Optimal inventory policy under continuous unit cost and risk of sudden obs

Optimal Inventory Policy under Continuous Unit Cost
Decrease and Risk of Sudden Obsolescence
Moutaz Khouja, Cem Saydam, F. Elizabeth Vergara
The Belk College of Business Administration
The University of North Carolina at Charlotte
Charlotte, NC, USA
Volume 12, Number 2 (mjkhouja@email.uncc.edu), (saydam@uncc.edu)
June 2006, pp. 1-18 (fvergara@email.uncc.edu)
Received: October 2005
Hari K. Rajagopalan
Accepted: May 2006 School of Business, Francis Marion University
Florence, SC, USA
(hrajagop@fmarion.edu)

High technology industries face decreasing component unit costs and a risk of
their component parts and products becoming obsolete. This is especially true in the
personal computer industry where the unit costs for some components decrease at a
rate of one percent per week, and these components have short lifetimes, typically a
year or less. This paper develops inventory models for products experiencing
continuous unit cost decrease and are subject to the possibility of sudden
obsolescence.
We develop two models. In the first model we assume constant probability of
obsolescence and declining unit costs, and solve the model for 1) varying lot sizes,
and 2) constant lot size. Results show that the penalty incurred from the simpler
constant lot size approach is negligible. In the second model we analyze a case
where probability of obsolescence is increasing and consider both constant unit
costs and declining unit costs.
Keywords: Inventory, Optimization, Manufacturing, Obsolescence
1. Introduction
In many industries, particularly in the electronics and computer industries, the rate of
innovation coupled with shortened product life cycles result in continually declining
component unit costs and increasing obsolescence. For example, computer costs are
driven by the unit cost of the CPU. Intel, the market leader in manufacturing CPUs,
follows the strategy of pricing their latest processors high, then progressively
lowering the price while newer, faster processors come into the market (Magee
2000). After a period of price decrease, earlier processors become obsolete and are
phased out. Both of these forces, decreasing unit costs and obsolescence, cause
firms to practice just-in-time (JIT) inventory management. Nevertheless, firms still
have to determine how much to order, when to order, and whether to maintain a
constant lot size or vary lot sizes over time. It is also important to distinguish
between the concepts of obsolescence and deterioration. An inventoried item subject
to obsolescence incurs little or no physical damage, whereas an inventoried item
subject to deterioration (perishability) will degrade overtime, thereby reducing its
marketable value (Goyal and Giri 2001). Hence, for an obsolete item, the market
value of the item decreases due to a decrease in market demand for the item, not due

2 OPTIMAL INVENTORY POLICY

to any inherent change in the quality of the item itself (Cobbaert and Van
Oudheusden 1996).
There are several approaches to developing inventory models with obsolescence
(Arcelus, Pakkala and Srinivasan 2002; Masters 1991) and with continuous unit cost
decrease (Khouja and Park 2003). However, to our knowledge the two models in
this paper are the first to consider products facing both continuous unit cost decrease
and risk of obsolescence. In the first model we assume constant probability of
obsolescence and declining unit costs. We solve this model for varying and constant
lot sizes and conduct numerical experiments to test the performance of the solution
algorithms. Our results show that constant lot size solution incurs a very small
penalty over the variable lot size solution. In the second model we analyze a case of
increasing probability of obsolescence. The model can deal with both constant and
decreasing unit cost. The second model is solved under only variable lot sizes (or
cycle times) because the increasing obsolesce probability makes the constant lot size
restriction suboptimal for many problems.
The paper is organized as follows: In the next section we review the literature. We
then formulate and solve a model with constant probability of obsolescence followed
by a model with increasing probability of obsolescence. The performance of the
algorithms, comparisons of varying and constant lot size approaches, and numerical
results are then discussed. We close with a summary of our findings and suggestions
for future research.
2. Literature Review
Since the proposed models analyze inventory decisions under changing unit cost and
obsolescence, we review inventory models in these two areas. Economic order
quantity (EOQ) models with unit cost changes have been studied under infinite
versus finite horizon, and single unit cost change versus continuous unit cost change.
Naddor (1966) developed an EOQ model for a single unit cost change that occurs at
the end of a cycle. Similar models can be found in inventory text books (Peterson
and Silver 1979). Taylor and Bradley (1985) allowed unit cost increases to occur
anytime during the cycle. Goyal, Srinivasan, and Arcelus (1991) reviewed models
with announced one time unit cost increases. Models that incorporate a finite
horizon and unit cost changes have been developed by Lev and Soyster (1979),
Goyal (1979) and Markowski (1990). Lev and Weiss (1990) studied the optimal
inventory policy with a single change in any EOQ cost parameter for both finite and
infinite horizons. Later, Goyal (1992) and Gascon (1995) improved the algorithms
developed by Lev and Weiss (1990).
Few papers have dealt with continuous cost changes. Buzacott (1975) assumed
that both the setup cost and unit cost continually increase in an infinite horizon. In a
note on Buzacott’s paper, Goyal (1976) incorporated inflation in conjunction with
time value of money. Buzacott (1976) showed through cash flow analysis that his
solution, without consideration of time value of money, can result in a better
solution. Erel (1992) revisited the EOQ model with continuous unit cost increases,
but fixed setup costs.
Among the EOQ models with unit cost change, some researchers dealt with
decreasing cost (Goyal, Srinivasan, and Arcelus, 1991; Lev and Weiss 1990),
however these models are limited to a single cost change. Chen and Min (1999)

3 KHOUJA, SAYDAM, VERGARA, RAJAGOPALAN

analyzed a buyer’s decision when a temporary future sale is announced by the
supplier. Arcelus, Srinivasan and Shah (2001) examined the profitability of forward
buying practices for a retailer confronted with a temporary unit cost decrease of
perishable items. Khouja and Park (2003) extended the EOQ model to products with
continuous unit cost decrease. They derived an accurate closed-form approximate
solution. Their findings show that declining unit costs lead to substantial decreases
in the optimal cycle time. Teunter (2005) developed a net present value formulation
of Khouja and Park’s model resulting in a simpler modified EOQ solution that
provides an alternative to Khouja and Park’s solution. Khouja and Goyal (2006)
developed an algorithm for the problem solved by Khouja and Park (2003) and
Teunter (2005) but allowed for varying cycle times. The use of variable cycle time
may allow for longer cycles in the late stages of the planning horizon. The longer
cycles may be a result of smaller unit holding cost brought about by decreasing unit
cost.
Traditionally, costs associated with obsolescence have been categorized as part of
the risk of holding inventory and have been included by adding an obsolescence
component to the holding cost (Naddor 1966; Silver and Peterson 1985; Tersine
1988). Lambert and Londe (1976) established norms for components of the holding
cost. They estimated the value of all obsolete inventories written off over a year and
expressed this quantity as a fractional value of average inventory. Lambert and
Mentzer (1980) and Walter (1988) suggest that this is a common practice, and that
the obsolescence component is typically 1% or less of the total holding cost.
Hadley and Whitin (1963), argued that since obsolescence occurs at a specific
point in time its cost should not be included in the holding cost. Alternative
techniques, such as dynamic programming, have also been employed to solve these
problems (Cobbaert and Van Oudheusden 1996). Barankin and Denny (1965) and
Pierskalla (1969) assumed that an item becomes obsolete according to a given
sequence of probabilities. Brown, Lu, and Wolfson (1964) used a Bayesian process
to model obsolescence. Since a precise probability distribution of the moment of
obsolescence is required, these approaches proved very difficult to implement.
Under gradual obsolescence an item becomes gradually obsolete over time,
whereas with sudden obsolescence the point of total obsolescence is reached at a
finite point in time. Brown et al. (1964) and Arcelus et al. (2002) dealt with both
gradual and sudden obsolescence. In the first model by Brown et al., demand is
constant until obsolescence. In the second model they allowed for both sudden and
gradual obsolescence. They showed that an iterative dynamic programming
approach could lead to an optimal solution.
Masters (1991) investigated the legitimacy of including obsolescence cost within
the holding cost in the basic EOQ model. The model employs a negative exponential
probability distribution to model an item’s lifetime, which implies that the age of the
item does not influence the probability of obsolescence during any subsequent time
interval. Under such assumption, the optimal order quantity does not depend on the
age of the item. For this reason, Masters’ model can be directly compared to those
from other models without obsolescence.
Starting with Masters’ model, Joglekar and Lee (1993) developed an exact total
cost formula and used the software package GINO to find the optimal order quantity.


They showed that under short order cycles, Masters’ approximation is accurate, but
for long cycle times the error in total cost can be as high as 14.3%. Van Delft and
Vial (1996) proposed four EOQ extensions for items subject to high obsolescence
rates using discounted total cost criterion. The economic order quantity turns out to
be much smaller than the one obtained using the EOQ. Van Delft and Vial’s first
model focused only on costs and assumed sudden and total obsolescence. Their
second model assumed that obsolete inventory has a positive value. The third model
examined progressive obsolescence where the risk “jumps” at certain periods. Their
final model investigated cases of obsolescence with planned stock-outs.
Pierskalla (1969) modeled a stochastic inventory problem that takes into account
the probability that items will become obsolete within a particular period. Menipaz
(1988) assumed discrete Poisson demand, instantaneous replenishments, and no
shortages. Items are assumed to have a constant obsolescence risk and used a
continuous review inventory (S, s) model.
Cobbaert and Van Oudheusden (1996) extended the works of Brown et al. (1964)
and Masters (1991) for fast moving products. They developed three models which
build upon Masters’ (1991) model. Their first model assumed a constant
obsolescence risk, and no shortages. The second model assumed that obsolescence
is unlikely to occur early in the product’s lifetime and that the risk of obsolescence
starts increasing substantially thereafter. This resulted in a modified EOQ formula
with a time dependent obsolescence factor added to the holding cost. Their third
model extended their second model by allowing shortages. David and Greenshtein
(1996) generalized Joglekar and Lee’s model by incorporating the variability of
inventory levels. This model tries to minimize the expected inventory cost when
faced with obsolescence. The inventory level is modeled by a Brownian motion
since the production and demand processes are not deterministic.
The above review and the summary of the literature presented in Table 1 show that
models on obsolescence assume constant unit cost until the moment of obsolescence
(Masters, 1991). Also, models dealing with decreasing unit cost assume no risk of
obsolescence (Khouja and Goyal 2006; Khouja and Park 2003; Teunter 2005).
However, many products that face sudden obsolescence, also experience decreasing
cost over time. This paper combines both aspects and formulates and solves
inventory models for products experiencing continuous decreasing unit cost and
facing sudden obsolescence. This contribution is important because the rate of unit
cost decrease is significant for many products facing obsolescence. For example, in
the PC industry where many components face sudden obsolescence, some
components’ costs are decreasing by as much as 1% per week (Anonymous, 2001).
Examples of these components include flash memory, central processing units, and
video cards.
3. Model 1: Constant Probability of Obsolescence and Declining
Unit Cost
We develop an inventory model for products experiencing continuous decrease in
unit cost by a certain percentage per unit time and facing a risk of sudden
obsolescence. We follow Masters’ (1991) assumptions on obsolescence. Masters
defines sudden obsolescence as an abrupt and complete disappearance of demand at
a random point in time t, where t is negative exponentially distributed with a mean of


Table 1
Summary of Literature
Price Obsolescence
Change None Sudden Gradual
None N/A Brown, Lu, & Wolfson Brown, Lu, &
(1964); Barankin & Wolfson (1964)
Denny (1965); Menipaz (1988);
Pierskalla (1969); Arcelus, Pakkala
Masters (1991); & Srinivasan
Joglekar & Lee (1993); (2002)
Van Delft & Vial
(1996); Cobbaert &
Oudheusden (1996);
David & Greenshtein
(1996)
One Naddor (1966); Lev
Time & Soyster (1979);
Goyal (1979); Taylor
& Bradley. (1985);
Lev & Weiss (1990)
Goyal Srinivasan, &
Arcelus (1991); Chen
& Min (1999);
Arcelus et al. (2001)
Contin- Buzacott (1975); Proposed models
uous Goyal (1976);
Buzacott (1976);
Erel (1992); Khouja
& Park (2003);
Teunter, (2005);
Khouja & Goyal
(2006)
T years. However, unlike Masters’ model, the optimal lot size will depend on the
age of the product (because unit cost depends on age) and restricting cycle times to
be equal may incur a penalty in total cost. Therefore, we initially formulate model 1
to allow for varying lot sizes, and then modify it by restricting lot sizes (cycle times)
to be equal.
We retain all the classic assumptions of the EOQ model except that the unit cost is
assumed to be decreasing over time and product lifetime is a random variable. The
following classic assumptions of the EOQ model are used: (1) Demand is
deterministic and uniform over a planning horizon, (2) No quantity discounts are
given, (3) All of the order quantity is delivered in one shipment, (4) Holding cost is
linear in the number of units held, (5) No shortages are allowed, (6) Lead time is
constant. The notation is as follows:
D = demand in units per year until obsolescence,
S = ordering cost,
r = holding cost per unit time expressed as a percentage of an item’s value,
T = expected life of the product,
n = the number of cycles during the planning horizon,


i = 1, 2, 3,…, n, cycle index,
Qi = order quantity in the ith cycle, a decision variable,
τi = the length of the ith cycle, a decision variable,
ki = the time of the beginning of the ith cycle, k1 = 0and kn+1 = T,
u = percent decrease in unit cost per unit time,
Ci = cost per unit of the product at the beginning of cycle i,
p = fraction of product value that can be salvaged at obsolescence, 0 ≤ p ≤ 1 ,
Cs = salvage value of the obsolete item ( pC1 ), and
TC = the total cost over the planning horizon, an effectiveness measure.

3.1.Varying Lot Size Model
The probability of obsolescence in any cycle can be expressed as:

τ
Ps = i . (1)
T
If unit cost decreases by u% per unit time, then at time t , unit cost is:
−α ki
Ci = C1e , (2)

where α is given by:

α = − Ln (1 − u / 100) . (3)

Therefore, the total cost over the planning horizon is composed of the ordering,
purchasing, holding, and obsolescence cost.

n⎡ τ DCi (1 − ρ )τ i DCi τ i ⎤
TC = n S + ∑ ⎢τ i DCi + i rτ i + ⎥, (4)
i =1⎣ 2 2 T⎦

The terms inside the brackets are the purchase cost, the holding cost, and the
−α ki
obsolescence cost, respectively. Simplifying and replacing Ci with C1e results
in the following problem:
n
⎡ −α ki ⎛ rτ i (1 − ρ ) τ i ⎞ ⎤ . (5)
Min TC = n S + ∑ ⎢τ i DC 1 e
⎣
⎜1 +
⎝ 2
+
2 T ⎠⎥
⎟
⎦
i =1

−α k
The term e i can be approximated using the Taylor series expansions of
exponential functions given by

α 2 k i2 α 3 k i3 α x k ix
e − α ki = 1 − α k i + − + .... + + ... (6)
2! 3! x!


Using the first two terms in the expansion, it results in an error of less that 1% for
horizons of up to one year, and an annual unit cost decrease rate of up to 30%. Also
τ i can be written as:
τ i = k i +1 − k i . (7)

Substituting from equations (6) and (7) into equation (5) gives:
n ⎡ ⎛ α 2 k i2 ⎞⎛ R (k i + 1 − k i ) ⎞ ⎤
Min TC = n S + ∑ ⎢ (k i +1 − k i )DC 1 ⎜ 1 − α k i +
⎜ 2
⎟⎜1 +
⎟ 2
⎟⎥ (8)
i =1 ⎣ ⎝ ⎠⎝ ⎠⎦

where
1− ρ . (9)
R = r+
T
The structure of this problem closely resembles the problem of determining the
optimal variable cycle times under decreasing demand (Khouja and Goyal 2006).
Several heuristics have been developed to solve this type of problem, including
Silver’s (1979) least unit cost (LUC) algorithm, and the continuous least unit cost
(CLUC) method by Tsado (1985). Recently, Zhao, Yang, and Rand (2001) noticed
that some of these algorithms, such as the above mentioned heuristic methods, share
a similar optimality condition that differs only by a single parameter. They have
published papers with the aim of developing an eclectic algorithm so that these
heuristics can be applied by only varying the parameter vector. More recently,
Goyal and Giri (2003) introduced a backward search procedure which can be
modified to use the optimality conditions of Silver’s LUC, CLUC, and others. The
procedure is both straightforward and outperforms forward search procedures.
Khouja and Goyal (2006) showed that the backward search approach can be
successfully modified to solve inventory models for products experiencing
continuous change in unit cost. We adapt Goyal and Giri’s (2003) approach to solve
Model 1 with variable cycle times.
In order to determine the last order cycle interval, we note that the total cost for the
nth cycle (last cycle) is:

⎛ α 2 k n2 ⎞⎛ R (T − k n ) ⎞ .
TC n = S + (T − k n )DC 1 ⎜ 1 − α k n +
⎜ ⎟⎜1 +
⎟ ⎟ (10)
⎝ 2 ⎠⎝ 2 ⎠

Therefore, the total cost per unit time (TCPUT) during the last cycle is obtained by
dividing TC n by the length of the cycle (T − k n ) which gives:

S ⎛ α 2 k n2 ⎞⎛ R (T − k n ) ⎞ . (11)
TCPUT = + DC 1 ⎜ 1 − α k n +
⎜ ⎟⎜1 +
⎟ ⎟
n
(T − k n ) ⎝ 2 ⎠⎝ 2 ⎠

In order to determine the optimal kn we set the first derivative of TCPUTn w.r.t.
k n to zero and obtain the following necessary condition for optimality:


4S 3 (12)
+ 2 D α C1 [2 R + α (2 + T R )] k N − 2 D C1 (2α + R + αT R ) − D α 2 C1 R k n = 0
2

(T − k n ) 2 4

For convenience, let
x = 2 D C 1 ( 2α + R + α T R ) , (13)

y = 2 D α C 1 [ 2 R + α ( 2 + T R )] , (14)

equation (12) can be rewritten as:
4S 3
+ yk − x − D α 2 C 1 R k n2 = 0 . (15)
(T − k n ) 2
n
4

The term 3 D α 2 C R is a very small value relative to other terms because both α
4 1
and R are small. Therefore, we approximate the solution to equation (15) by
removing this last term. Since the first two terms of equation (15) are increasing in
k n and x is positive, equation (15) has one positive real root. The experiment in
Section 5, which compares the results of the algorithm with those obtained using the
Steepest Descent method used on the original total cost function, shows that this
approximation is very accurate.
3.1.1 Algorithm
Step 1. Set k n+1 = T , determine τ n first by solving equation (15) and setting
τ n = k n +1 − k n then successively determine τ n−1 , τ n − 2 ,L,τ 1 ,τ 0 such
n −1 n
that ∑τ j=0
n− j < T and ∑τ
j =0
n− j >T .

Set y = 0 .

Step 2. Set y = y + 1 .

Modify the individual replenishment intervals according to:

T i = 1, 2, L , n − y .
τ i* = τ i n− y
,

∑τ
j =0
n− j

* * * * *
Step 3. Set k 0 = 0 , k n − y = T , and k j +1 = k j + τ j .

Step 4. If y = 1

Compute TC ( n − y ) using equation (5) (or its approximation given by equation (8))

Go to Step 2.


If TC ( n − y − 1 ) > TC ( n − y ) , then go to Step 2.

Step 5. Compare TC ( n − y ) computed using equation (5) for all values of y and
select the one with the minimum total cost.
3.2. Constant Lot Size Model
In this formulation, we restrict cycle times to be equal, therefore τ i = τ ∀i . Since
the planning horizon is divided into n equal-length cycles, each has a length of:
τ =T /n. (16)
The probability of obsolescence is given in equation (1) and the order quantity can
be expressed as a function of n as:

DT
Q= . (17)
n

With the assumption of an exponentially distributed item lifetime, Masters
indicates that a fixed order quantity (or alternatively a fixed order cycle) is optimal,
because the age of the item does not help predict its remaining life. However, in this
model, item unit cost decreases over time. Therefore, the expression for the total
expected lifetime inventory costs is essentially the same as that in (Masters 1991),
but in addition to the inventory ordering, holding, and obsolescence costs,
purchasing cost must also be included. The total cost as a function of the number of
order cycles, n, is:
α ( i −1 ) T α ( i −1 ) T α ( i −1 ) T
⎛ − − − ⎞
⎜ DTC 1 e
n n
DT 2 rC 1 e n
DTC l e n
τ ⎟ (18)
TC = nS + ∑ ⎜ + + ⎟
i =1 ⎜ n 2n 2 2n T ⎟
⎝ ⎠
where C = (1 − ρ )C s . The first term of equation (18) is the ordering cost,
l
followed by the purchasing, holding, and obsolescence costs. Setting d TC / d n = 0 ,
along with further manipulation and simplification, which are shown in the
Appendix, gives the following approximation to the necessary condition for
optimality:

(Sα3T 2 + 2α DTC1(e−αT −1)+ 2αrDTC1(e−αT −1)+ 2dCl (e−αT −1) + 4αS(n2 + Tα n) = 0 . (19)
Solving equation (19) for n yields:

n=−
αT
+
(
D (α + r )C1T + Cl e )( αT − 1) . (20)
2 eαT 2α S

DT
Substituting n = into equation (20) yields the optimal order quantity:
Q


* DT

)( α T − 1)
Q = . (21)

−
αT
+
(
D (α + r )C1T + Cl e
αT
2 e 2α S

4. Model 2: Increasing Probability of Obsolescence
In this model we consider a product with an increasing probability of obsolescence
and decreasing unit cost. Obsolescence within each cycle is modeled using a
Markov chain. With each unit time, or with the sale of every unit, the product may
become obsolete with a probability qi . This assumption is similar to Porteus’ (1986)
with respect to the time a production process may shift out of control and start
producing defective products. The following additional notation is needed:
qi = the Markov probability of obsolescence at the beginning of cycle i .

Let q0 be the Markov obsolescence probability at time t = 0 . If the probability
increases linearly, then it is given by:
qt = q0 + b t . (22)

Obviously, the speed of obsolescence is reflected by the slope, b . At the
i −1
beginning of cycle i , the total time elapsed is ∑τ
j =1
j and the Markov obsolescence

probability is:
i −1
qi = q0 + b ∑τ j . (23)
j =1

Q
For small values of qi , Porteus (1986) showed that E(Ui ) = Qi − βi (1 − β i ) / qi ,
i
where U i is the number of obsolete units per lot of size Qi and β i = 1 − qi .
Porteus (1986) proved that E (U i ) is a strictly increasing, strictly convex function of
Qi . If qi is small, then Porteus (1986) further showed that
Q 1 2 1 2
(Qi − βi (1 − βi i ) / qi ) ≈ qi Qi = qi (τ i D) . Also, since q t depends on the age of
2 2
the product, the probability of obsolescence occurring in a cycle (which is given by
τ i / T in the first model) must be modified to reflect the increasing q t as shown in
the first fraction of the last term in the total cost function:
n⎡ τ DC [( q + qi ) / 2]τ i (1 − ρ )Ci 2 2⎤
TC = n S + ∑ ⎢τ i D Ci + i i rτ i + i −1 qiτ i D ⎥ , (24)
i =1⎢
⎣ 2 qM T 2 ⎥
⎦


where q M is the value of qt at time T / 2 . The terms inside the brackets are the
purchase cost, the holding cost, and the obsolescence cost, respectively. Therefore,
the probability of obsolescence per cycle is computed as qt at the midpoint of the
cycle multiplied by the cycle time divided by qt at the midpoint of the total horizon
multiplied by T . Substituting for qi from equation (23) into equation (24) and
simplifying gives the total expected annual cost, including obsolescence cost, as:
⎡ i−1 ⎤
[q0 + b( ∑ τ j + τ i / 2)]τ i
n⎢ τ i DCi j=1 (1 − ρ )Ci i−1 2 2⎥
TC = n S + ∑ ⎢τ i DCi + rτ i + (q0 + b ∑ τ j )τ i D ⎥ (25)
i=1⎢ 2 (q0 + bT / 2) T 2 j=1 ⎥
⎢
⎣ ⎥
⎦
Again, we use a backward search procedure to minimize the total cost. In order to
determine the last order cycle interval, we obtain the total cost for the nth cycle (last
cycle) as:

2 D Cn r 2 2 (1 − ρ )[ q0 + b(T − τ n )]Cn
TC n = S + τ n D Cn + τ n +τn D . (26)
2 2
Therefore, the total cost per unit time during the last cycle is obtained by dividing
TC n by the length of the cycle τ n which gives:

S D Cn r 2 (1 − ρ )[ q0 + b(T − τ n )]C n
TCPUTn = + D Cn + τ n + τ nD . (27)
τn 2 2

Let α1 = α T − 1 . As shown in the Appendix, the optimal τn can be approximated
by finding the solution to:
2 2
− 4 S + [ 2 D r + α D ( 4αC nα1 + Tr (α1 − 1)) + C n D ( 2 + αT (α1 − 1))(bT + q0 ) ρ ]τ n +
. (28)
2 2 3
[ −4αD (α C n − r + αTr ) − 2C n D (b ( 2 + 3αT (3αT − 4)) + 2αα1q0 ) ρ ]τ n = 0

The algorithm in the previous section can be used to solve the problem in this
section by replacing equation (15) with equation (28) and replacing equation (8) with
equation (25).
5. Experiments and Examples
5.1. Numerical Examples
Consider a product with an annual demand D = 12,000 units, ordering cost S = $200
per order, annual fraction holding cost r = 0.07, and an initial unit cost C1 = $40. The
value of α used is 0.25 which corresponds to annual unit cost decrease rate of
22.12%. Product life has a negative exponential distribution with a mean of T = 0.25
years and a salvage value of 50% of the original value. Under these conditions,
equations (20), (21) and (18) give n = 12.96, an order quantity of Q = 231 units, and
expected total annual cost of $121,528, respectively. However, if we had not taken


Table 2
Cycle Times and Expected Total Cost for Proposed Algorithm versus Steepest Descent
Constant unit cost Decreasing unit cost
Algorithm Steepest Algorithm Steepest
Descent Descent
Optimal 10 10 11 11
n
Total cost $123,129.98 $123,127.58 $119,772.09 $119,771.09
Cycle Cycle times
number
1 0.0255 0.0267 0.0231 0.0238
2 0.0254 0.0262 0.0230 0.0235
3 0.0252 0.0258 0.0229 0.0233
4 0.0251 0.0254 0.0228 0.0231
5 0.0250 0.0251 0.0228 0.0229
6 0.0249 0.0247 0.0227 0.0226
7 0.0248 0.0244 0.0226 0.0224
8 0.0247 0.0241 0.0225 0.0222
9 0.0246 0.0237 0.0224 0.0220
10 0.0245 0.0235 0.0224 0.0219
11 N/A N/A 0.0223 0.0217
Total time 0.25 0.25 0.25 0.25
obsolescence into account (i.e., Cl = 0 ) then equations (20), (21) and (18) give n =
4.79, an order quantity of Q = 627 units. Without obsolescence this would result in
an expected total annual cost of $118,259, however if the there is obsolescence then
the expected total annual cost for n = 4.79 and Q = 627 units would be $124,368
which is a 2.33% increase in costs.
Assuming an increasing probability of obsolescence with q0 = 0.001 and
b = 0.005 and a constant unit cost, the algorithm, which uses equations (5) (or (8))
and (15) gives the cycle times shown in Table 2 and an expected total annual cost of
$123,129.98. Assuming unit cost is decreasing at a rate of 22.12% annually ( α =
0.25) the algorithm with equations (28) and (25) gives the cycle times shown in
Table 2 and an expected total annual cost of $119,772.09.

5.2. Numerical Experiments
To test the performance of the algorithm of Model 1, we compare its solutions to
those obtained using the Steepest Descent method in Mathematica (Wolfram 1996)
for problems with different parameter values. First, we obtain the best value for n
and τ i using the algorithm. We then obtain the Steepest Descent algorithm solution
for the value of n identified by our algorithm. Then a search is conducted using
Mathematica by increasing and decreasing the values of n . As long as the total cost
is decreasing in a given direction the search is continued in that direction.
We test the proposed algorithm using two values for each of the five parameters:
planning horizon (0.25 and 0.4 years), annual demand (12,000 and 24,000 units), rate
of unit cost decrease (22% and 39% per year which results in α = 0.25 and


α = 0.50 ), holding cost ($0.07 and $0.14 per $1 worth of inventory per year), and
ordering cost ($200 and $400 per order). This results in 2×2×2×2×2 = 32 problems.
The salvage value is set to 50 % of the initial unit cost, which was set to C1 = $40 .
We compared the algorithm’s solution to the solution obtained using the Steepest
Descent method in Mathematica. As shown in Table 3, the algorithm consistently
matched solutions found by the Steepest Descent method. We also compare the
results between Steepest Descent solutions with variable cycle times and the equal-
cycle time solutions. Table 3 summarizes the accuracy of the algorithm and the equal
cycle time model over the steepest descent algorithm. As shown in the table, on
average, the difference in total cost is only 0.00078% and in the worst case it is
0.00271%, which is negligible. This means that practitioners can use equal cycle
times since the increase in total cost is negligible. In addition, equal cycle times are
easier to compute and implement and are more appealing to suppliers.
To test the performance of the algorithm of Model 2 with constant unit cost, we
also compare its solutions to those obtained using the Steepest Descent method in
Mathematica for problems with different parameters and following the same
procedure as in the previous test. The same parameters of the previous experiment
(except the unit cost decrease, α ) are used here with the addition of two values for
b : 0.001 and 0.005, which results in 2×2×2×2×2 = 32 problems. In testing the
performance of the algorithm, the exact cost function, and not the approximation (of
e −α ki ), was used to ensure complete assessment of the performance of the
algorithm. As Table 3 shows, the average penalty in total cost of the algorithm over
the Steepest Descent method is 0.02574%, while the maximum penalty is 0.07589%.
To test the performance of the algorithm of Model 2 under decreasing unit cost,
we use the same parameter combinations as in the constant unit cost experiment with
the addition of two unit cost decrease rates (22% and 39% per year which results in
α = 0.25 and α = 0.50 ). This results in 2×2×2×2×2×2 = 64 problems. As Table 3
shows, the average penalty in total cost of the algorithm over the Steepest Descent
method was 0.01547%, while the maximum penalty was 0.05829%.
Table 3
Numerical Experiments Results (% Total Cost Penalty)
% TC penalty for % TC penalty for equal
Range algorithm over Steepest cycle times over
Model
of n Descent Steepest Descent
Average Maximum Average Maximum
Model 1 9 - 26 0.00000 0.00000 0.00078 0.00271
Model 2
(constant unit 8 - 28 0.02574 0.07589 N/A N/A
cost)
Model 2
(decreasing unit 8 - 29 0.01547 0.05829 N/A N/A
cost)


6. Conclusion and Suggestions for Future Research
In this paper, we developed two inventory models for products experiencing
continuous decrease in unit cost and at risk of becoming obsolete. The first model
assumes a constant rate of obsolescence and is solved for both varying and constant
lot sizes. The second model deals with increasing probability of obsolescence and
uses varying lot sizes. The results from the algorithms of both models were
compared to solutions found using the Steepest Descent method for problems with
different parameters. The solutions identified by the algorithm were as good or very
close in all instances to solutions obtained using Steepest Descent. It is noteworthy
that restricting cycle times to be equal under constant obsolescence rate was found to
cause only a negligible increase in total cost, especially when the planning horizon is
short. For practitioners, this implies that simpler fixed-length cycle solutions which
are easier to compute and implement with suppliers are appropriate for most items
experiencing decreasing cost facing a chance of obsolescence.
A limitation of the proposed model is that it does not consider decreasing demand.
For many products, such as those in the electronics industry, as the product
progresses in its life cycle there are newer, more advanced, and expensive
alternatives available. Central processing units are good examples of such a product.
Not only the probability of obsolescence increases overtime, but the demand also
decreases. The model developed in this paper does not take into account this
common type of trending demand for some products. While the varying-lot size
algorithm in this paper is a modification of a decreasing-demand algorithm (Goyal
and Giri, 2003), we have not tested its performance when demand is decreasing, in
addition to decreasing unit cost and obsolescence.
Useful future research directions may examine multi-item inventory policy under
both unit cost decrease and obsolescence. In this case, several products are ordered
from a single supplier and, therefore their replenishment must be coordinated. Some
of these products may have stable unit cost and no risk of obsolescence, while others
may experience significant unit cost decrease and face a high risk of obsolescence.
Such a problem may be addressed within the Joint Replenishment Problem, which is
well researched in inventory management. Another direction may examine the
problem when unit cost experiences sudden drops at times which are random
variables with known distributions.

7. References
1. Anonymous (May 14, 2001) "How Dell Keeps from stumbling" Business
Week 3732 38B-D
2. Arcelus, F. J., T. P. M. Pakkala and G. Srinivasan (2002). "A myopic policy
for the gradual obsolescence problem with price dependent demand."
Computers and Operations Research 29: 1115-1127.
3. Arcelus, F. J., G. Srinivasan and N. H. Shah (2001). "Forward Buying
Policies for Deteriorating Items Under Price Sensitive Demand and
Temporary Price Discounts." International Journal of Operations and
Quantitative Management 9(2): 87-101.


4. Barankin, E. W. and J. Denny (1965). "Examination of an Inventory Model
Incorporating Probabilities of Obsolscence." Logistics Review and Milatary
Logistics Journal 1(1): 11-25.
5. Brown, G. W., J. Y. Lu and R. J. Wolfson (1964). "Dynamic modeling of
inventories subject to obsolescence." Management Science 11(1): 51-63.
6. Buzacott, J. A. (1975). "Economic order quantities with inflation."
Operational Research Quarterly 26(3): 553.
7. Buzacott, J. A. (1976). "Rejoinder to Goyal's Letter." Operational Research
Quarterly 27(4): 905-907.
8. Chen, C.-K. and K. J. Min (1999). "Optimal Inventory Policies in Response
to a Pre-Announced Sale." International Journal of Operations and
Quantitative Management 5(2): 151-169.
9. Cobbaert, K. and D. Van Oudheusden (1996). "Inventory Models for fast
moving spare parts subject to obsolescence." International Journal of
Production Economics 44: 239-248.
10. David, I. and E. Greenshtein (1996). "Brownian analysis of a buffered flow
system in the face of sudden obsolescence." Operations Research Letters
19: 43-49.
11. Erel, E. (1992). "The effect of continuous price change in the EOQ."
OMEGA International Journal of Management Science 20(4): 523-527.
12. Gascon, A. (1995). "On the finite horizon EOQ model with cost changes."
Operations Research 43(4): 716-717.
13. Goyal, S. K. (1976). "A Note on Economic Order Quantities with Inflation."
Operational Research Quarterly 27(4): 902-905.
14. Goyal, S. K. (1979). "A note on the paper: An inventory model with finite
horizon and price change." Journal of Operational Research Society 30(9):
839-840.
15. Goyal, S. K. (1992). "A Note on Inventory Models with Cost Increases."
16. Goyal, S. K. and B. C. Giri (2001). "Recent Trends in modeling of
deteriorating inventory." European Journal of Operational Research 134:
1-16.
17. Goyal, S. K. and B. C. Giri (2003). "A simple rule for determining
replenishment intervals of an inventory item with linear decreasing demand
rate." International Journal of Production Economics 83: 139-142.
18. Goyal, S. K., G. Srinivasan and F. J. Arcelus (1991). "One time only
incentives and inventory policies." European Journal of Operational
Research 54: 1-6.
19. Hadley, G. and T. M. Whitin (1963). Analysis of inventory Systems.
Englewood Cliffs New Jersey, Prentice Hall Inc.
20. Joglekar, P. and P. Lee (1993). "An exact formulation of inventory costs
and optimal lot size in face of sudden obsolescence." Operations Research
Letters 14: 283-290.
21. Khouja, M. and S. K. Goyal (2006). "Optimal Lot Sizing Under Continuous
Price Decrease." International Journal of Production Economics 102: 87-
94.


22. Khouja, M. and S. Park (2003). "Optimal lot sizing under continuous price
decrease." OMEGA International Journal of Management Science 31: 539-
545.
23. Lambert, D. M. and L. Londe (1976). "Inventory carrying costs."
Management Accounting 58(2): 31-35.
24. Lambert, D. M. and J. T. Mentzer (1980). "Is Integrated physical
distribution management a reality?" Journal of Business Logistics 2(1): 18-
34.
25. Lev, B. and A. L. Soyster (1979). "An inventory model with finite horizon
and price change." Journal of Operational Research Society 30(1): 43-53.
26. Lev, B. and H. J. Weiss (1990). "Inventory Models With Cost Changes."
27. Magee, M. (2000) Pentium 4 prices and speeds, January to June. The
Register http://www.theregister.co.uk/content/3/15599.html
28. Markowski, E. P. (1990). "Criteria for evaluating purchase quantity
decisions in response to future price increases." European Journal of
Operational Research 47: 364-370.
29. Masters, J. M. (1991). "A note on the effect of sudden obsolescence on the
optimal lot size." Decision Sciences 22: 1180-1186.
30. Menipaz, E. (1988). "An Inventory model with product obsolescence with
implications to the high technology industry." Engineering Costs and
31. Naddor, E. (1966). Inventory Systems. New York, John Wiley & Sons.
32. Peterson, R. and E. A. Silver (1979). Decision Systems for Inventory
Management and Production Planning. New York, John Wiley & Sons.
33. Pierskalla, W. P. (1969). "An inventory problem with obsolescence." Naval
Research Logistic Quarter 16: 217-228.
34. Porteus, E. L. (1986). "Optimal lot sizing, process quality improvement and
setup cost reduction." Operations Research 34: 137-144.
35. Silver, E. A. (1979). "A simple inventory replenishment decision rule for a
linear trend in demand." Journal of the Operational Research Society 30:
71-75.
36. Silver, E. A. and R. Peterson (1985). Decision Systems for Inventory
Management and Production Planning. New York, John Wiley and Sons.
37. Taylor, S. G. and C. E. Bradley (1985). "Optimal ordering strategies for
announced price increases." Operations Research 33(2): 312-325.
38. Tersine, R. J. (1988). Principles of inventory and materials management.
New York, Elsevier Science Publishing Co.
39. Teunter, R. (2005). "A note on "Khouja and Park, optimal lot sizing under
continuous price decrease, Omega 31(2003)"." OMEGA International
Journal of Management Science 33: 467-471.
40. Tsado, A. (1985) Evaluation of the performance of lot-sizing techniques on
deterministic and stochastic demand PhD Thesis University of Lancaster,
UK
41. Van Delft, C. and J. P. Vial (1996). "Discounted costs, obsolescence and
planned stockouts with the EOQ formula." International Journal of


42. Walter, C. K. (1988). "The inventory carrying cost methodology - A review
and application to farm equipment." Logistics Spectrum 22(2): 25-32.
43. Wolfram, S. (1996). The Mathematica Book. Champaign, IL, Wolfram
Media.
44. Zhao, G. Q., J. Yang and G. K. Rand (2001). "Heuristics for replenishment
with linear decreasing demand." International Journal of Production
Economics 69: 339-345.

Appendix
1. Optimal Number of Cycles for Equal-Length Cycle Times
Setting d TC / d n = 0 yields the following sufficient condition for optimality:

⎛ 2
⎞
⎜ 2n 4 Seα T ⎛ e n − 1⎞ − DTC e n (eαT − 1)⎛ 2n 2 ⎛ e n − 1⎞ − 2nT⎛ b + r − reα n ⎞ − αrT 2 ⎞ ⎟
αT αT αT αT
e −αT
⎜ ⎟ ⎜ ⎜ ⎟ ⎜ ⎟ ⎟
⎜ ⎜ ⎟ 1
⎜ ⎜ ⎟ ⎜ ⎟ ⎟⎟
⎝ ⎝ ⎠ ⎝ ⎝ ⎠ ⎝ ⎠ ⎠⎠
2
⎛ αT ⎞ (A.1)
2n 4 ⎜ e n − 1⎟
⎜ ⎟
⎝ ⎠
⎛ αT
⎛ ⎛ αT ⎞ ⎞⎞
e −αT ⎜ DCl Te n (eαT − 1)⎜ 2n⎜ e n − 1⎟ + αT ⎟ ⎟
⎜ ⎜ ⎜ ⎟ ⎟⎟
⎝ ⎝ ⎝ ⎠ ⎠⎠
− 2
=0
⎛ αT ⎞
2n 4 ⎜ e n − 1⎟
⎜ ⎟
⎝ ⎠

α T /n
The term e can be approximated using the series
2 2 3 3 4 4 x x
α T /n αT α T α T α T α T
e = 1+ + + + +L+ x +L (A.2)
2 3 4 x! n
n 2n 6n 24 n

The accuracy of the approximation depends on the number of terms used in the
series in equation (A.2). Khouja and Park (2003) examined the accuracy of the
approximation using two terms of the series in equation (A.2) and showed that the
error is very small especially for short horizons and small unit cost decrease rates.
Using the two-term approximation to estimate eα T / n gives the following necessary
optimality condition:
5 2 4
4α Sn + 4 STα n + DrC1α T e (
3 4 −α T
) 3 3 −α T
− 1 + DClα T e (
−1 + )
+ n (Sα T + 2αDTC1 (e − 1)+ 2α rDTC1 (e − 1)+ 2 dCl (e − 1))
3 3 2 −α T −α T −αT
(A.3)
2
(
2 2 −αT
+ n 2 DC1α T e ( 2 −αT
− 1 + 4αDrC1T e )
− 1 + 4αDTCl e
−αT
−1 )) ( ) (
+ n (DC1α T (e − 1) + 3DrC1α T (e − 1) + 3DClα T (e − 1))
3 3 −αT 2 3 −αT 2 2 −αT


Equation (A.3) is a fifth degree polynomial for which analytical solutions are
difficult to obtain. Dividing equation (A.3) by n 3 gives:
2 2 3 2
(
4αSn + 4STα n + Sα T + 2αDTC e
1
−αT
(
−1 + 2αrDTC e
1 )
−αT
−1 + 2dC e
l
−αT
(
−1 ) ( )
+
DrCα T e
1 (
3 4 −αT
)3 3 −αT
−1 + DCα T e
l (
−1 )
3
n

+
(
DCα T e
1 (
3 3 −αT
) 2 3 −αT
−1 + 3DrCα T e
1 ( l )
2 2 −αT
−1 + 3DCα T e −1 ( ) (A.4)
2
n

+
( 2 2 −αT
2DCα T e
1 (
−1 + 4bDrCT e
1 )2 −αT
(
−1 + 4αDTC e
l )
−αT
−1
=0
( )
n
The coefficients of the last three terms are decreasing in n . Therefore, for large
values of n and realistic values of α , r , and S , the solution to equation (A.4) can
be approximated by equation (19).
2. Necessary Conditions for Increasing Probability of Obsolescence

The first derivative of TCPUTn is given by

2 2
− 4S + [2 Dr + α D(4αCnα1 + Tr(α1 − 1)) + Cn D (2 + αT (α1 − 1))(bT + q0 ) ρ ]τ n +

2 2 3 2
[−4αD(α Cn − r + αTr) − 2Cn D (b(2 + 3αT (3αT − 4)) + 2αα1q0 ) ρ ]τ n + [3α D(r + (A.5)

2 4 2 5
Cn D(3bT + q0 ) ρ ) − 6αbCn D ρ ]τ n − 4αbCn D ρτ n

For products with possible obsolescence, τn is expected to be small. In addition,
α and b are small. Therefore, the solution to setting (A.5) to zero is approximated
by the solution to setting the third degree polynomial part which is given by equation
(28) to zero.

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