Applied Mathematical Sciences, Vol. 8, 2014, no. 73, 3607 - 3618
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ams.2014.43178
Deteriorated Economic Production Quantity (EPQ)
Model for Declined Quadratic Demand with Time-
value of Money and Shortages
Bhanupriya Dash
Kamala Nehru Womens’ College
Bhubaneswar, India
Monalisha Pattnaik
Dept. of Business Administration
Utkal University, Bhubaneswar
India-751004
Hadibandhu Pattnaik
Dept. of Mathematics
KIIT University, Bhubaneswar, India
Copyright © 2014 Bhanupriya Dash, Monalisha Pattnaik and Hadibandhu Pattnaik. This is an
open access article distributed under the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
Abstract
A deteriorating inventory model using time-value of money with price
dependent declined quadratic demand is developed for a deterministic inventory
system. This study applied the discounted cash flows (DCF) approach for problem
analysis. The objective of this model is to maximize the net present value profit so
as to determine the optimal time period and order quantity. The numerical
analysis shows that an appropriate policy can benefit the retailer and that policy is
important, especially for deteriorating items. Finally, sensitivity analysis of opti-
3608 Bhanupriya Dash, Monalisha Pattnaik and Hadibandhu Pattnaik
mal solution with respect to the major parameters are also studied to draw some
decisions with managerial implications for competitive advantage.
Keywords: Economic Production quantity (EPQ), Declined quadratic demand,
Deteriorating items, time-value of money, shortages
1 Introduction
Inventory modelling is an important part of operations research, which
may be used in variety of problems. To make it applicable in real life situation
researchers are engaged in modifying existing models on different parameters
under various circumstances. Inventories play a major role in the united stages
economy and have been in excess of 22% of the nation’s gross national product
over the past few decades. As millions of dollars are tied up in inventories, proper
management of these inventories can prove to be very profitable. A major concern
of inventory management is to know when and how much to order or manufacture
so that the total cost per unit time is minimized. The total cost consists of
carrying, shortage, replenishment or setup cost and the purchase or production
cost. Usually the time value of money is not considered explicitly in analysing
inventory systems, although the cost of capital tried up in inventories is included
in the carrying cost. Most of the literature on inventory control and production
planning has dealt with the assumption that the demand for product will continue
infinitely in the future either in a deterministic or in a stochastic fashion. This
assumption does not always hold true. Inventory management plays a significant
role for production system in business since it can help companies reach the goal
of ensuring prompt delivery, avoiding shortages, helping sales at competitive
prices and so forth for achieving competitive advantage in the globe. However,
excessive simplification of assumptions results in mathematical models that do
not represent the inventory situation to be analyzed.
The classical analysis builds a model of an inventory system and
calculates the EOQ which minimize the costs satisfying minimization criterion.
One of the unrealistic assumption is that items stocked preserve their physical
characteristics during their stay in inventory for long run. Items in stock are
subject to many possible risks, e.g. damage, spoilage, dryness, vaporization etc.,
those results decrease of usefulness of the original one and a cost is incurred to
account for such risks of the product. The problem of deteriorating
inventory has received considerable attention in recent years. This is a realistic
trend since most products such as medicine, diary products and chemicals starts to
deteriorate once they are produced.
Most researches in inventory do not consider the time-value of money.
This is unrealistic, since the resource of an enterprise depends vary much on when
Deteriorated economic production quantity (EPQ) model 3609
it is used and this is highly correlated to the return of investment. Therefore,
taking into account the time-value of money should be critical especially when
investment and forecasting are considered. Wee (2001) developed a replenishment
policy for items with a price-dependent demand and a varying rate of
deterioration. Sarkar et al. (1994) assumed a finite replenishment model and
analysed the effects of inflation and time-value of money on order quantity when
shortages and allowed. Hariga (1995) extended the study to analyze the effect in
inflation and time-value of money of an inventory model with time-dependant
demand rate and shortages. Bose et al. (1993) developed an EOQ model for
deteriorating items with linear time-dependent demand rate and shortages under
inflation and time discounting. Abad (1996), Chang et al. (1999), Goyal et al.
(2001) Raafat (1991), Singh et al. (2013) and Chung et al. (1993) investigated
deterministic deteriorated economic order quantity models. Vandeltt et al. (1995)
proposed a simple economic order quantity for inventory with a short and
stochastic life time. Their approach was performed in the framework of the total
discounted cost criterion. Pattnaik (2011), (2012) and (2013) derived different
types of deterministic inventory models for deteriorating items in finite horizon.
The objective of this paper is to maximize the net present-value profit so
as to determine the optimal period and order quantity by using time-value of
money and price dependent declined quadratic demand allowing shortages.
Replenishment decision during planning horizon due to deterioration for
maximizing the net present value profit in response to change the market demand
and is also considered. Controlling the demand of customer through manipulation
of selling price for maximizing total net value profit. The major assumptions used
in the above research article summarized in Table-1. The remainder of paper
organized in section 2 is assumptions and notations for development of the model.
The mathematical model is developed in section 3. Optimization is given in
section 4. Numerical example is presented to illustrate the development of model
in section 5. The sensitivity analysis carried out in section 6 to observe the
changes in optimal solution. Finally section 7 deal with conclusion.
Table 1 Summary of Related Researches Authors
Publishe
d Year
Model
Structur
es
Demand Demand
Pattern
Deteriorati
on
Allowin
g
shortag
e
Time
value
of
Mone
y
Planni
ng
Model New form
Wee et
al. 2001
EOQ Price Declined
Linear
Yes
Weibull
Yes Yes Finite Profit DCF
approach
Pattnaik
2013
EOQ Constant
Deteriorati
on
Constant Yes
Constant
Wasting
No. No. Finite Profit Unit lost due
to
deterioration
and various
ordering cost
Hariga
1995
EOQ Time Decreas
e
No Yes Yes Finite Profit Inflation
Present
Paper
2014
EPQ Price Declined
Quardrat
ic
Yes
Weibull
Yes Yes Finite Profit DCF
approach
3610 Bhanupriya Dash, Monalisha Pattnaik and Hadibandhu Pattnaik
2 Assumption and Notation
The distribution of time until deterioration the item follows a tow-parameter
Weibull distribution.
Deterioration occurs as soon as the items received into inventory.
There is no replacement or repair of deteriorating items during the period
under consideration.
The demand rate is a decreasing quadratic function of selling price.
The replenishment rate is instantaneous; the order quantity and the
replenishment cycle is same for each period.
The system operates for a prescribed period of a planning horizon.
Shortages are completely back ordered.
The order quantity, inventory level, replenishment epoch and demand are
treated as continuous variables while the number of replenishments is
restricted to an integer variable.
Continuous cost compounding is implemented throughout the analysis.
Product transactions are followed by instantaneous cash flow.
S Per unit selling price of the items (s/unit) where m
lsc
d(s) Demand rate, 2)( smslsd ,
m,l Constant values where l>0 and m>0
C Per unit cost of items (s/unit)
)( 11 tI Inventory level at any time 1t : 110 Tt (Positive inventory)
)t(I 22 Inventory level at any time, 120 TTt (Negative inventory)
H Planning horizon
T Replenishment cycle
N Number of replenishment during planning horizon; T
HN
1T Time with positive inventory
1TT Time when shortage occurs
R Interest rate
Q The 2nd
, 3rd
, .... Nth replenishment size (units)
Im maximum inventory level
Cost per replenishment when t= 0 ($)
Per unit holding cost per unit time (s unit / unit time)
Per unit shortage cost per unit time (s / unit / unit time)
One distribution that has been used extensively in literature to pattern a varying
rate of deterioration is the Weibull distribution. The two parameter Weibull
density function is e .)( t-1 ttf (1)
here t is the time to deterioration, t>0, f(t) probability density function the scale parameter, t>0 and the shape parameter, 0 . This probability
Deteriorated economic production quantity (EPQ) model 3611
density function represent the hand inventory deterioration that may have an
increasing, decreasing or constant rate depending on the value of . When
1 , deteriorating rate increases with time e.g. Fish and vegetables. When
1 deteriorating rate decreases with time e.g.: light bulb where the initial
deterioration breakdown rate may be higher due to irregular voltages and
handling. When 1 deterioration rate is constant; e.g. electronic products.
Here, the two-parameter Weibull distribution is reduced to an exponential
distribution. The instantaneous rate of deterioration of the hand inventory is
given by 1 t .
3 Mathematical Model
The changes in inventory level against time are depicted in Fig. 1.
Fig. 1 The Inventory system when shortage is allowed.
The first replenishment lot size of Im is replenished at t=0. During the
period 1T , the inventory level decreases due to demand an deterioration until it is
zero at 1Tt . During the time interval, )TTT(T 122 , shortages occurred are
accumulated until Tt before they are backordered. The inventory system at any
time t can therefore be represented by the following equations.
1111
1
1
1
11 0),()()(
TtsdtItdt
tdI (2)
12
2
22 0 TTt),s(ddt
)t(dI
(3)
The first-order differential equation can be solved by using the boundary
conditions 021 )O(IIm,)O(I ,
110
11 0;)(Im
)(1
1
Tte
duesdtI
t
tu
(4)
T1
Q
0 t2
t1
T
T=H
3612 Bhanupriya Dash, Monalisha Pattnaik and Hadibandhu Pattnaik
12222 0 TTt;t)s(d)t(I (5)
Since 011 )T(I , one can derive from (4) the maximum inventory level as
1 1
0 0 0
1
1
0 )1(!)(
!)()(
T T
n
nn
n
nnu
mnn
Tsddu
n
usdduesdI
(6)
Assuming a very small value ).( 050 the approximate solution is
found by neglecting the second and higher – order terms of , one has
1)(Im
1
11
TTsd (7)
The total cost in this model includes the replenishment cost, material cost,
holding cost and shortage cost. The objective is to maximize the total profit when
the time-value of money with compounding interest rate is considered. The
detailed analysis of each cost function is given below.
Present – Value Sales Profit
During the period T1, the replenished inventory is being consumed due to
demand and deterioration. At t=T1, all the shortages during the period T-T1, are
backordered with an instantaneous cash transactions during sales, the present-
value sale profit is
1
0 0
21
11 1
11
)()( TTer
esdsdtsdsedtesdsR rt
rTT TT
rTrt
(8)
Assuming a very small r value 080.r approximate solutions can be
found by neglecting the second and higher-order terms of r.
TrTrT
rTT)s(dsR 1
22
1
2 (9)
Present-Value Ordering Cost
Since replenishment is each cycle is done at the start of each cycle, the
present-value replenishment cost is 10 cC 10)
Present Value Inventory Cost
Inventory occurs during period 1T , therefore, the present value inventory
cost during the period is
1
0
1
00
1
0
1
1
1
12
01
00
20
1112
!!1!)(
)()(
1
11
1
11
11
dtn
rt
n
t
nn
tTsdc
dtee
dueduesdcdtetIcC
n
nT
n
n
n
nnn
Trt
t
tu
Tu
Trt
H
Deteriorated economic production quantity (EPQ) model 3613
Assuming very small and r value (see above) the approximate solution
is found by neglecting second and higher-order terms of and r and terms
containing r . Consequently,
2162
21
31
21
2
TrTT)s(dCCH
(11)
Present-value shortage cost
The maximum shortage level )TT)(s(dIb 1 . All shortage during
)TT( 1 be completely backordered at T. The present value shortage cost for the
period is
]1[)(
))(()(
1
121
121
12
3
02
)(
230
2
)(
223
rTrT
TTtTr
TTtTr
S
erTrTer
sdc
dtetsdcdtetIcC
Assuming a very small r value (see above) approximate solutions are
founded by neglecting second and higher order terms of r, on has
31
211
231
23 332636
rTTTrTrTTTT)s(dC
CS 12)
Present-value item cost
Replenishment is done at t=0 and T; the replenishment items are
consumed by demand as deterioration during 1T . The present-value cost, Cp,
therefore includes item cost and deterioration cost, one has
1
02
TTrTmP dt)s(dCeCIC (13)
Assuming very small and r values (see above) the approximate solution
is found by neglecting the second and higher order terms of and r and the
terms containing r. Consequently,
1
1
11
2
TTrTrTT)s(CdCP
(14)
The first cycle present-value net profit is 1 PSHO CCCCR (15)
There are N cycles during the planning horizon. Some inventory is
assumed to start and end at zero, an extra replenishment at T=H is required to
satisfy the back orders of the last cycle in the planning horizon. Therefore, the
total number of replenishment = N+1 times; the first replenishment lot are = Im
and the 2nd
, 3rd
, Nth
replenishment lot size
1
02
TTdt)s(dIml
(16)
and the last or (N+1)th
replenishment lot size 1
02
TTdt)s(d (17)
The time-value of money affects all the replenishment periods and
therefore must be considered separately, the total net present-value profit for the
planning horizon is
3614 Bhanupriya Dash, Monalisha Pattnaik and Hadibandhu Pattnaik
rH
rT
rnTN
n
rHrnT
rHrTNrTrTrT
ece
eece
eceeeeNTt
11
1
0
1
1
132
11
1
1
.....1),,(
(18)
where, T=H/N and 1 is derived by substituting (8) to (14) into (15). The
optimization problem of this study can be formulated by maximizing (18) subject
to m
lSc and TT 10 .
4 Model Analysis
The following heuristic technique is derived the optimal S, T1 and N
values;
Step 1:Start by choosing a discrete variable N, where N is any integer number
equal or greater than 1;
Step 2:Take the partial derivatives of ),,( NTS with respect to 1T and S, and
equate the results to zero, the necessary conditions for optimality are
0),,(1
NTS
T and
0),,( 1
NTS
S
Step 3:For different integer N values, derive *T1 and *S from above two equations
substitute N,T,S **1 into (18) to derive NTS ,, *
1
* Step 4:Repeat step 2 and 3 for all possible N values within the lower and upper
bound until the maximum NTS ,, *
1
* is found. The *** N,T,S 1 and
**
1
* ,, NTS values constitute the optimal solution and they satisfy the
following conditions:
1,,0,, **
1
***
1
* NTSNTS (19)
Where **
1
***
1
***
1
* ,,1,,,, NTSNTSNTS substitute *** N,T,S 1
into (16) to derive the 2nd
, 3rd
, ..., Nth replenishment lot size. If the
objective function is concave, the following sufficient condition must
be satisfied;
02
2
2
1
22
1
STTS
(20)
and any one of the following 02
1
2
T
, 0
2
2
S
(21)
Deteriorated economic production quantity (EPQ) model 3615
Since the total net present-value profit for the planning horizon ~ is a very
complicated function due to high-power expression of the exponential function, it
is not possible to show analytically the validity of the above sufficient conditions,
a search procedure is used instead. The computation al results are show in the
following illustrative example.
5 Numerical Examples
Optimal replenishment and pricing policies for the maximum present-
value profit may be derived by using the methodology given in the preceding
sections; this will help managers to improve their replenishment and pricing
decisions. The replenishment cost, c1 is $80/order, the annual inventory cost c2 is
$0.6/unit/year, the annual shortage cost c3 is $1.4/unit/year. The unit item cost c is
$5/unit, the scale and the shape parameters of the deterioration rate are =0.05
and =1.5 respectively. The annual interest rate, r is 0.08, the yearly demand rate
d(s) is 1000-4S-S2 unit/year and the planning horizon, H is 10 years.
Table – 2 Optimal Values of the Proposed Model Model Iteratio
n
N S T1 T d(s) Q Profit =
Quadrati
c d(s)
58 25.9315
6
18.5235
5
0.264951
2
0.385630
4
570.379
8
220.083
3
74703.1
3
Linear
d(s)
83 57.5486
8
127.539
6
0.158247
8
0.173765
9
489.841
7
85.2273
3
591039.
6
% change - 121.93 588.53 -40.27 -54.94 -14.12 -61.27 691.18
For the given data the total net value profit for the planning horizon is
Rs.74703.13, the number of replenishment during planning horizon N, is 26, per
unit selling price of the items is Rs. 18.52, time with positive inventory T1, is
0.2649512, replenishment cycle T is 0.3856304, selling price dependent declining
quadratic demand d(s) is 570.3798 and the replenishment size Q is 220.0833. The
total number of order is therefore N+1=27. All the decision parameters are
compared with the other model related to the declined demand d(S) which is also
related linearly to the selling price. It is observed that demand rate d(s), order
quantity Q, time with positive inventory 1T and replenishment cycle T are more
than the compared model. But only the number of replenishment during planning
horizon, N, selling price S, unit selling prices and the total present value profit are less from the compared model. Fig.2 represents the relationship between the
unit selling price S and declined quadratic demand rate d(S). Similarly Fig. 3
depicts the mesh plot of T, T1 and total present value profit .
3616 Bhanupriya Dash, Monalisha Pattnaik and Hadibandhu Pattnaik
Fig. 2 Unit Selling Price S and Demand Rate d(S). Fig. 3 Mesh Plot of T, T1 and .
Table – 3 Sensitivity Study
Parameter Value Iteration N S T1 T d(s) Q Profit =
Change in
Profit
1.3 56 26.04150 18.82446 0.2610136 0.3840024 570.3421 219.2651 74680.85 0.029824720
1.7 57 25.83612 18.82296 0.2681136 0.3870550 570.4044 220.8428 74718.97 0.021203930
1.9 53 25.75498 18.82257 0.2707048 0.3882745 570.4204 221.5129 74730.31 0.036384017
R 0.06 42 23.43569 18.82469 0.2903966 0.4266995 570.3324 243.4881 75090.75 0.518805834
0.07 47 24.71444 18.82401 0.2767682 0.4046217 570.3605 230.9078 74891.83 0.252599857
0.09 66 27.09511 18.82325 0.2545707 0.3690703 570.3924 210.6424 74523.27 0.240766350
H 8 57 20.74525 18.82355 0.2649512 0.3856304 570.3798 220.0833 59746.50 20.021423470
11 89 28.52772 18.82355 0.2649512 0.3856304 570.3798 220.0833 82181.44 10.01070504
12 92 31.11787 18.82355 0.2649512 0.3856304 570.3798 220.0833 89659.76 20.021423470
c1 75 67 26.77670 18.82198 0.2567588 0.3734590 570.4451 213.1654 74835.87 0.177690011
85 93 25.16201 18.82507 0.2728813 0.3974246 570.3164 226.7853 74574.42 0.172295324
90 62 24.45739 18.82655 0.2805722 0.4088743 570.2548 233.2901 74449.40 0.339651096
c2 0.5 53 25.61640 18.82151 0.2756066 0.3903750 570.4649 222.8228 74756.55 0.071509721
0.8 91 26.50602 18.82707 0.2459799 0.3772728 570.2333 215.2610 74606.14 0.129833917
0.9 81 26.76853 18.82858 0.2374960 0.3735730 570.1701 213.1276 74561.97 0.188961292
c3 1.2 76 25.78575 18.82142 0.2595902 0.3878111 570.4687 221.3617 74725.32 0.029704243
1.5 83 25.99868 18.82451 0.2674014 0.3846350 570.3399 219.5002 74692.94 0.013640660
1.7 61 26.12278 18.82624 0.2719040 0.3828076 570.2677 218.4303 74674.15 0.038793555
C 4 79 26.29748 18.44723 0.2706690 0.3802646 585.9108 222.9321 80442.99 7.683560247
6 59 25.51993 19.20582 0.2598668 0.3913507 554.3131 217.3319 69125.82 7.465965616
8 59 24.56800 19.98819 0.2514508 0.4070322 520.5196 211.9847 58479.41 21.717590680
0.04 52 25.83192 18.82325 0.2678795 0.3871179 570.3921 220.8820 74714.09 0.014671406
0.06 54 26.02814 18.82384 0.2621229 0.3841996 570.3679 219.3363 74692.42 0.014336748
0.08 54 26.21276 18.82438 0.2567432 0.3814937 570.3452 217.9960 74671.72 0.042046430
6 Sensitive Analysis
It is interesting to investigate the influence of major parameters
.,,,,,,, 321 ccccHr
N , S , and d(s) are insensitive to the parameter . T1 , T , Q and are
moderately sensitive to the parameter .
N, T1 , T and Q are moderately sensitive to parameter r. S and d(s) are
insensitive to r but is sensitive to the parameter r.
0 10 20 30 40 50 60 70 80-2.5
-2
-1.5
-1
-0.5
0
0.5x 10
4
s: Unit Selling Price
d(s): D
eclined Q
uadratic D
em
and R
ate, d(s)=
1000-4s-s
2
Deteriorated economic production quantity (EPQ) model 3617
N , T1 and T are moderately sensitive to parameter H but S , d(s) and Q are
insensitive to the parameter H and is sensitive to the parameter H.
N, S, 1T and d(s) are insensitive to the parameter C1 but T is moderately
sensitive and Q and are sensitive to the parameter C1.
N, S and d(s) are insensitive to the parameter C2 but T1 and T are moderately
sensitive to C2 and Q and are sensitive to the parameter C2.
N, S, and d(s) are insensitive to the parameter C3 but T1 and T, Q and are
moderately sensitive to the parameter C3.
N is sensitive, S, T1 and T and Q are moderately sensitive to parameter C. d(s)
and are sensitive to the parameter c.
N, S and d(s) are insensitive to the parameter . T1 , T , Q and are
moderately sensitive to the parameter .
7 Conclusion
In this paper, an EPQ model is introduced which investigates the optimal
replenishment quantity. Unit selling price, replenishment cycle, time with positive
invention the total value net profit with finite planning horizon for deteriorating
items. The model considers the impact of price dependant quadratic demand, and
shortages and varying rate of deterioration. The model can be used for electronics
and other luxury products which are more likely to have the above characteristics.
This paper provides a useful property for finding the optimal net present value
profit with finite planning horizon for deteriorating items. A new mathematical
model with decline quadratic demand is developed and compared to the other
EPQ model with decline linear demand and numerically. The economic
replenishment quantity Q* and net present value profit
* for the present model
were found to be more than that of the compared model respectively. Hence the
utilization of selling price dependent declined quadratic demand makes the scope
of application broader. Lingo 13.0 version software is used to derive the optimal
number of replenishment and unit price to maximize the total present value net
profit. Further, a numerical example is presented to illustrate the theoretical
results, and some observations are obtained from sensitivity analysis with respect
to the major parameters controlling the market demand through the manipulation
of selling price is an important strategy for increasing profit. This can be achieved
by using the joint optimal replenishment and pricing strategy developed in this
study. In the future study, it is hoped further incorporate the proposed model into
several situations such as stochastic market demand, fuzzy decision parameters,
partial back logging, and selling price.
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3618 Bhanupriya Dash, Monalisha Pattnaik and Hadibandhu Pattnaik
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Received: March 15, 2014
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