering policies of deteriorating items under conditions of permissible delay in payments, J. Operat. Res.Soc., 46, (1995) 658-662.

2. Chung, K.J.: An algorithm for an inventory model with inventory-level-dependent demand rate,Comput. Operat. Res. (2003), 30, 1311-1317.

3. Datta, T.K. and Pal, A.K.: Effects of inflation and time-value of money on an inventory model with linear time-dependent demand rate and shortages, Eur. J.

Operat. Res.(1991), 52: 326-333.

4. Goyal, S.K.: Economic order quantity under conditions of permissible delay in payments,J. Operat. Res.Soc., 36, (1985) 335-338.

5. Hariga, M.A.: Effects of inflation and time-value of money on an inventory model with time-dependent demand rate and shortages, Eur. J. Operat. Res. (1995), 81, 512-520.

6. Hariga, M. and Ben-Daya, M.: Optimal time-varying lot-sizing models under inflationary conditions.Eur. J. Operat. Res.(1996), 89, 313-325.

7. Hou, K.L. and Lin, L.C.: A cash flow oriented EOQ Model with deteriorating items under permissible delay in payments, Journal of Applied Science (2009), 9(9), 1791-1794.

8. Hwang, H. and Shinn, S.W.: Retailer’s pricing and lot sizing policy for exponentially deteriorating product under condition of permissible delay in payments,Computer & Operations Research, 24, (1997) 539-547.

9. Jamal, A.M., Sarker, B.R. and Wang, S.: An ordering policy for deteriorating items with allowable shortage and permissible delay in payment, J. Operat. Res.Soc., 48, (1997) 826-833.

10. Maity, A.K.: One machine multiple-product problem with production-inventory system under fuzzy inequality constraint. Applied Soft Comput. (2010), 10.1016/j.asoc. 2009. 12.029.

11. Pal, M. and Ghosh, S.K. (2006): An inventory model with shortage and quantity dependent permissible delay in payment, Aust. Soc. Operat. Res. Bull. (2006), 25(3).

12. Pal, M. and Ghosh, S.K. (2007): An inventory model for deteriorating items with quantity dependent permissible delay in payment and partial backlogging of shortage,CSA Bulletin, 59, September & December.

13. Shah, N.H. and Shah, Y.K.: A discrete-in-time probabilistic inventory model for deteriorating items under conditions of permissible delay in payments, Inter. J. Syst. Sci.(1998), 29, 121-126.

14. Shinn, S.W., Hwang, H.P. and Sung, S.: Joint price and lot size determination under conditions of permissible delay in payments and quantity discounts for freight cost,Euro. J. Operat. Res., 91, (1996) 528-542.

15. Uthayakumar, R. and Geetha, K.V.: Replenishment policy for single item inventory model with money inflation,OPSEARCH(2009), 46(3), 345‐357. 16. Vrat, P. and Padmanabhan, G.: An inventory model under inflation for