Journal of Applied Mathematics and Stochastic Analysis, 15:3 (2002), 235-245.
FILTERING AND PREDICTING THE COST OF HIDDEN
PERISHED ITEMS IN AN INVENTORY MODEL
LAKHDAR AGGOUN and LAKDERE BENKHEROUF
Sultan Qaboos University
Department of Mathematics and Statistics
P.O. Box 36, Al-Khod 123, Sultanate of Oman
(Received August 2001; Revised June 2002)
This paper is concerned with a discrete time, discrete state inventory model for
items of changing quality. Items are assumed to be in one of a finite number, Q ,
of quality classes that are ordered in such a way that Class 1 contains the best
quality and the last class contains the pre-perishable quality. The changes of
items' quality are dependent on the state of the ambient environment.
Furthermore, at each epoch time, items of different classes may be sold or moved
to a lower quality class or stay in the same class. These items are priced
according to their quality, and costs are incurred as items lose quality. Based on
observing the history of the inventory level and prices, we propose recursive
estimators as well as predictors for the joint distribution of the accumulated losses
and the state of the environment.
Key words: Hidden Markov Models, Optimal Filtering, Inventory Control.
AMS subject classifications: 60K30, 60J10, 90B05.
1. Introduction
This paper is concerned with a discrete time, discrete state inventory model for items of
changing quality (or aging like in [4]). However, in [4], movement between classes was
deterministic, i.e. each item, at the end of each epoch, moved to an older category or class
provided that it did not perish. In the present situation, items are allowed to remain within the
same class for an indefinite length of time, with some probability of loss of quality, and move
to a lower quality class (see dynamics equation (1.2), which is substantially different from
dynamics equation (1.2) in [4]). Another new feature here is the quantification of the
financial loss incurred by the loss of quality in the inventory (see equation (1.4) and Theorem
1). Finally, Theorem 2, which provides a one-step ahead recursive predictor, has no analog in
[4].
Items are assumed to be in one of a finite number, Q , of quality classes that are ordered in
such a way that Class 1 contains the best quality and the last class contains the pre-perishable
quality.
1.
The vector M8 œ ÐM8" ß á ß M8Q Ñ represents the inventory level at time 8 where M83
refers to the inventory level of Class 3 at time 8, 3 œ "ß á ß Q and 8 œ !ß "ß á .
2.
At time 8, items in Class 3 are assumed to have a unit price T83 such that T83 œ )3 :8
for some nonnegative real number :8 , with )" œ " and )"  )#  á  )Q  !.
The latter condition reflects the order of the quality classes. We also assume that
the price process ÖT8 × defined by T8 œ ÐT8" ß á ß T8Q Ñ is predicable with respect
to the history of the inventory model.
Printed in the U.S.A. ©2002 by North Atlantic Science Publishing Company
235
236
3.
4.
5.
L. AGGOUN and L. BENKHEROUF
The random vector H8 œ ÐH8" ß á ß H8Q Ñ, with distribution FÐ † Ñ and marginals
ÖG38 ×8− , 3 œ "ß ß á ß Q , defined on ™Q
 represents the demand process at time 8.
New arriving items are assigned to Class 1 and these are represented by a
predictable process ÖY8 × defined on ™ .
Define the operator “ ‰ " such that for any nonnegative random variable \ and
α − Ð!ß "Ñ,
\
α‰\ œ
^4 ß
Ð"Þ"Ñ
4œ"
where ^4 is a sequence of iid random variables, independent of \ , such that
T Ð^4 œ "Ñ œ "  T Ð^4 œ !Ñ œ αÞ
Also, let Ö\8 ×8− be a stochastic process with a finite state space W\ of size R ,
which we identify without loss of generality with the set of unit vectors
Ö/" ß á ß /R × in ‘R . The stochastic process Ö\8 ×8− represents the state of the
ambient environment. We assume that at each time 8, items in the stock may
experience some kind of change of quality depending on the state of the
environment Ö\8 × where the stock is held and these changes follow the dynamics
"
M8" œ J " Ð\8" Ñ ‰ M8"
 H8"  Y8
3
3"
M83 œ J 3 Ð\8" Ñ ‰ M8"
 Ö"  J 3" Ð\8" Ñ× ‰ M8"
 H83 ß
Ð"Þ#Ñ
3 œ #ß á ß Q Þ
3
where “ ‰ " is defined in (1.1) and J 3 Ð\8" Ñ ‰ M8"
represents the number of items
3"
of quality 3 that survived from period 8  " and Ö"  J 3" Ð\8" Ñ× ‰ M8"
is the
number of items of quality 3 that moved to quality Class 3 from Class 3  ". The
function J 3 Ð\8" Ñ plays the same role as parameter α in (1.1). The dependence
on \8" is implicit here to stress the dependence of the process affecting the
change of quality on the environment. Note we may write
J 3 œ ÖJ 3 Ð/" Ñß á ß J 3 Ð/R Ñ×À œ ÖJ"3 ß á ß JR3 ×Þ
6.
The process Ö\8 ×8− is a Markov chain with semimartingale representation (see
[6])
\8 œ E\8"  Q8 ß
7.
Ð"Þ$Ñ
where ÖQ8 ×8− is a martingale with respect to the complete filtration
ÖY8 ×8− generated by Ö\8 ×8− and E denotes the transition probability
matrix of the Markov chain Ö\8 ×.
The accumulated losses up to time 8 due to loss of qualities may be written as
Q
-8 œ -8" 
3œ"
with -! œ !.
Ö"  J 3 ÐB8" Ñ× ‰ M83 )3 :8 , 8 œ "ß á ß
Ð"Þ%Ñ
The Cost of Hidden Perished Items in an Inventory Model
237
Clearly, having an estimate of Ð-8 ß \8 Ñ and a forecast of the quantity Ð-8" ß \8" Ñ would
be desirable in a sense that managers interested in stock control would have key information
available to help in deciding the course of action to be taken on the number of items to stock.
The aim of this paper is to derive recursive expressions for the conditional distribution of
Ð-8 ß \8 Ñ and Ð-8" ß \8" Ñ given the complete filtration Öl8 ×8− generated by the observed
processes ÖM83 ×8− , 3 œ "ß á ß Q and ÖT8 ×8− .
Inventory models for perishable items have been in the past, apart from [1, 2] and [3],
examined from a different perspective. We are not aware of discrete models where items are
allowed to lose quality. Also, interest in previous studies focused on investigating optimal
operating characteristics (see [7, 8]).
In the next section, we derive a recursive conditional probability distribution of the cost
process. In Section 3, a parameter updating algorithm is discussed.
2. The Model
Assume that all random variable are initially defined on an `ideal' probability space ÐHß Y ß UÑ
such that:
1.
The inventory levels ÒM83 ×8− are Q sequences of independent random variables,
with probability distributions ÖG38 ×8− ß 3 œ "ß á ß Q .
2.
All other processes are defined in 1-7 of Section 1.
Let -! œ " and, for 5 ", define
-5 œ
‚#
Q
3œ#
"
G"5 ÐJ " Ð\5" щM5"
M5" Y5 Ñ
G"5 ÐM5" Ñ
3
3"
G35 ÐJ 3 Ð\5" щM5"
Ð"J 3" Ð\5" ÑщM5"
M53 Ñ
Þ
G35 ÐM53 Ñ
Ð#Þ"Ñ
Set
A8 œ # -5 .
8
Ð#Þ#Ñ
5œ!
Let ÖZ ×8− be the complete filtration generated by the processes ÖM8 ×8− ; ÖT8 ×8− ;
3
ÖH8 ×8− ; Ö\8 ×8− ; and Ö^8ß6
×8− , 3 œ "ß á ß Q , up to time 8. Here
3
^8ß6 refers to the state of the 6th item in the inventory at the end of period 8. Thus,
3
^8ß6
œœ
!ß
"ß
if the item is moved to class Ð3  "Ñ
otherwise.
It can be shown that the process ÖA8 ×8− is a ÖZ ×-martingale.
Define a probability measure T such that
A8 œ
.T
.U ¹Z
ß
Ð#Þ$Ñ
8
where A8 is given by (2.2). The existence of T is guaranteed by Kolmogorov's Extension
Theorem.
238
L. AGGOUN and L. BENKHEROUF
The next lemma asserts that we can recover our original model under T .
Lemma 1: On ÐHß Y Ñ and under probability measure T , the demand processes ÖH83 ×,
8 œ "ß #ß á ß 3 œ "ß á ß Q , form
sequence of random variables with probability
distributions ÖG38 ×8− , where
"
H8" œ J " Ð\8" Ñ ‰ M8"
 Y8  M8" ß
3
3"
H83 œ J 3 Ð\8" Ñ ‰ M8"
 Ð"  J 3" Ð\8" ÑÑ ‰ M8"
 M83 ß 3 œ #ß á ß Q Þ
Proof: Let 1À ™ Ä ‘ be a Borel test function and for # Ÿ 5 Ÿ Q , consider the
expectation
IT Ò1ÐH85 Ñ ± Z8" Óß
Ð#Þ%Ñ
where IT Ð † Ñ is the expectation with respect to probability measure T .
Now, using the abstract Bayes Theorem (see [6]), expression (2.4) gives
IT Ò1ÐH85 Ñ ± Z8" Ó œ
œ
IU Ò1ÐH58 ÑA8 ±Z8" Ó
IU ÒA8 ±Z8" Ó
IU Ò1ÐH58 Ñ-8 ±Z8" Ó
IU Ò-8 ±Z8" Ó À
œ
R
Hß
where -8 and A8 are given by (2.1) and (2.2), respectively. Using (2.1) we have
H œ IU ’
G ÐJ
‚# 8
Q
3
3œ#
3
"
G"8 ÐJ " Ð\8" !‰M8"
M8" Y8 Ñ
G"8 ÐM8" Ñ
3
3"
ÐB8" щM8"
Ð"J 3" Ð\8" ÑщM8"
M83 Ñ
ºZ8" •
G38 ÐM83 Ñ
œ
D " ßáßD Q
"
G"8 ÐJ " Ð\8" ‰ M8"
 D "  Y8 Ñ
3
3"
‚ # G38 ÐJ 3 Ð\8" Ñ ‰ M8"
 Ð"  J 3" Ð\8" ÑÑ ‰ M8"
 D 3 Ñ œ "Þ
Q
Mœ#
The numerator R of (2.5) is equal to
5
5!"
IU Ò1ÐJ 5 Ð\8" Ñ ‰ M8"
 Ð"  J 5" Ð\8" ÑÑ ‰ M8"
 M83 Ñ-8 ± Z8" ÓÞ
Again, using (2.1), the above is
5
5"
IU Ò1ÐJ 5 Ð\8" ‰ M8"
 Ð"  J 5" Ð\8" ÑÑ ‰ M8"
 M85 Ñ
Ð#Þ&Ñ
The Cost of Hidden Perished Items in an Inventory Model
‚
‚#
Q
3œ#
239
"
G"8 ÐJ " Ð\8" щM8"
M8" Y8 Ñ
G"8 ÐM8" Ñ
3
3"
G38 ÐJ 3 Ð\8" щM8"
Ð"J 3" Ð\8" ÑщM8"
M83 Ñ
¹Z8" Ó
G38 ÐM83 Ñ
œ IU ’ G8 ÐJ
"
"
Ð\8" щM8" M8" Y8 Ñ
G"8 ÐM8" Ñ
Ð#Þ'Ñ
‚#
Q
3œ#
3
3"
G38 ÐJ 3 Ð\8" щM8"
Ð"J 3" Ð\8" ÑщM8"
M83 Ñ
G38 ÐM83 Ñ
5
5"
‚ IU Ò1ÐJ 5 Ð\8" Ñ ‰ M8"
 Ð"  J 5" Ð\8" ÑÑ ‰ M8"
 M85 Ñ
‚
5
5"
G58 ÐJ 5 Ð\8" щM8"
Ð"J 5" Ð\8" ÑщM8"
M85 Ñ 3
kM8 ß 3
G58 ÐM85 Ñ
Á 5ß Z8" kZ8" “.
The inner expectation in the above relation is equal to
D5
5
5"
1ÐJ 5 Ð\8" Ñ ‰ M8"
 Ð"  J 5" Ð\8" ÑÑ ‰ M8"
 D5Ñ
‚
5
5"
G58 ÐJ 5 Ð\8" щM8"
Ð"J 5" Ð\8" ÑщM8"
D 5 Ñ
G58 ÐD 5 Ñ
1Ð?5 ÑG5 Ð?5 ÑÞ
œ
?5
It is easily seen that the expectation with respect to U of the remaining factors in (2.6) is 1
from which the lemma follows.
Remark: Note that (1.3) and (1.4) are unchanged with respect to probability measure U.
Recall that the aim of the paper is to derive the joint distribution of Ð-8 ß \8 Ñ. For that,
consider IT ÒØ\8 ß /4 Ù1Ð-8 Ñ ± l8 Ó where 1 is an arbitrary Borel test function. Here Ø † ß † Ù
denotes the inner product in ‘R .
Again, using the abstract Bayes Theorem (see [6]), we have
IT ÒØ\8 ß /4 Ù1Ð-8 Ñ ± l8 Ó œ
IU ÒØ\8 ß/4 Ù1Ð-8 ÑA8 ±l8 Ó
Þ
IU ÒA8 ±l8 Ó
Ð#Þ(Ñ
The numerator in (2.7) represents the unnormalized conditional expectation. Write
IU ÒØ\8 ß /4 Ù1Ð-8 ÑA8 ± l8 ÓÀ œ
-
1Ð-Ñ;84 Ð-ÑÞ
Ð#Þ)Ñ
240
L. AGGOUN and L. BENKHEROUF
This is a measure-valued process where the normalizing denominator in (2.7) is simply
R
IU ÒA8 ± l8 Ó œ
5
6œ"
;86 Ð5ÑÞ
Theorem 1: The value of ;84 Ð † Ñ is given by the recursion
;84 œ
‚#
Q
3œ#
M8"
á
5" œ!
M8Q
"
M8"
á
5Q œ! 6" œ"
3"
G38 Ð63 M8"
563" M83 Ñ
G38 ÐM83 Ñ
Q
M8"
R
6Q
<œ!
"
8
Q
3
3
3œ"
Q
3
5
8
3
63
# Š M8 ‹ Š M8" ‹ÐJ<3 Ñ63 Ð"  J<3 ÑM8"
53
63
<
‚ Ð"  J43 Ñ53 ÐJ43 ÑM8 53 ;8"
Ð- 
with normalized form :84 Ð-Ñ œ
"
8 Y8 Ñ
E4< G8 Ð6G" M
" ÐM " Ñ
53 )3 :8 Ñ,
3œ"
;84 Ð-Ñ
.
;86 Ð5Ñ
6
4
Also, the marginal distribution of the cost is :8 Ð-Ñ œ R
4œ" :8 Ð-Ñ.
Proof: In view of (1.4), (2.1) and (2.2) and for any test function 1, expression (2.8) is
equal to
R
<œ"
E4< IU ”Ø\8" ß /< Ù1œ-8" 
R
œ
M8"
á
<œ" 5" œ!
‚#
Q
3œ#
Q
"
M8"
M8Q
Ð"  J43 Ñ ‰ M83 )3 :8 A8" -8 ± l8 •
3œ"
á
5Q œ! 6" œ"
3"
G38 Ð63 M8"
563" M83 Ñ
G38 ÐM83 Ñ
Q
M8"
6Q
R
<œ!
"
8
Q
3
3
3œ"
3
R
M8"
<œ" 5" œ!
á
M8Q
8
3
63
# Š M8 ‹ Š M8" ‹ÐJ<3 Ñ63 Ð"  J<3 ÑM8"
53
63
‚ Ð"  J43 Ñ53 ÐJ43 ÑM8 53 IU ÒØ\8" ß /< Ù1Ð-8" 
œ
"
8 Y8 Ñ
E4< G8 Ð6G" M
" ÐM " Ñ
"
M8"
5Q œ! 6" œ"
á
Q
M8"
R
6Q
<œ!
Q
53 )3 :8 Ñ ± l8" Ó
3œ"
"
"
8 Y8 Ñ
E4< G8 Ð6G" M
" ÐM " Ñ
8
8
The Cost of Hidden Perished Items in an Inventory Model
‚#
Q
3œ#
3
63
# Š M8 ‹ Š M8" ‹ÐJ<3 Ñ63 Ð"  J<3 ÑM8"
53
63
Q
3"
G38 Ð63 M8"
63" M83 Ñ
G38 ÐM83 Ñ
3œ"
Q
3
M8"
R
á
<œ" 5" œ!
‚#
Q
3œ#
3
3
‚ Ð"  J43 Ñ53 ÐJ43 ÑM8 53
œ
1Ð? 
?
<
53 )3 :8 Ñ ± ;8"
Ð?Ñ
3œ"
"
M8"
M8Q
Q
M8"
á
5Q œ! 6" œ"
3"
G38 Ð63 M8"
63" M83 Ñ
G38 ÐM83 Ñ
241
6Q
R
"
<œ!
"
8 Y8 Ñ
E4< G8 Ð6G" M
" ÐM " Ñ
8
8
3
63
# Š M8 ‹ Š M8" ‹ÐJ<3 Ñ63 Ð"  J<3 ÑM8"
53
63
Q
3
3
3œ"
3
‚ Ð"  J43 Ñ53 ÐJ43 ÑM8 53
<
1Ð-Ñ;8œ"
Ð- 
-
Q
53 )3 :8 Ñ.
3œ"
This holds for all test functions 1. So the recursion for ;84 in the theorem is true.
Now, we shall be interested in the joint distribution of Ð-8" ß \8" Ñ given the information
accumulated up to time 8. That is, we wish to predict the behavior of Ð-ß \Ñ one period of
the future. Needless to say, this can be helpful for planning purposes of inventory
management.
Consider the unnormalized conditional expectation 38ß8" Ð † Ñ such that
IU ÒØ\8" ß /7 Ù1Ð-8" ÑA8" ± l8 Ó œ
7
1Ð-Ñ38ß8"
Ð-ÑÞ
-
Ð#Þ*Ñ
Then, we have
7
Theorem 2: The value of 38ß8"
Ð † Ñ is given by the recursion:
7
38ß8"
Ð-Ñ
œ
á
2" "
2"
2Q " 5" œ!
2Q
M8"
5Q œ!
M" œ!
á
á
M8Q
R
E74 G"8" Ð6"  2"  ?8" Ñ
MQ œ! 4œ"
‚ # G38" Ð63  M83"  63"  23 Ñ# Š 2533 ‹ Š M683 ‹ ÐJ43 Ñ63 Ð"  J43 ÑM8 63
Q
Q
3œ#
3œ"
‚ Ð"  J73 Ñ23 ÐJ73 Ñ23 53 ;84 Ð- 
where ;84 Ð † Ñ is given recursively in Theorem ".
Proof: The left-hand side of (2.9) is equal to
3
Q
53 )3 T8" Ñß
3œ"
3
242
L. AGGOUN and L. BENKHEROUF
Q
E74 IU ÒØ\8 ß /4 Ù1Ð-8 
4
3
Ð"  J73 Ñ ‰ M8"
)3 :8 ÑA8 -8" ± l8 ÓÞ
3œ"
3
Note that M8"
, 3 œ "ß á ß Q are known at time 8. So to get the result, use the fact that under
probability measure U, they are sequences of independent random variables with probability
distribution G38" , respectively. This completes the proof.
It is worth noting at this stage that the estimates obtained in Theorems 2 and 3 are optimal
in the sense that they minimize the mean squares of the errors. For more details on similar
estimates, see [5].
3. Parameter Estimation
This section is concerned with estimating the parameters
) œ ÖJ43 ß E=< ß G3 Ð.Ñß 3 œ "ß á ß Q ; 4ß =ß < œ "ß á ß R à ! Ÿ . Ÿ H×Þ
Here, we assume for simplicity that the marginal distribution G3 of the demand H3 has finite
support and that T ÒH83  HÓ œ ! for all 8 œ "ß á and 3 œ "ß á ß Q .
Recall that initially we assumed that we have an `ideal' probability space ÐHß Y ß T Ñ. We
shall denote T by T) to emphasize the dependence of T on ).
Our estimation procedure is based on the (EM) algorithm (see [5]). The method starts with
a guess (prior) value of ). Then this value is updated at each time 8 based on the information
gathered in the 5 -field generated by the inventory levels and the prices.
Let
s=< ß G
s) œ šJ
s 34 ß E
s3 Ð.Ñß 3 œ "ß á ß Q à 4ß =ß < œ "ß á ß R à ! Ÿ . Ÿ H›Þ
In order to update ) to s) at time 8, define the martingale
J
>8 œ # # # # Œ 3 44
8
Q P8 R
5œ" 3œ" 6œ" 4œ"
3
MÐ^56
œ"Ñ
s3
J
s=<
‚ # ŠE
E=< ‹
R
s 34
"J
Œ "J43
# # Œ Gs3 Ð.Ñ
G Ð.Ñ
MÐ\5" œ/< ß\5 œ/= Ñ Q
<=œ"
H
3
3
MÐ^5ß6
œ!Ñ
MÐH35 œ.Ñ
.
Ð$Þ"Ñ
3œ" .œ!
It can be shown that IT) Ò>8 Ó œ " so that we can set
.Ts)
.T) ¹Z
œ >8 Þ
8
Let
[8 œ l8 ” 5ÖH53 ß 3 œ "ß á ß Q ß 5 Ÿ 8×ß
and
Ð$Þ#Ñ
The Cost of Hidden Perished Items in an Inventory Model
243
s † ÓÀ œ ITs Ò † ÓÞ
IÒ
)
Then it follows from (3.1) and (3.2) that
R
s og>8 ± [8 Ó œ
IÒ
R
4œ" 3œ"
R
R

4œ" 3œ"
R
R

4œ" 3œ"
s
I
–
P5
8
P5
8
5œ" 6œ"
H
8
P5
8
3
s4
MÐ^5ß6
œ "Ñ ± [8 — logJ
3
5œ" 6œ"
5œ" 6œ"
s
I
–
Q
s
I
–
3
s 34 Ñ
MÐ^5ß6
œ !Ñ ± [8 — logÐ"  J
s=<
Ø\5" ß /< ÙØ\5 ß /= Ù ± [8 — logE
3

3œ" .œ! 5œ"
s Ð.Ñ  VÐ)Ñß
MÐH85 œ .Ñ logG
where VÐ)Ñ is an expression which does not depend on s)Ð8Ñ.
Write
^83 À œ
5œ" 6œ"
^83" À œ
N8=< À œ
P5
8
8
3
MÐ^5ß6
œ "Ñß
P5
5œ" 6œ"
3
MÐ^5ß6
œ !Ñß
8
Ø\5" ß /< ÙØ\5 ß /= Ù
5œ"
08α Ð^83 ÑÀ œ IU Ò^83 Ø\5 ß /α ÙA8 ± [8 Óß
08 Ð^83 ÑÀ œ
R
αœ"
08α Ð^83 Ñ œ IU Ò^83 A8 ± [8 Ó
;8α À œ /U ÒØ\5 ß /α ÙA8 ± [8 Ó
Ð$Þ$Ñ
244
L. AGGOUN and L. BENKHEROUF
;8 ÐN8=< \8 ÑÀ œ IU ÒN8=< A8 \8 ± [8 Ó
;8 ÐN8=< ÑÀ œ Ø;8 ÐN8=< \8 Ñß Ð"ß "ß á ß "ÑÙÞ
Maximizing the conditional log-likelihood in (3.3) with respect to s), we obtain the following
result.
Theorem 2:
s3 Ð.Ñ œ
G
s 34 Ð8Ñ œ
J
8
3
5œ" MÐH5 œ.Ñ
8
IÒ^83 ±[8 Ó
IÒ^83 ±[8 ÓIÒ^83" ±[8 Ó
s8=< œ
E
œ
;8 ÐN8=< Ñ
R
=<
=œ" ;8 ÐN8 Ñ
ß
08 Ð^83 Ñ
38 Ð^83 Ñ08 Ð^83" Ñ
ß
Ð$Þ%Ñ
Ð$Þ&Ñ
Ð$Þ'Ñ
and the unnormalized conditional probability distribution of 08 Ð^83 Ñ; 08 Ð^83" Ñ and ;8 ÐN8=< Ñ
are given by the following recursions:
08 Ð^83 Ñ œ
08α Ð^83 Ñ œ M83 ;8α 
R
" œ"
R
αœ"
08α Ð^83 Ñ,
"
3
Ð^8"
Ñ-8 Ð" ÑE"α ß
08"
Ð$Þ(Ñ
where
3
3"
"
Ð"J"3" щM8"
M83 Ñ
G"8 ÐJ"" ‰M8"
M8" ?8 Ñ Q G38 ÐJ"3 ‰M8"
#
G8" ÐM8" Ñ
G8" ÐM83 Ñ
3œ#
-8 Ð" Ñ œ
;8 ÐN8=< \8 Ñ œ
R
=<
<
Ø;8" ÐN8"
ß \8" Ñß /" ÙE/" -8 Ð" Ñ  E<= /= ;8"
-8 Ð<Ñß
" œ"
<
;8"
œ IU ÒØ\8" ß /< ÙA8" ± [8" ÓÞ
Proof: Straightforward algebra gives (3.4)-(3.6). To derive (3.7), write
3
^83 œ ^8"

P8
6œ"
3
3
3
MÐ^8ß6
œ "Ñ œ ^8"
 M8Þ
Hence,
08α Ð^83 Ñ œ IU Ò^83 Ø\8 ß /α ÙA8 ± [8 Ó
Ð$Þ)Ñ
3
œ IU Ò^8"
ØE\8" ß /α ÙA8" -8 ± [8 Ó  M83 IU ÒØ\8 ß /α ÙA8 ± [8 ÓÞ
The Cost of Hidden Perished Items in an Inventory Model
245
The second expression in (3.8) is M83 ;8α . The scalar product in the first expression in (3.8) is
equal to " Ø\8" ß /" ÙE"α , which leads to
08α Ð^83 Ñ œ
R
" œ"
3
E"α -8 Ð" ÑIU ÒØ\8" ß /" Ù^8"
A8" ± [8" Ó  M83 ;8α
R
œ
" œ"
"
3
E"α -8 Ð" Ñ08"
Ð^8"
Ñ  M83 ;8α
which is expression (3.7). This completes the proof.
4. Summary
In this paper, we proposed another version of a discrete-time, discrete-state inventory model
for items of changing quality. The quality alteration of the items is affected by random
changes in the ambient environment. Also, prices of the items of different quality are allowed
go change from period to period in a random fashion. Based on observing the history of the
inventory level and prices, recursive estimators as well as predictors for the joint distribution
of the accumulated losses and the state of the environment were derived. Further, parameters
estimation were discussed.
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[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
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