International Journal of Statistics and Systems
ISSN 0973-2675 Volume 12, Number 3 (2017), pp. 457-474
© Research India Publications
http://www.ripublication.com
An Optimal Replenishment Policy for Deteriorating
Items with Power Pattern under Permissible delay in
payments
Nalini Prava Behera
P.G. Department of Statistics, Utkal University, India.
Prof. Pradip Kumar Tripathy
P.G. Department of Statistics, Utkal University, India.
Abstract
In this paper, Economic order quantity (EOQ) model based for time dependent
deteriorating items with power demand pattern is presented. Demand is related
to shortage under permissible delay in payments. The optimal cycle time is
determined to minimize the total inventory cost. Furthermore, sensitivity
analysis of the optimal solution is studied with respect to changes in different
parameter values and to draw managerial insights of proposed model.
Keywords: Power Demand Pattern, Salvage Cost, Permissible Delay in
Payment, Partial backlogging.
AMS Mathematics Subject Classification (2010):90B05
INTRODUCTION:
The Economic Order Quantity (EOQ) model proposed by Harris [10] has been widely
used by enterprises in order to reduce the cost of stock. Due to the variability in
458 Nalini Prava Behera and Prof. Pradip Kumar Tripathy
economic circumstances, many scholars constantly modify the basic assumptions of
the EOQ model and consider more realistic factors in order to make the model
correspond with reality. One such modification is the inclusion of the deterioration of
items. The effect of deterioration is very important in many inventory systems.
Deterioration is defined as decay or damage such that the item cannot be used for its
original purpose. Food items, drugs, pharmaceuticals and radioactive substances are
example of items in which sufficient deterioration can take place during the normal
storage period of the units and consequently this loss must be taken into account while
analyzing the system.
During the past few years, many research papers dealing with deteriorating inventory
problems have appeared in various research journals. In this paper the deterioration
rate is time dependent. Inventory of deteriorating items first studied by Whitin [20],
he considered the deterioration of fashion goods at the end of prescribed storage
period. Ghare and Schrader [7] extended the classical EOQ formula to include
exponential decay, wherein a constant fraction of on hand inventory is assumed to be
lost due to deterioration. Covert and Philip [4] then Ghare and Schrader’s model for
variable rate of deterioration by assuming two parameter weibull distribution
function.
In traditional EOQ model, it is assumed that buyer must pay for the items purchased
as soon as the items are received. But supplier permits the buyer a period of time (say
credit period), to settle the total amount owed to him. Usually, interest is not charged
for the outstanding amount if it is paid within the permissible delay period. However,
if the payment is not paid within the permissible delay period, then interest is charged
on the outstanding amount under the previously agreed terms and conditions. The
extensive use of permissible delay has been addressed by Goyal [9] who developed an
EOQ model under the condition of permissible delay in payments. Chand and Ward
[2] studied Goyal’s model under assumptions of the classical EOQ model. Goyal’s
model was extended by Aggarwal and Jaggi [1] for deteriorating items. Chung [3]
presented discounted cash flow (DCF) approach for the analysis of the optimal
inventory policy in the presence of trade credit.
An Optimal Replenishment Policy for Deteriorating Items with Power Pattern… 459
Backlogging occurs due to shortages. Sometimes, researchers assumed partial
backlogging while others considered full backlogging. In reality, if all customers are
prepared to wait until the arrival of the next order, then it is called completely
backlogged else, all the customers leave the system. However, in certain situations,
some customers will be able to wait for the next order in order to satisfy their
demands during the stock out period, while others do not wish to or cannot wait,
hence they meet their demands from other sources (the partial backlogging case).
Dye [6] have developed a optimal selling price and lot size with a varying rate of
deterioration and exponential partial backlogging. Skouri and Papachristors [11] have
developed an inventory model with deteriorating items, time-varying demand, linear
replenishment cost and partially time varying backlogging. Sushil and Rajput [17]
have introduced a Partially Backlogging Inventory Model for Deteriorating Items with
Ramp Type Demand Rate. Dave and Patel [5] proposed an EOQ model for time
proportional demand with constant deterioration. Geol and Aggarwal [19] formulated
an order level inventory system with power demand pattern for deteriorating items.
Datta and Pal [18] proposed an order level inventory system with power demand
pattern for items with variable rate of deterioration.
Giri et al. [8] developed an inventory models with time dependent deterioration.
Mishra and Shah [12] have developed an inventory management of time dependent
deteriorating items with salvage value. Rajeswari and Vanjikkodi [13] proposed an
inventory model for items with two parameter weibull distribution deterioration and
backlogging. Mohan and Venkateswarlu [15] have discussed an inventory
management model with quadratic demand, variable holding cost with salvage value.
With this motivation, this paper attempts to investigate an EOQ model assuming the
existence of a suitable power demand pattern, time dependent deterioration and time
dependent holding cost. The salvage value is associated to the deterioration units.Only
deterministic case of demand is considered. Shortages are allowed. Permissible delay
in payments is also considered for the optimal total cost. Suitable numerical example
and sensitivity analysis is also done.
460 Nalini Prava Behera and Prof. Pradip Kumar Tripathy
Table 1. Major characteristic on inventory models on selected areas:
Authors
&Publication
year
Deterioration Varying
Demand
Backlogged
Allowed
Permissible
Delay in
Payments is
allowed
Dave & Patel
(1981)
Constant Time
Proportional
No No
Geol & Aggarwal
(1981)
Constant Power
Demand
No No
T.K. Datta et.al.
(1988)
Variable rate of
Deterioration
Power
Demand
Yes No
Giri et.al. (1996) Linear Time
Varying
No No
Skouri &
Papachristos
(2002)
Constant Time varying Partial No
Dye C. (2007) Variable rate of
Deterioration
Exponential Partial No
Poonam Mishra
et.al (2008 )
Weibull Constant No No
N. Rajeswari et.al.
(2012)
Weibull Power
Demand
partial No
Roy & Chaudhuri
(2012)
Constant Stock
Dependent
No No
Venkateswaralu R.
et. al (2013)
Weibull Quadratic No No
Sushil & Rajput
(2015)
Constant Ramp type partial No
Present Paper
(2017)
Time Dependent Power
Demand
Partial Yes
An Optimal Replenishment Policy for Deteriorating Items with Power Pattern… 461
ASSUMPTIONS AND NOTATIONS:
To develop the mathematical model the following assumptions and notations are
being made:
Assumptions:
(i)The inventory consists of only one type of item.
(ii) The demand up to time t is assumed to be n
Td
1
1
, where d is the demand size
during the fixed cycle time T and n (0 < n < ) is the pattern index. n
nn
nT
dttD1
1
is
the demand rate at time t. Such pattern in the demand rate is called power demand
pattern.
(iii) A variable fraction 𝜃(t) of the on hand inventory deteriorates per unit time. In the
present model, the function 𝜃(t) is assumed in the form
tt 0 ; 10 0 , t > 0
(iv) The demand rate is deterministic and constant.
(v) The lead time is zero.
(vi) The planning horizon is infinite.
(vii) The holding cost is time dependent i.e tth , where 0 .
(viii) Shortages are allowed and demand is backlogged at the rate of
.1
1
tT The
backlogging parameter is a positive constant and (T-t) is the waiting time
.1 Ttt
(ix) During the trade credit period, M, the account is not settled; generated sale
revenue is deposited in an interest bearing account. At the end of the period, the
retailer’s pays off all units bought, and start to pay the capital opportunity cost for the
items in stock.
462 Nalini Prava Behera and Prof. Pradip Kumar Tripathy
Notations:
t : Inventory level at time t.
tQ : Order quantity at time t=0.
A: Ordering cost per order.
P: The purchasing cost per unit.
S: The selling price per unit, with S > P.
T: The length of order cycle.
2C : The deterioration cost per unit per year.
3C : The Shortage cost for backlogged per unit per year.
4C : The unit cost of lost sales per unit.
10 : The salvage value associated with deteriorated units during a cycle
time.
eI : The interest earned per dollar per year, where ce II .
cI : The interest charged in stock by the supplier.
M: Trade credit period.
1t : Length of time in which the inventory has no shortage.
TC: The total cost of the system.
Formulation and Solution of the Model:
The inventory system is developed as follows: Q units of items arrive at the inventory
system at the beginning of each cycle. The inventory level is dropping to zero owing
to demand and deterioration during the time interval 1,0 t . Finally, a shortage occurs
due to demand and partial backlogging during the time interval Tt ,1 .
Based on the above description, during the time interval 1,0 t , the differential
equation representing the inventory status is given by
An Optimal Replenishment Policy for Deteriorating Items with Power Pattern… 463
tDtt
dttdI
, 10 tt (1)
With the boundary condition 01 t and QI 0 the solution of (1) is
nn
nnnn
n
ttn
tttT
dt1212
1
011
1
2
01 12221
(2)
and the order quantity is
n
nn
n
tn
tT
dQ12
1
01
11 122
(3)
During the second interval Tt ,1 , shortage occurred and the demand is partially
backlogged.
That is, the inventory level at time t is governed by the following differential
equation:
,1 tT
tDdt
tdI
Ttt 1 (4)
With the boundary condition 01 t , the solution of (4) is
n
nnn
nn
n
ttn
ttTT
dt1
1
11
1
1
1 11
(5)
Ordering cost per cycle time is given by
OC = A (6)
464 Nalini Prava Behera and Prof. Pradip Kumar Tripathy
The deteriorating cost DC during the period [ 0,T] is given by
DC= 1
0
t
dttDQ
DC=
21
11
0
2
122
n
n
tnT
dC
(7)
The holding cost HC during the period [0,T] is given by
HC = 1
0
t
dttt
=
41
1
041
10
21
11 122142
1
1428
1
122
1nnn
n
tnn
ntnnt
nn
T
d
(8)
The salvage cost SV during the period [0,T] is given by
SV =
1
0
t
dttDQ =
21
11
0
2
122
n
n
tnT
dC
(9)
The total amount of shortage cost during the period Tt ,1 is given by
SC = T
t
dttC1
3
=
21
21
1
21
1
11
1
1
1
11
2
1
1
1
13
121
111
nn
nnnnn
n Ttnn
n
tTtn
tTn
nn
TntTTtTt
T
dC
(10)
An Optimal Replenishment Policy for Deteriorating Items with Power Pattern… 465
The amount of lost cost during the period (0, T) is given by
LC =
n
nn
T
t nT
dttT
C1
1
4
11
11
=
11
1
11
1
1
14 ntTt
nnT
T
dCnn
nnn
n
(11)
The total average cost of the system per unit time is given by
12
11
,
0,
tMTCtMTC
TC (12)
Where 1TC and 2TC are discussed as follows.
Case 1: 10 tM
In this case the length of delay in payment (M) is less than equal to the period with
positive inventory 1t . The retailers can sale units during (0, M) at a sale price (S) per
unit which he can bring an interest rate eI per unit per annum in an interest bearing
account. So the total interest earned during (0, M) is
Mn
n
e
n
nn
e MnT
dPItdt
nT
dtSIIE0
11
11
1
1
1
(13)
466 Nalini Prava Behera and Prof. Pradip Kumar Tripathy
During 1,tM , the supplier will charge the interest to the retailer on the remaining
stock at the rate cI per unit per annum. Hence total interest charges payable by the
retailer during
1,tM is
1
11
2
2
2111
21
21
1
22
1
1
1
11
1
11
1
1
11
1
nMt
n
MttMtn
nn
tMtt
T
dPIdttIPIIC
nn
nnnn
n
ct
Mc
(14)
So, the total variable cost per unit time is
SVIEICLCSCDCHCOCT
TC 111
1 (15)
21
11
0
2
11
1
21
21
1
22
1
1
1
11
11
11
1
1
1
11
11
1
11
14
21
21
1
21
1
11
1
1
1
11
2
1
1
1
13
21
11
0
2
41
1
041
10
21
11
11
1221
1
11
2
2
2
111
11
1211
11
122
122142
1
1428
1
122
1
1,
n
n
n
n
e
nn
n
nnn
n
c
nn
n
n
nnnn
nnn
n
n
n
nnn
n
tnT
dCM
nT
dPI
nMt
nMt
t
Mtn
nn
tMtt
T
dPInt
Ttn
nT
T
dC
Ttnn
ntTtn
tTn
nn
TntTTtTt
T
dCtnT
dC
tnn
ntnnt
nn
T
dA
TTtTC
An Optimal Replenishment Policy for Deteriorating Items with Power Pattern… 467
The solutions for the optimal values of t1 and T can be found by solving the
following equations simultaneously.
0
,
1
11 dt
TtdTC and
0
,11 dT
TtdTC. (16)
Provided
0,
2
1
11
2
dt
TtTCd and
0
,2
11
2
dT
TtTCd (17)
Case 2: 1tM
In this case, the period of delay in payment (M) is more than period with positive
inventory 1t . The retailer earns interest on the sales revenue up to the permissible
delay period and no interest is payable during the period for the item kept in stock.
Interest earned for the time period (0, T) is
1
1
1
1
11
0
11121
1
nn
ne
t
e tn
nMtnT
dSItDttMdttDSIIE (18)
Here, the interest charges is zero i.e. 02 IC (19)
So, the total variable cost per unit time is
SVIEICLCSCDCHCOCT
TC 222
1 (20)
468 Nalini Prava Behera and Prof. Pradip Kumar Tripathy
21
11
0
2
11
1
1
11
11
11
1
11
14
21
21
1
21
1
11
1
11
1
11
2
1
1
1
13
21
11
02
41
104
1
10
21
11
12
122111
1211
11
122
122142
1
1428
1
122
1
1,
n
n
nn
ne
nn
n
n
nnnn
nnn
n
n
n
nnn
n
tnT
dCt
nnMt
nT
dPIntTt
nnT
T
dC
Ttnn
ntTtn
tTn
nn
TntTTtTt
T
dCtnT
dC
tnn
ntnnt
nn
T
dA
TTtTC
The solutions for the optimal values of t1 and T can be found by solving the
following equations simultaneously.
0
,
1
12 dt
TtdTC and
0
,12 dT
TtdTC. (21)
Provided
0,
2
1
12
2
dt
TtTCd and
0
,2
12
2
dT
TtTCd
(22)
Numerical Example:
Case 1: 10 tM
Consider an inventory system with the following data: In appropriate units
25.0,12.0,13.0
,08.0,15.0,30,8.0,1,30,01.0,15,10,35,30,100 432
e
c
IMIPndCCCPA
Then we obtained the optimal values as 9041.0*
1 t , 2628.1* T and
848.165,*
11 TtTC
An Optimal Replenishment Policy for Deteriorating Items with Power Pattern… 469
Case 2: 1tM
Consider an inventory system with the following data: In appropriate units
25.0,12.0,13.0
,22.0,15.0,30,17.0,1,30,01.0,15,10,35,30,100 432
e
c
IMIPndCCCPA
Then we obtained the optimal values as 7894.0*
1 t , 2212.2* T and
293.225,*
12 TtTC
Sensitivity Analysis:
On the basis of the data given in example above we have studied the sensitivity
analysis by changing the following parameters one at a time and keeping the rest
fixed.
Table 2: Case 1: (0 ≤ M ≤ 𝒕𝟏)
Parameter % change 1t T TtTC ,11
+50 1.2664 0.4967 637.433
+25 1.5651 2.2388 333.770
0 0.9041 1.2628 168.848
-25 0.9084 1.2723 164.538
-50 0.9130 1.2825 163.155
+50 0.8884 1.2316 170.724
+25 0.8960 1.2466 168.342
0 0.9041 1.2628 168.848
-25 0.9124 1.2787 163.347
-50 0.9214 1.2959 160.683
M +50 0.9313 1.3310 157.909
+25 0.9171 1.2962 161.935
470 Nalini Prava Behera and Prof. Pradip Kumar Tripathy
0 0.9041 1.2628 168.848
-25 0.8917 1.2291 169.853
-50 0.8802 1.1965 173.870
+50 1.6164 1.7381 226.117
+25 1.4626 1.0802 211.377
0 0.9041 1.2628 168.848
-25 1.4775 0.9299 162.147
-50 2.1210 0.9585 156.434
+50 1.2648 0.9050 165.572
+25 1.2638 0.9043 165.781
0 0.9041 1.2628 168.848
-25 1.2618 0.9037 165.972
-50 1.2608 0.9032 166.125
Table 3: Case 2: M > 𝒕𝟏
Parameter % change 1t T TtTC ,11
+50 0.4055 2.1366 343.098
+25 0.5259 2.4027 336.453
0 0.7894 2.2212 225.293
-25 0.5626 2.3963 323.851
-50 0.6013 2.4159 313.749
+50 0.5488 2.4247 337.714
+25 0.5531 2.4117 331.804
0 0.7894 2.2212 225.293
-25 0.5614 2.3815 328.819
-50 0.5654 2.3687 322.406
M +50 0.7871 2.4287 245.827
+25 0.6706 2.4139 290.804
An Optimal Replenishment Policy for Deteriorating Items with Power Pattern… 471
0 0.7894 2.2212 225.293
-25 0.4511 2.3793 357.478
-50 0.3594 2.3616 384.331
+50 0.2789 2.7351 452.149
+25 0.3956 2.1025 342.264
0 0.7894 2.2212 225.293
-25 0.1508 2.7883 396.211
-50 0.1348 2.8561 308.465
+50 0.5589 2.3970 325.133
+25 0.5581 2.3971 325.403
0 0.7894 2.2212 225.293
-25 0.5565 2.3974 325.957
-50 0.5557 2.3975 326.227
DISCUSSION:
From the Table 2 we observed that
Decrease in results in decreases in inventory periods, decrease in total cost
per unit time.
Decrease in results in increases in inventory periods, decrease in total cost
per unit time.
Decrease in results in decreases in inventory periods, increase in total cost
per unit time.
Decrease in results in decreases in inventory periods, decrease in total cost
per unit time.
Decrease in results in decreases in inventory periods, marginal increases in
total cost per unit time.
472 Nalini Prava Behera and Prof. Pradip Kumar Tripathy
From the Table 3 we observed that
Decrease in results in increases in inventory periods, decrease in total cost
per unit time.
Decrease in results in increases in inventory periods, decrease in total cost
per unit time.
Decrease in results in decreases in inventory periods, increase in total cost
per unit time.
Decrease in results in increases in inventory periods, decrease in total cost
per unit time.
Decrease in results in increases in inventory periods, marginal increases in
total cost per unit time.
CONCLUSION
In this paper, we have developed a deterministic inventory model for time
proportional deteriorating items with associated salvage value. Shortages are allowed.
It is assumed that the retailer generates revenue on unit selling price which is
necessarily higher than the unit purchase cost when a power demand pattern has been
assumed with demand rate. The effect of delay period offered by the supplier to
retailer is analysed. It has been observed from the sensitivity analysis that decreases in
delay period M results in increase in total inventory cost. Decrease in salvage value
result in marginal increase in total inventory cost.
ACKNOWLEDGEMENT
The research work is supported by DST INSPIRE Fellowship, Ministry of Science
and Technology, Government of India, and P.G. Department of Statistics, Utkal
University, India.
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An Optimal Replenishment Policy for Deteriorating Items with Power Pattern… 473
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474 Nalini Prava Behera and Prof. Pradip Kumar Tripathy
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