Download Inventory models for deteriorating items with discounted selling price

Inventory models for deteriorating items with discounted
selling price and stock dependent demand
Freddy Andrés Pérez ( [email protected]) Universidad de los Andes,
Departamento de Ingeniería Industrial
Fidel Torres ( [email protected]) Universidad de los Andes, Departamento de
Ingeniería Industrial
Abstract
New families of inventory models are developed with pre- and post-deterioration discounts when
demand is initially stock dependent and then becomes constant. Shortages are allowed and
partially backordered depending on the waiting time for the next replenishment. Furthermore, the
time value of money is considered over a fixed planning horizon.
Keywords: Inventory, Deteriorating, Inflation
Introduction
Deteriorating inventory models accounting for partial back-ordering and financial aspects such
as pricing strategies, inflation and time value of money are well covered in the current literature;
however, there are some characteristics in the modeling of inventory systems with deteriorating
goods hardly studied. For instance, incorporating a known price discount is very common and
corresponds very well with the real world situations for deteriorating items, but a few authors
consider two deterioration phases associated to two subsequent price discounts.
In this sense, several authors over time have investigated different pricing strategies for
deteriorating inventory when shortages are allowed and partially backordered depending on the
waiting time (Abad 2001, Abad 2003, Dye 2012, Dye and Hsieh 2011, 2013, Dye et al. 2007,
Ghosh et al. 2011, Maihami and Abadi 2012, Maihami and Nakhai Kamalabadi 2012, Shavandi
et al. 2012, Soni and Patel 2012, Teng et al. 2007, Valliathal and Uthayakumar 2010) , as well as
inventory systems dealing with time value of money and shortage (Dye and Hsieh 2011, Berk et
al. 2009, Mirzazadeh et al. 2009, Abdul and Murata 2011, Chauhan and Kumari 2011, Gaur et al.
2011, Jaggi et al. 2011, Mishra et al. 2011, Valliathal and Uthayakumar 2011, Jain and Aggarwal
2012, Yang 2012, Ali et al. 2013, Bhunia et al. 2013, Taleizadeh and Nematollahi 2013, Yang
and Chang 2013). Most of the studies with exception of a few (Dye and Hsieh 2011, Berk et al.
2009, Valliathal and Uthayakumar 2011) do not endeavor to unify pricing decisions and
deterioration research streams with studies where shortages are allowed, and/or the time value of
money is taken into account. Berk et al. (2009) considered the presence of price-sensitive
renewal demand processes to maximize the discounted expected profit by determining the selling
1
prices dynamically for items with fixed lifetime, but shortages are not considered . Dye and
Hsieh (2011) developed an inventory model under inflation for deteriorating items with priceand stock-dependent demand and limited capacity. Meanwhile, Valliathal and Uthayakumar
(2011) presented a production lot-sizing model for deteriorating item under inflation by
considering a coordinated pricing and production decisions as well as price decision disregarding
the production cost. The major assumptions used in the existing works with pricing decision and
partially backordered depending on the waiting time have been summarized in Table 1.
Table 1- Summary of related literature for inventory models with partially backlogging
Time
Pricing decision
Time
Author(s) and year
Deterioration
Demand
value of
per inventory
horizon
money
cycle
Constant
(Abad 2003)
Price dependent
No
Infinite
Equals
(instantaneously)
Constant
(Ghosh et al. 2011)
Price dependent
No
Infinite
Equals
(instantaneously)
Two-phase
Time dependent
Time and price
(Dye 2012)
No
Finite
(instantaneously)
dependent
pricing
(Maihami and Abadi
Constant (nonTime and price
No
Infinite
Equals
2012)
instantaneously
dependent
(Maihami and Nakhai
Constant (nonTime and price
No
Infinite
Equals
Kamalabadi 2012)
instantaneously
dependent
(Valliathal and
Constant (nonTime and price
No
Infinite
Equals
Uthayakumar 2010)
instantaneously
dependent
(Dye and Hsieh
Constant
Stock level and
Yes
Finite
Dynamic pricing
2011)
(instantaneously) price dependent
(Dye and Hsieh
Constant
Price dependent
No
Infinite Two-phase pricing
2013)
(instantaneously)
Constant (non(Soni and Patel 2012)
Price dependent
No
Infinite
Equals
instantaneously
(Shavandi et al.
Constant
Prices
One pricing
No
Finite
2012)
(instantaneously)
dependent
decision Per item
Constant
(Teng et al. 2007)
Price dependent
No
Infinite
Equals
(instantaneously)
Time dependent
(Abad 2001)
Price dependent
No
Infinite
Equals
(instantaneously)
Time dependent
(Dye et al. 2007)
Price dependent
No
Infinite
Equals
(instantaneously)
(Valliathal and
Time dependent
Price and time
Yes
infinite
Equals
Uthayakumar 2011)
(instantaneously)
dependent
Two-phase
Constant (nonStock dependent
Present paper
Yes
Finite
instantaneously
and constant
price reduction
As a consequence, according to the present authors’ knowledge, no one has attempted to
introduce a temporary price reduction on selling price before the start of deterioration as well as
a discount on selling price as the deterioration starts in order to boost the inventory depletion rate
when inventory systems are dealing with both waiting time dependent backlog rate and
deteriorating inventory accounting for inflation and time value of money.
2
So, the current paper represents above issue in detail extending the model in Panda et al.
(2009). In what remains, this paper is organized in the following order. First, it formulates the
basic mathematical model with pre- and post-deterioration discounts on selling price. Second, the
special cases from the formulation of the basic model are submitted. Third, a numerical example
is presented to exemplify the model developed. And finally, the conclusion of this study is given.
Mathematical Model
In this paper, a deterministic inventory model with pre- and post-deterioration discounts on
selling price in order to investigate how much discount on selling price may be given to
maximize the profit over a finite horizon H is considered. The total time horizon H has been
divided into m equal parts of length
(i.e.,
). At the beginning of the replenishment
cycle the inventory level raises to . As time progresses it decreases due to instantaneous stock
dependent demand up to the time . After deterioration starts and the inventory level decreases
for deterioration and constant demand until it reaches zero level at . Ultimately, backorders are
accumulated from time to . So, the evolution of inventory level occurs according to the
following system of linear differential equations
( )
(1)
( )
( ))
(
(
=
)
(2)
( )
(3)
,
(4)
Where a > 0 is the initial demand rate independent of stock level, b > 0 is the stock
(
)
sensitive demand parameter, I(t) is the instantaneous inventory level at time t,
(
)
(
), is the effect of
discount over demand,
(
), is the effect of
(
)
discounted selling price over demand, is the deterioration rate (
), and
is the fraction rate of backordered items at time t with
. Note that the fraction of
shortages backordered is a decreasing function of time t, and it reflects the fact that less waiting
time implies more backordered items. Also note that
and are determined from prior
knowledge of the seller depending on the influence caused by the reduction rate of selling price
and the waiting time over demand rate in each case respectively.
Solving the differential equations with the initial and boundary conditions we get
( )
(
( )
(
( )=
( )
)
[
(
(5)
(
)
(
)
]
[
(
)
(
)
)
(6)
(7)
)
]
(8)
Now, at the point t = τ we have from Equations (6) and (7)
3
(
(
(
)
(
)
)
)
(9)
The present value of the holding cost and disposal cost during the entire horizon H are
( )
[ ∫
( )
∫
(
( ) ]
)∫
(10)
Where and are the respective holding and disposal costs per unit and m denotes the
number of replenishment periods during the time horizon H.
The remaining costs during the entire time horizon H are found as follow:
Let c the per unit purchase cost of the items. Because shortage during the last cycle is
replenished at time
, and shortage during the first replenishment cycle should be
backordered during the next replenishment cycle, the present worth of the purchasing cost is
given by
{∑
(
)
∑[
(
)]
}
[
(
(
)
)
]
(11)
Let be the shortage cost for backordered items. The present value of the total shortage
cost during the entire time horizon H is equal to
∑[
( )
∫
]
(12)
(
)
(
[
)
(
(
)
)
[
(
)
(
)
]]
(
)
Defining as the per unit shortage cost for lost items, the present value of the total
shortage cost of the items being lost in the entire time horizon H is given by
∑[
∫
(
[
)
] ]
(13)
[
(
(
)
)
(
)]
Defining s as the per unit selling price of the items. The present worth of the total sales
revenue in the entire horizon H can be found by equation (14).
4
[∫
( ))
(
[
(
(
)∫
( ))
(
(
)∫
(14)
)]]
Thus, the net present value of the system during the entire horizon H (including set up
) is
cost
(
(15)
)
On integration and simplification of the relevant costs, the net profit value during the
entire horizon H becomes (16). Where, PC, SC, LC are equal to equations (11), (12), (13)
respectively
(
[
)
,
(
(
)
)
)
( )
)∫
(
(
(
)]
(
)∫
( )
(16)
( )
∫
Where:
(
) (
∫
[ ( )]
∫
( )
)
(
;
(
)(
(
( )
)
*
(
(
(
(
)
)(
(
∫
) (
)
(
)
(
)
)
)
)
(
)
)
(
)
)+
Now, considering the constraints in which the pre- and post-deterioration discounts on
selling price is given in such a way that the discounted selling price is not less that the unit cost
of the product, the objective here is to determine the optimal values of
and
which maximize the function
constrained by
;
and
. Note that it’s very difficult to derive the results analytically. Thus, some of the numerical
algorithms for constrained nonlinear optimization must be applied to derive the optimal values
of
, and m. There are several methods to cope with constraint optimization problem
numerically. But here it’s uses The Differential Evolution Algorithm (Price et al. 2005) to derive
the optimal values of the decision variables.
5
Some special cases
Let
be the above model described. The following situations can be easily deduced from
equations (9) and (16) as follow:
 Model
with only post-deterioration discount on unit selling price (
)
In this case only discount on selling price will be given as soon as the deterioration starts. So, the
order quantity is obtained from equation (9) by Substituting
and
as
(
(
(
)
)
)
(17)
Also, from equation (16) the net present value of the system is found as
(
)
)
*(
(
(
)
(
)
) [(
(
)(
)
)+
)(
(
(
[
(
)
(
)(
)
(
)
)
(
)]
(18)
)]
Therefore, the goal here is to find the optimal values of
maximize the function
constrained by
and
equal to equations (11), (12), (13) respectively, bearing in mind that
and
to
. Where PC, SC, LC are
in (11) now.
 Model
)
for no discount on unit selling price (
In this model there is no pre-deterioration as well as no post deterioration discount on unit selling
price. So, the order quantity is found substituting
,
from Equation (9) as
( (
(
)
(19)
)
)
And from equation (16) the net present value becomes (20). Where PC, SC, LC are equal
to equations (11), (12), (13) respectively, taking into account that
in (11) now
(
)
[(
)
(
(
(
)]
) [(
)(
)
[
(
)(
(
)
(
)(
)
(
6
)
)]
)
(
)]
(20)
So, the goal in (20) is to find the optimal values of
function
constrained by
.
 Model
, and
to maximize the
for instant deterioration with post deterioration discount (
)
In this case pre-deterioration discount is no option because the items start to deteriorate as soon
as it’s received in the stock. So, from equations (9) the order quantity can be found by
substituting
as
(
(21)
)
Similarly, from equation (16) net present worth becomes (22). Where PC, SC, LC are
equal to equations (11), (12), (13) respectively, considering that
in (11) now.
(
)
*
(
(
)
)
(
)+
(22)
(
)(
)
*(
(
)
)+
Hence, the aim here is to determine the optimal values of
maximize the function
constrained by
and
 Model
with instant deterioration and without discount (
and
which
)
,
The initial inventory level can be obtained from equations (21) by substituting
as
(
)
(23)
So, from equation (22) the net present value becomes (24). Where PC, SC, LC are equal
to equations (11), (12), (13) respectively with
in (11). Thus, we have to determine
and from which the function (24) is maximized and meet
(
)
(
)
(24)
(
 Model
)(
)
*(
)
(
for fixed life time products with pre deterioration discount (
)+
)
In this scenario there is only pre-deterioration discount on selling price and cycle length must be
less than life time of items. So the order quantity (25) and net present worth (26) for fixed life
7
time products where
as follows
(
(
(
)
can be found from equation (9) and (16), by letting
)
)
*
(
(
(25)
(
(
)
)
)
) [(
(
(
)
(
)(
)+
)
(
)(
)
(
)
)]
(26)
(
(
)
(
) *(
)
(
(
(
)
)
)
)+
Thus, the aim is to determine
and
to maximize (26) constrained by
, Where PC, SC, LC are equal to (11),(12),(13) respectively,
bearing in mind that
in (11) now.
 Model
for fixed life time products with no discount (
)
Letting
, the initial inventory level and the net present value becomes (27)
and (28) from (25) and (26) respectively. So, the objective here is to found
and
to
maximize (28) constrained by
. Where PC, SC, LC are equals to (11),(12),(13)
respectively, taking in account that
in (11) now.
(
(
)
)
(
(27)
)
(28)
(
) [(
)(
)(
(
)
)
(
)]
Numerical example
In order to illustrate the proposed model, an example has been solved with the following
parameter values:
;
. As was mentioned it used
The Differential Evolution Algorithm (Price et al. 2005) to derive the optimal values of the
decision variables. For model with both pre- and post-deterioration discounts ( ), the predeterioration discount on unit selling price is 0.59 ( ) and discount starts at time
and continued to
.Then the product start to deteriorates. During this time in order to boost
inventory depletion rate, 59% discount is provided for remaining time of replenishment
cycle ( ). The net present worth is 3994 ( ). The optimal order quantity is 2494 ( ) and the
cycle length is 1.75( ). These results and the others subcases (
) are listed in Table 1.
8
As can be noted, the net present value in model
is greater than model
and model , but
there are combinations in the parameters where it can be different. So, a more deep analysis is
needed to know when it is useful for managers each models in order to maximize the profit when
the time value of money is taken into consideration.
Table 2- Optimal values of the decision variables, order quantity and profit of the models
Models
0.59
0.59
0.856602
1.75
1.75
2494.22
3994.08
0
0.777778 0.777778
75.4906
3615.75
1.75
1.75
266.17
1973.32
0.59
4.56344
7
2395.04
168 828
0.7
0.7
63.6662
91 409.9
0.59
0.340424
1.3
1.4
4070.68
828 408
1.3
1.4
177.221
1569.13
Conclusion
Panda et al. (2009) presented a practical inventory model with pre- and post-deterioration
discounts on selling price. The mathematical model is developed in order to investigate how
much discount on selling price may be given to maximize the profit per unit time when demand
is both stock dependent and constant, but the time value of money and shortage are not
considered. In this paper, that model has been extended to make it a little more applicable when
inventory systems are dealing with the inventory replenishment problem for partial backlogging
and time value of money.
Although this research represent an important contribution of existing inventory models for
deteriorating items with temporary price discounts , the model developed here can be further
improved by employing more elaborate inventory models. Potential topics for notable future
research include the incorporation of stochastic demand rate, multi-echelon inventory and multiitem approaches.
References
Abad, P.L. 2001. Optimal price and order size for a reseller under partial backordering. Computers & Operations
Research 28(1): 53–65.
Abad, P.L. 2003. Optimal pricing and lot-sizing under conditions of perishability, finite production and partial
backordering and lost sale. European Journal of Operational Research 144(3 ): 677–685.
Abdul, I. and Murata, A. 2011. Optimal production strategy for deteriorating items with varying demand pattern
under inflation. International journal of industrial engineering computations 2(3): 449-466.
Ali, S.S., Madaan, J., Chan, F.T.S. and Kannan, S. 2013. Inventory management of perishable products: a time
decay linked logistic approach. International Journal of Production Research 51(13): 3864-3879.
Berk, E., Gürler, Ü. and Yıldırım, G. 2009. On pricing of perishable assets with menu costs. International Journal of
Production Economics 121(2): 678-699.
Bhunia, A.K., Shaikh, A.A. and Gupta, R.K. 2013. A study on two-warehouse partially backlogged deteriorating
inventory models under inflation via particle swarm optimisation. International Journal of Systems Science.
DOI:10.1080/00207721.2013.807385: 1-15.
Chauhan, S.S. and Kumari, R. 2011. Optimal Policy for Deteriorating Inventory with Permissible Delay in Payments
Under Inflation. International Transactions in Applied Sciences 3(4): 615-636.
9
Dye, C.-Y. 2012. A finite horizon deteriorating inventory model with two-phase pricing and time-varying demand
and cost under trade credit financing using particle swarm optimization. Swarm and Evolutionary
Computation 5: 37-53.
Dye, C.-Y. and Hsieh, T.-P. 2011. Deterministic ordering policy with price- and stock-dependent demand under
fluctuating cost and limited capacity. Expert Systems with Applications 38(12): 14976-14983.
Dye, C.-Y. and Hsieh, T.-P. 2013. Joint pricing and ordering policy for an advance booking system with partial
order cancellations. Applied Mathematical Modelling 37(6), 3645-3659.
Dye, C.-Y., Hsieh, T.-P. and Ouyang, L.-Y. 2007. Determining optimal selling price and lot size with a varying rate
of deterioration and exponential partial backlogging. European Journal of Operational Research 181(2),
668-678.
Gaur, N.K., Kumar, S. and Singh, S.R. 2011. An Integrated Production-Inventory Model for Deteriorating Items
under the effect of Inflation. International Transactions in Applied Sciences 3(3): 543-562.
Ghosh, S.K., Khanra, S. and Chaudhuri, K.S. 2011. Optimal price and lot size determination for a perishable product
under conditions of finite production, partial backordering and lost sale. Applied Mathematics and
Computation 217(13), 6047-6053.
Jaggi, C.K., Khanna, A. and Verma, P. 2011. Two-warehouse partial backlogging inventory model for deteriorating
items with linear trend in demand under inflationary conditions. International Journal of Systems Science
42(7): 1185-1196.
Jain, D. and Aggarwal, K.K. 2012. The effect of inflation-induced demand and trade credit on ordering policy of
exponentially deteriorating and imperfect quality items. International Transactions in Operational Research
19(6): 863-889.
Maihami, R. and Abadi, I.N.K. 2012. Joint control of inventory and its pricing for non-instantaneously deteriorating
items under permissible delay in payments and partial backlogging. Mathematical and Computer Modelling
55(5-6): 1722-1733.
Maihami, R. and Nakhai Kamalabadi, I. 2012. Joint pricing and inventory control for non-instantaneous
deteriorating items with partial backlogging and time and price dependent demand. International Journal of
Production Economics 136(1): 116-122.
Mirzazadeh, A., Seyyed Esfahani, M.M. and Fatemi Ghomi, S.M.T. 2009. An inventory model under uncertain
inflationary conditions, finite production rate and inflation-dependent demand rate for deteriorating items
with shortages. International Journal of Systems Science 40(1): 21-31.
Mishra, S., Misra, U.K., Barik, S. and Paikray, S.K. 2011. Optimal control of an inventory system for weibull
ameliorating, deteriorating items under the influence of inflation. Bulletin of Pure & Applied SciencesMathematics 30E(1): 85-94.
Panda, S., Saha, S. and Basu, M. 2009. An EOQ model for perishable products with discounted selling price and
stock dependent demand. Central European Journal of Operations Research 17(1): 31-53.
Price, K.V., Storn, R.M. and Lampinen, J.A. 2005. Differential Evolution: A Practical Approach to Global
Optimization. Springer Berlin Heidelberg.
Shavandi, H., Mahlooji, H. and Nosratian, N.E. 2012. A constrained multi-product pricing and inventory control
problem. Applied Soft Computing 12(8): 2454-2461.
Soni, H.N. and Patel, K.A. 2012. Optimal pricing and inventory policies for non-instantaneous deteriorating items
with permissible delay in payment: Fuzzy expected value model. International journal of industrial
engineering computations 3(3): 281-300.
Taleizadeh, A.A. and Nematollahi, M. 2014. An inventory control problem for deteriorating items with backordering and financial considerations. Applied Mathematical Modelling 38(1):93-109
Teng, J.-T., Ouyang, L.-Y. and Chen, L.-H. 2007. A comparison between two pricing and lot-sizing models with
partial backlogging and deteriorated items. International Journal of Production Economics 105(1): 190-203.
Valliathal, M. and Uthayakumar, R. 2010. Optimal pricing and replenishment policies of an EOQ model for noninstantaneous deteriorating items with shortages. The International Journal of Advanced Manufacturing
Technology 54(1-4): 361-371.
Valliathal, M. and Uthayakumar, R. 2011. Simple approach of obtaining the optimal pricing and lot-sizing policies
for an EPQ model on deteriorating items with shortages under inflation and time-discounting. Istanbul
University Journal Of The School Of Business Administration 40(2): 304-320.
Yang, H.-L. 2012. A Partial Backlogging Inventory Model for Deteriorating Items with Fluctuating Selling Price
and Purchasing Cost. Advances in Operations Research 2012: 1-15.
Yang, H.-L. and Chang, C.-T. 2013. A two-warehouse partial backlogging inventory model for deteriorating items
with permissible delay in payment under inflation. Applied Mathematical Modelling 37(5): 2717-2726.
10