International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 3, March 2013
ISSN 2319 - 4847
EFFECT OF PERMISSIBLE DELAY ON
TWO-WAREHOUSE INVENTORY MODEL
FOR DETERIORATING ITEMS WITH
SHORTAGES
1
Dr. Ajay Singh Yadav, 2Ms. Anupam Swami
1
Assistant Professor, Department of Mathematics, SRM University NCR Campus, Ghaziabad, U.P
2
Assistant Professor, Department of Mathematics, Govt. Degree College, Sambhal, U.P
ABSTRACT
In this paper we developed an inventory system with the effect of permissible delay in payments and stock dependent demand.
The occurrences of shortages are natural phenomenon allowed in inventory. Therefore, shortages are occurring with partial
backlogging. Backlogging rate is taken as waiting time for the next replenishment. Holding cost is variable and it is linear
increasing function of time. Numerical example is presented to illustrate the model and the sensitivity analysis of the optimal
with respect to parameters of the system is also carried out.
1. INTRODUCTION
In today's business transactions, it is frequently observed that a customer is allowed some grace period before settling
the account with the supplier or the producer. The customer does not have to pay any interest during this fixed period
but if the payment gets beyond the supplier will charge the period interest. This arrangement comes out to be very
advantageous to the customer as he may delay the payment till the end of the permissible delay period. During the
period he may sell the goods, accumulate revenues on the sales and earn interest on that revenue. Thus, it makes
economic sense for the customer to delay the payment of the replenishment account up to the last day of the settlement
period allowed by the supplier or the producer. This concept is known as permissible delay in payments. Goyal (1985)
was the first to develop the economic order quantity under conditions of permissible delay in payments. Author has
assumed that the unit selling price and the purchase price are equal. The unit selling price should be greater than the
unit purchasing price. Aggarwal and Jaggi (1995) developed ordering policies of deteriorating items under permissible
delay in payments. The demand and deterioration were consumed as constant. Jamal et al. (1997) developed a model
to determine an optimal ordering policy for deteriorating items under permissible delay of payment and allowable
shortage. Different facets of the permissible delays in payment are discussed, and this generalized model exhibits a set
of solutions that reduces to an existing model. Kun-Jen Chung (1998) discussed the economic quantity under
conditions of permissible delay in payments. Jamal et al. (2000) presented optimal payment time for a retailer under
permitted delay of payment by the wholesaler. The wholesaler allowed a permissible credit period to pay the dues
without paying any interest for the retailer. In the study, a retailer model was considered with a constant rate of
deterioration. Dye (2002) developed a deteriorating inventory model with stock-dependent demand and partial
backlogging. The conditions of permissible delay in payments were also taken into consideration. Chung and Liao
(2004) deals the problem of determining the economic order quantity for exponentially deteriorating items under the
conditions of permissible delay in payments. In addition, the objective function is modeled as a total variable costminimization problem. Teng et al. (2005) developed various EOQ models for a retailer when the supplier offers a
permissible delay in payments. In this paper, they complement the shortcoming of the previous models by considering
the difference between the selling price and the purchase cost. Soni et al. (2006) formulate optimal ordering policies for
the retailer when the supplier offers progressive credit periods to settle the account. The objective function to be
optimized is considered as present value of all future cash-out-flows. Singh, S.R. and Singh, T.J. (2008) developed the
perishable inventory model with quadratic demand, partial backlogging and permissible delay in payments. Soni, H. et
al. (2008) developed a mathematical model to formulate optimal ordering policies for retailer when demand is partially
constant and partially dependent on the stock, and the supplier offers progressive credit periods to settle the account.
This chapter proposed a two storage inventory model for deteriorating items with inventory level dependent demand.
Shortages are allowed and partially backlogged. Backlogging rate is taken as waiting time for the next replenishment.
The effect of permissible delay in payments is also taken in this study. Holding cost is variable and it is linear
Volume 2, Issue 3, March 2013
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 3, March 2013
ISSN 2319 - 4847
increasing function of time. Numerical example is presented to illustrate the model and the sensitivity analysis of the
optimal with respect to parameters of the system is also carried out, which is followed by concluding remarks.
2. ASSUMPTIONS AND NOTATIONS
The mathematical model is based on the following assumptions:
1. Lead-time is zero.
2. The initial inventory is zero.
3. The demand rate D (t) is deterministic and is a known function of instantaneous stock level; the function D (t)
is given by:
D
   I (t ),

  ,
 ,

0  t  t1
t1  t  t 2
t2  t  T
Where  > 0 and 0 <  < 1.
Replenishment rate is infinite and replenishments are instantaneous.
The owned warehouse (OW) has a fixed limited capacity of W units.
The rented warehouse (RW) has unlimited capacity.
The items of OW are started to consume when RW is empty.
The inventory costs (including holding cost and deterioration cost) in RW are higher than those in OW.
Shortages are permitted and the backlogging rate is defined to be 1/[1+δ(T-t)] when the inventory is negative.
The backlogging parameter δ is positive constant.
In addition, the following notations are used throughout this paper:
L1
represents an inventory system with an OW only.
L2
represents an inventory system with both OW and RW.
c0
the replenishment cost per order.
cd
deterioration cost per unit.
ch1
the inventory holding cost per unit per unit time in OW.
ch2
the inventory holding cost per unit per unit time in RW.
4.
5.
6.
7.
8.
9.
Note that implies assumption 6, ch2 +  cd > ch1 +  cd.
cs shortage cost per unit time.

the deterioration rate in OW, where 0 <  < 1.
the deterioration rate in RW, where 0 <  < 1.

S the highest stock level at RW and OW.
B the maximum shortage level.
P purchase cost per unit.
M permissible delay period in settled the accounts.
Ic interest charges per rupee per year.
Ie interest that can be earned on the sales revenue of units sold during the permissible delay period (Ie < Ic).
W storage capacity of OW, fixed constant and W < S.
I0(t) the inventory level in OW at any time t.
Ir(t) the inventory level in RW at any time t.
3. MATHEMATICAL FORMULATION
Here, we discuss the deterministic inventory model for deteriorating items with two-warehouse where shortages occur
at the end of the cycle. For a L2 system (see fig. 1(a)), at time t=0, a lot size of S units enters the L2 system in which W
units are kept in OW and S-W units in RW. The goods of OW are consumed only when RW is empty. During the time
interval [0, t1], the inventory S-W in RW decreases due to demand and deterioration and it vanishes at t=t1. In OW, the
inventory W decreases during [0, t1] due to deterioration only, but during [t1,t2] the inventory is depleted due to both
demand and deterioration . At time t=t2. The inventory in OW reaches zero and thereafter the shortages occur during
the time interval [t2, T]. The shortage quantity is supplied to customers at the beginning of the next cycle. The objective
of the inventory system is to determine the timings of t1, t2 and T in order to keep the total relevant cost per unit of time
as low as possible. As to a L1 system (see fig. 1(b)), the firm receives W units in OW at t=0. The inventory W depleted
due to both demand and deterioration, and reaches zero at t=t2, and thereafter the shortages occurs during [t2, T]. Note
that the L1 system here is, in fact, equivalent to the L2 system with t1=0.
Volume 2, Issue 3, March 2013
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Volume 2, Issue 3, March 2013
ISSN 2319 - 4847
For a L2 system, the inventory level at RW during the time interval [0,t1] is depleted by the combined effect of demand
and deterioration, the inventory level at time t € [0,t1], Ir(t), is governing by the following differential equation:
d I r t 
dt
     I r  t     I r  T  , 0  t  t1
… (7.1)
with the boundary condition the Ir(t1)=0. Solving the differential equation (1), we have
I r t  

 e      t1  t   1  , 0  t  t1

  
… (7.2)
During the time interval [0,t1], as the demand is meet from RW, the stock at OW decreases due to deterioration only.
Thus, the inventory level at time t € [0,t1], I0(t) is governed by the following differential equation:
d I 0 t 
… (7.3)
   I 0  T  , 0  t  t1
dt
with the initial condition I0(0)=W. Again, during the time interval [t1,t2], the inventory level at OW is depleted by the
combined effect of demand and deterioration, the inventory level at time t € [t1,t2], I0(t), is governed by the following
differential equation:
dI 0  t 
dt
… (7.4)
     I 0  t  , t1  t  t 2
with the boundary condition I0(t2)=0. Solving the differential equation (7.3) and (7.4), we have
… (7.5)
I 0  t   W e   t , 0  t  t1
I 0 t  
    t2  t 
e
 1  , t1  t  t 2

 
… (7.6)
Due to continuity of I0 (t) at t=t1, if follows eq. (7.5) and (7.6), we have
I 0  t1   W e   t1 
    t 2  t1 
e
 1

 
… (7.7)
Furthermore, during the period [t2, T], the behavior of the inventory system can be described by
dI 0  t 
dt


, t2  t  T
1   T  t 
… (7.8)
with initial condition I0(t2)=0, we have
I 0 t   

In 1   T  t 2   In 1   T  t   , t 2  t  T

… (7.9)
From the equations (7.2), (7.5), (7.6) and (7.10), the total per cycle consists of the elements:
1.
Ordering cost per cycle = c0
2.
Holding cost per cycle in RW  t ( F  h t ) I  t  d t
1
r
0

3.
F
h
e (    ) t1  (    ) t1  1 
2 t1  (    ) t1 2
(   )2
2(   )2




Holding cost per cycle in OW
 t1
   ( H   t ) I 0  t  dt 
0
W
 H 
 
1  e
  t1
t2

t1

 ( H   t ) I 0 t  dt 
  t 2  t1 
  t 2  t1 
2
2


  2 ( e  t 2  t1     t 2  t1   1)      t2  t1 e   12  e  2  t 22  t12 



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Volume 2, Issue 3, March 2013
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Shortage cost per cycle  s T  I  t  d t
 0
4.
t2
s
  T  t 2   In 1    T  t 2  
2



The amount of deteriorated items in both RW and OW are D
And
D 0  I 0 0   
 t1   W  
t 2

r



2
e 
    t1
 1       t1


t 2
 t1
e 
    t1
5. Deterioration cost per cycle
  
 P  Dr  D0   P 
    
6.

2

 1       t1  W    t 2  t1  


Opportunity cost due to lost sale per cycle
T
OC  

  1 
t2


1

 dt 

 1    T  t   
  T  t
–
2
In  1    T  t 2  

Case I: when M  t2
In this situation, since the length of period with positive stock is larger then the permissible delay period, the buyer can
use the sale revenue to earn interest at an annual rate Ie in (0, t2). The interest earn IE1 is
t2
 t1

I E 1  P I e    t1  t     I  t   d t    t 2  t   d t 
t1
0

P I e
3
2

2         t1  t 2       
3
2    

… (7.10)

2
t 12  2  e 
    t1
 1  

   t1 
However beyond the permissible delay period, the unsold stock is supposed to be financial with an annual rate Ir and
interest payable is given by
t
P I r
… (7.11)
I P  P I r  I 0 t  d t 
 e  t  M   1    t 2  M  
2
2

M
2
Therefore total average cost per unit time is
O C  H C R W  H O O W  S C  O C  D C  IP  IE 1
T
1
F
h
(    ) t1
 {c 0 
e
 (    ) t1  1 
2 t1  (    ) t1 2
T
(   )2
2(   )2
T C 1  t1 , T  

W
H 
 


1  e
 s    

2
  t1
  
2
 T


(e
  t 2  t1 
 t t
    t2
t e   t 2  t1 
t2
1
e  2 1  t2 2
   t 2  t1   1)  
 1
 2 

 1 

2
  



2
2 

  
 t 2   In 1    T  t 2    P 
    
PI e

2         t  t 
2   
3
1
3

2
2

2
       t12  2  e 
2
e 
    t1
    t1

 1       t1  W    t 2  t1  


  1       t1  
PI r    t2  M 
e
 1    t2  M
2


 

… (7.12)
For minimizing the total relevant cost per unit time, the approximate optimal values of t 1 and T (denoted by t1* and T*)
can be obtained by solving the following equations:
 TC 1
 TC 1
… (7.13)
 0 and
0
 t1
T
which also satisfies the conditions:
 2 TC1
 2 TC1
|

0
and
|
0
*
*
t12 t1 ,T 
T 2  t1* ,T * 
  2TC    2 TC    2TC  2 
1
1
1
and 


 | * * 0
2
2 
t ,T
t1

  T
   t1 T    1 
Next by using the optimal values t1* and T*, the approximate optimal values of t2 (denoted by t2*) and the approximate
minimum total cost per unit time can be obtained from (13) respectively.
Case-II: when M>t2
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Volume 2, Issue 3, March 2013
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Since M>t2 the buyer pays an interest but earns interest at an annual rate Ie during the period (0, M), interest earns in
this case, denoted by IE2, is given by
t2
t2
 t1
 t1

IE 2  P I e    t 1  t     I( t )  d t    t 2  t   d t   M  t 2        I  t   d t    d t  


0
t1
t1
0


PIe
3
2
2
2
    t 1

2      t 1  t 2         t 1  2   2  e
  1      t 1 
3
2    

 M  t 2

  2      t 1
3
 t 2         t 12  2   2  e 
2
    t1
  1       t  
1
… (7.14)
Then the total average cost per unit time is
1
 O C  H C R W  H C O W  S C  O C  D C  IE 2 
T


1 
F

 e (    ) t1  (    ) t1  1   2 ( h  ) 2  2 t1  (    ) t1 2   H  W 1  e   t1   2 ( e  t 2  t1    t 2  t1   1) 
 c0 
T 
(   )2


t1 e   t 2  t1 
t22
t1 2   s     
   t2
1
e   t 2  t1 


 2 




  

2
2 

 2
2
T C 2 t1 , T



T
 t2
P I e

2


M
3


I n  1    T  t 2    P 
    


2
 

 t2
2 
  

3

3
  t1
t1  t 2
 t2

2
 
   

  


2
e 
2

2
    t1
 1    
t 12  2   2  e 
t 12  2   2  e  
 
   t1
t 1
 t1   W

 1   
 1   

 
t 2

 t1 

 t1 

 t1  
… (7.15)
For minimizing the total relevant cost per unit time, the approximate optimal values of t 1 and T (denoted by t1* and T*)
can be obtained by solving the following equations:
T C 2
T C 2
… (7.16)
 0 and
 0
 t1
T
which also satisfies the conditions:
 2T C 2
 2T C 2
| t * ,T *  0 a n d
|
 0
1 
 t 12
 T 2  t1* ,T * 
and
   2 T C    2 T C    2 T C  2 
2
2
2


 
  | * *  0
2
2
  T
   t1  T    t1 ,T 
   t1
Next by using the optimal values t1* and T*, the approximate optimal values of t2 (denoted by t2*) and the approximate
minimum total cost per unit time can be obtained from (7.15) respectively.
4. NUMERICAL EXAMPLES
To illustrate the results, we apply the proposed method to solve the following numerical example:
Let α = 350,
β = 0,
co = 60,
ch1 = 8,
ch2 = 10,
W = 100, γ = 0.05,
θ = 0.06,
cs = 3,
Ir = 0.15,
Ie = 0.12,
P = 68,
M = 0.31,
cd = 0.25.
The optimal values of t1, t2, T, TC1 and TC2 have been computed. Computed results are displayed in table 7.1.
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Table 1:
Parameters
C
co
D
W
ch1
M ≤ t2
M > t2
t1 = 0.9098
t2 = 2.0908
T = 4.7461
TC1 = 900.594
T1 = 0.0120
t2 = 0.2955
T = 1.0546
TC2 = 524.826
Percentage
change in
parameters
- 20
- 10
10
20
- 20
- 10
10
20
- 20
- 10
10
20
- 20
- 10
10
20
- 20
- 10
10
20
Table 2: Sensitivity analysis:
M ≤ t2
TC1
Percentage
change in
total cost
5011.92
456.512
2078.87
130.834
464.977
-48.3699
865.727
-3.8716
898.106
-0.2762
899.35
-0.1381
901.838
0.1381
903.082
0.2762
6123.2
579.906
2403.49
166.879
517.727
-42.5127
255.567
-71.6223
368.554
-59.0765
477.076
-47.0264
2702.23
200.05
6885.66
664.569
460.655
-48.8498
504.971
-43.9291
1750.96
94.4229
3253.15
261.222
TC2
587.841
550.619
501.239
477.667
515.205
519.961
529.806
534.906
508.592
512.097
535.231
531.50
446.182
490.627
561.396
605.767
477.436
503.004
544.974
564.779
M > t2
Percentage
change in total
cost
12.0069
4.9146
-4.4941
-8.9857
-1.8331
-0.9269
0.9487
1.9206
-3.0931
-2.4253
1.9825
1.2716
-14.9847
-6.5162
6.9680
15.4224
-9.0296
-4.1580
3.8390
7.6125
5. OBSERVATIONS
1. From table 7.1 and table 7.2, it is observed that TC2 is always less then TC1 with respect to the change in every
parameter. This is due to in the second case M > t2. So, we have not paid any interest and we earn some interest.
2. As the purchasing cost (P) increases, the total cost is decreases in both cases.
3. As the ordering cost increases (c0), the total inventory cost is increases in both cases.
4. As the demand rate increases (D), the total inventory cost is decrease in both cases.
5. As the capacity of the own warehouse increases, the total inventory cost is also increases in both cases.
6. As the holding cost of own warehouse increases, the total inventory cost is also increases in both cases.
7. The total inventory cost is very sensitive with respect to W and very less effected by the variation of c0.
6. CONCLUSION
In this study an inventory system is developed for decaying items with two-warehouses and stock dependent demand.
Shortages are permitting in this model and partially backlogged. And backlogging rate is time dependent and it is
waiting time for the next replenishment. The conditions of permissible delay in payments and time dependent holding
cost are also taken into account. Holding costs and deterioration costs are different in OW and RW due to different
preservation environments. The inventory costs (including holding cost and deterioration cost) in RW are assumed to
be higher than those in OW. To reduce the inventory costs, it will be economical for firms to store the goods in OW to
the maximum level and after that the remaining goods store in RW, but clear the stocks in RW before OW. So that rent
of rented warehouse is minimum. From the viewpoint of the costs, decisions rules to find the optimal order cycle time t2
contains two cases:
(i) M ≤ t2
(ii) M > t2.
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
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Volume 2, Issue 3, March 2013
ISSN 2319 - 4847
Finally, a numerical example in Table 1 is studied to illustrate the theoretical results. From the above table 1 and 2, it is
observed that the total inventory cost TC2 is always less then TC1 with respect to the change in every parameter. This is
due to in the second case M > t2. So, we have not paid any interest and we earn some interest. So, we conclude that the
effect of permissible delay cannot be ignored.
Thus, this model incorporates some realistic features that are likely to be associated with some kinds of inventory. The
model is very useful in their retail business. It can be used for electronic components, fashionable clothes, domestic
goods and other products which are more likely with the characteristics above.
In future research on this problem, it would be of interest to add effect of more realistic demand rate in the model (e. g.
time-varying and stock-dependent demand patterns). On the other hand, the possible extension of this work may relax
the assumption of constant deterioration rate.
REFERENCES
[1] Aggarwal, S.P. and Jaggi, C.K. (1995): “Ordering policies of deteriorating items under permissible delay in
payments”, Journal of Operational Research Society (J.O.R.S.), 46, 658-662.
[2] Chung, K.J. (1998): A theorem on the determination of economic order quantity under conditions of permissible
delay in payments, Computers & Operations Research, 25, 1, 49-52.
[3] Chung, K.J. and Liao, J.J. (2004): “Lot sizing decision under trade credit depending on the ordering quantity”,
C.O.R., 31, 909-928.
[4] Dye, C.Y. (2002): “A deteriorating inventory model with stock dependent demand and partial backlogging under
conditions of permissible delay in payments”, Opsearch, 39(3&4), 189-200.
[5] Goyal, S.K. (1985): “Economic order quantity under conditions of permissible delay in payments”, J.O.R.S., 36,
335-338.
[6] Jamal, A.M.M., Sarker, B.R. and Wang, S. (1997): “An ordering policy for deteriorating items with allowable
shortage and permissible delay in payment”, J.O.R.S., 48, 826-833.
[7] Jamal, A.M.M, Sarker, B.R. and Wang, S. (2000): “Optimal payment time for a retailer under permitted delay of
payment by the wholesaler”, I.J.P.E., 66, 59-66.
[8] Soni, H. et al. (2006): “An EOQ Model For Progressive Payment Scheme Under DCF Approach”, Asia-Pacific
Journal of Operational Research, 23, 4, 509-524.
[9] Soni, H. and Shah, N.H. (2008): “Optimal ordering policy for stock-dependent demand under progressive payment
scheme”, E.J.O.R., 184 (1), 91-100.
[10] Singh, S.R. and Singh, T.J. (2008): “Perishable inventory model with quadratic demand, partial backlogging and
permissible delay in payments”, International Review of Pure and Applied Mathematics, 1, 53-66.
[11] Teng, J.T., Chang, C.T. and Goyal, S.K. (2005): “Optimal pricing and ordering policy under permissible delay in
payments”, I.J.P.E., 97, 121-129.
Dr. Ajay Singh Yadav has done M.Sc. in Mathematics and Ph.D. in “inventory Modelling, he has over 6
years experience in teaching Mathematics in defferent Engineering Colleges. Presently he is Assistant
Professor in SRM University NCR Campus Ghaziabad
Ms. Anupam Swami has done M.Sc ,M.Phil. in Mathematics and pursing Ph.D in “inventory Modelling,
she has over 5 years experience in teaching Mathematics in defferent Degree Colleges. Presently she is
Assistant Professor in Department of Mathematics, Govt. Degree College, Sambhal, U.P
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