FILTERING AND PREDICTING THE COST OF HIDDENPERISHED ITEMS IN AN INVENTORY MODEL

Printed in the U.S.A. ©2002 by North Atlantic Science Publishing Company 235

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.

Hidden Markov Models, Optimal Filtering, Inventory Control.

Key words:

60K30, 60J10, 90B05.

AMS subject classifications:

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 M œ ÐM ß á ß M Ñ_{8 8}^" ₈^Q represents the inventory level at time where 8 M₈³ refers to the inventory level of Class at time , 3 8 3 œ "ß á ß Q and 8 œ !ß "ß á. 2. At time , items in Class are assumed to have a unit price 8 3 T₈³ such that T œ₈³ )_{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 ÖT ×8 defined by T œ ÐT ß á ß T Ñ8 " Q is predicable with respect

8 8

to the history of the inventory model.

3. The random vector H œ ÐH ß á ß H Ñ₈ ^"₈ ^Q₈ , 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 4. New arriving items are assigned to Class 1 and these are represented by a

predictable process ÖY ×₈ defined on ™_.

5. Define the operator “‰" such that for any nonnegative random variable and\ α− Ð!ß "Ñ,

α‰ \ œ ^\^ ß Ð"Þ"Ñ

4œ" 4

where ^₄ is a sequence of iid random variables, independent of , such that\ T Ð^ œ "Ñ œ "  T Ð^ œ !Ñ œ Þ_{4 4} α

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 , items in the stock may8 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

M œ J Ð\₈^{" "} _8"Ñ ‰ M_8"^"  H  Y^"_{8 8}

M œ J Ð\₈^{3 3} _8"Ñ ‰ M_8"³  Ö"  J^3"Ð\_8"Ñ× ‰ M_8"^3" H ß³₈ Ð"Þ#Ñ 3 œ #ß á ß Q Þ

where “‰" is defined in (1.1) and J Ð\³ _8"Ñ ‰ M_8"³ represents the number of items of quality that survived from period 3 8  " and Ö"  J^3"Ð\_8"Ñ× ‰ M_8"^3" is the number of items of quality that moved to quality Class from Class 3 3 3  ". The function J Ð\³ _8"Ñ plays the same role as parameter in (1.1). The dependenceα 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 œ ÖJ Ð/ Ñß á ß J Ð/ Ñ×À œ ÖJ ß á ß J ×Þ^{3 3} _" ³ _{R "}³ _R³

6. The process Ö\ ×_{8 8−} is a Markov chain with semimartingale representation (see [6])

\ œ E\_{8 8"} Q ß₈ Ð"Þ$Ñ where ÖQ ×8 8− is a martingale with respect to the complete filtration

ÖY_{8 8−}× _ generated by Ö\ ×_{8 8−} and denotes the transition probabilityE matrix of the Markov chain Ö\ ×₈ .

7. The accumulated losses up to time due to loss of qualities may be written as8 - œ -8 8" Ö"  J ÐB8"Ñ× ‰ M 3 8: 8 œ "ß á ß Ð"Þ%Ñ

Q 3œ"

3 3

8) ,

with .- œ !_!

Clearly, having an estimate of Ð- ß \ Ñ8 8 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- ß \ Ñ_{8 8} and Ð-_8"ß \_8"Ñ given the complete filtration Öl_{8 8−}× _ generated by the observed processes , ÖM ×_{8 8−}³ _ 3 œ "ß á ß Q and .ÖT ×_{8 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 Ð ß ß UÑH Y such that:

1. The inventory levels ÒM ×_{8 8−}³ are Q sequences of independent random variables, with probability distributions ÖG³_{8 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 ^{G"5ÐJ Ð\" 5"}_G^щM"_{ÐM Ñ}^5""^{" M Y Ñ5" 5}

5 5

‚#^Q Þ Ð#Þ"Ñ

3œ#

ÐJ Ð\ щM Ð"J Ð\ ÑщM M Ñ

ÐM Ñ G

G

3 3 3 3" 3" 3

5 5" 5" 5" 5" 5

3 3

5 5

Set

A₈ ⁸ -₅

5œ!

œ# . Ð#Þ#Ñ

Let Ö ×Z _8− be the complete filtration generated by the processes ÖM ×_{8 8−}; ÖT ×_{8 8−}; ÖH ×_{8 8−}; Ö\ ×_{8 8−}; and Ö^ ×_8ß6³ _8−, 3 œ "ß á ß Q, up to time . Here8

^_8ß6³ refers to the state of the th item in the inventory at the end of period . Thus,6 8

^ œ !ß Ð3  "Ñ

"ß

8ß63 œ if the item is moved to class otherwise.

It can be shown that the process ÖA8 8−× is a Ö ×Z -martingale.

Define a probability measure such thatT

A₈ œ^.T_.U¹ ß Ð#Þ$Ñ

Z8

where A₈ is given by (2.2). The existence of is guaranteed by Kolmogorov's ExtensionT Theorem.

The next lemma asserts that we can recover our original model under .T

Lemma 1: On Ð ß ÑH Y and under probability measure , the demand processes T ÖH ׳₈ , 8 œ "ß #ß á ß 3 œ "ß á ß Q, form sequence of random variables with probability distributions , ÖG³_{8 8−}× _ where

H œ J Ð\^"₈ ^" _8"Ñ ‰ M_8"^"  Y  M ß_{8 8}^"

H œ J Ð\³₈ ³ 8"Ñ ‰ M8"³  Ð"  J^3"Ð\8"ÑÑ ‰ M8"^3"  M ß 3 œ #ß á ß Q Þ₈³

Let be a Borel test function and for , consider the

Proof: 1À™^Ä‘ # Ÿ 5 Ÿ Q

expectation

I Ò1ÐH Ñ ±_T ⁵₈ Z_8"Óß Ð#Þ%Ñ where I Ð † Ñ_T is the expectation with respect to probability measure .T

Now, using the abstract Bayes Theorem (see [6]), expression (2.4) gives

I Ò1ÐH Ñ ±_{T 8}⁵ Z_8"Ó œ ^{I Ò1ÐH ÑU}_{I Ò}⁵⁸_±^{8± 8"}_Ó ^Ó

U 8 8"

A Z A Z

œ^{I Ò1ÐH ÑU}_{I Ò}⁵⁸_±^{8± 8"}_Ó ^ÓÀ œ _H^Rß Ð#Þ&Ñ

U 8 8"

- Z - Z

where -₈ and A₈ are given by (2.1) and (2.2), respectively. Using (2.1) we have

H œ IU ÐJ Ð\ !‰M M Y Ñ

’^{G"8 " 8"}_G^"₈ÐM Ñ₈^{8""" 8" 8}

‚#^Q º •

3œ#

ÐJ ÐB щM Ð"J Ð\ ÑщM M Ñ

ÐM Ñ 8"

G

G

3 3 3 3" 3" 3

8 8" 8" 8" 8" 8

3 3

8 8 Z

œ ÐJ Ð\ ‰ M  D  Y Ñ

D ßáßD

" " " "

8 8" 8" 8

" Q

G

‚#^Q ÐJ Ð\ Ñ ‰ M  Ð"  J Ð\ ÑÑ ‰ M  D Ñ œ "Þ

Mœ#

3 3 3 3" 3" 3

8 8" 8" 8" 8"

G

The numerator R of (2.5) is equal to

I Ò1ÐJ Ð\_U ⁵ _8"Ñ ‰ M_8"⁵  Ð"  J^5"Ð\_8"ÑÑ ‰ M_8"^5!"  M Ñ₈³ -₈±Z_8"ÓÞ Again, using (2.1), the above is

I Ò1ÐJ Ð\U 5 8"‰ M5  Ð"  J5"Ð\8"ÑÑ ‰ M5" M Ñ5

8" 8" 8

‚^{G"8 " 8"}_G" ^8""^{" 8" 8}

8 8

ÐJ Ð\ щM M Y Ñ

ÐM Ñ

‚#^Q ¹ Ó

3œ#

ÐJ Ð\ щM Ð"J Ð\ ÑщM M Ñ

ÐM Ñ 8"

G

G

3 3 3 3" 3" 3

8 8" 8" 8" 8" 8

3 3

8 8 Z

œ IU ÐJ Ð\ щM M Y Ñ

’^{G"8 " 8"}_G^"₈ÐM Ñ₈^{"8" 8" 8}

Ð#Þ'Ñ

‚#^Q

3œ#

ÐJ Ð\ щM Ð"J Ð\ ÑщM M Ñ

ÐM Ñ G

G

3 3 3 3" 3" 3

8 8" 8" 8" 8" 8

3 3

8 8

‚ I Ò1ÐJ Ð\U 5 8"Ñ ‰ M5  Ð"  J5"Ð\8"ÑÑ ‰ M5" M Ñ5

8" 8" 8

‚^{G58ÐJ Ð\5 8"щM8"5 Ð"J}_G⁵_{8ÐM Ñ8}^{55"Ð\8"ÑщM8"5"M Ñ85} kM ß 3 Á 5ß8³ Z8"kZ8"“.

The inner expectation in the above relation is equal to

D

5 5 5" 5" 5

8" 8" 8" 8"

5

1ÐJ Ð\ Ñ ‰ M  Ð"  J Ð\ ÑÑ ‰ M  D Ñ

‚^{G58 5 8" 8"5} _G5 5^{5" 8" 8"5" 5}

8

ÐJ Ð\ щM Ð"J Ð\ ÑщM D Ñ

ÐD Ñ

œ 1Ð? Ñ Ð? ÑÞ

?

5 5 5

5

G

It is easily seen that the expectation with respect to of the remaining factors in (2.6) is 1U

from which the lemma follows. …

Note that (1.3) and (1.4) are unchanged with respect to probability measure .

Remark: U

Recall that the aim of the paper is to derive the joint distribution of Ð- ß \ Ñ8 8 . For that, consider I ÒØ\ ß / Ù1Ð- Ñ ±T 8 4 8 l8Ó where is an arbitrary Borel test function. Here 1 Ø † ß † Ù denotes the inner product in ‘^R.

Again, using the abstract Bayes Theorem (see [6]), we have

I ÒØ\ ß / Ù1Ð- Ñ ±_{T 8 4 8 8}Ó œ I ÒØ\ ß/ Ù1Ð- Ñ ± ÓÞ Ð#Þ(Ñ

I Ò ± Ó

l ^{U 8 4 8 8 8}

U 8 8

A l A l

The numerator in (2.7) represents the unnormalized conditional expectation. Write I ÒØ\ ß / Ù1Ð- Ñ_{U 8 4 8 8}± ₈ÓÀ œ 1Ð-Ñ; Ð-ÑÞ Ð#Þ)Ñ

- 84

A l

This is a measure-valued process where the normalizing denominator in (2.7) is simply I Ò_{U 8}± ₈Ó œ ; Ð5ÑÞ

5 6œ"

R 6

A l 8

Theorem 1: The value of ; Ð † Ñ₈⁴ is given by the recursion

; œ⁴₈ á á E

M M

5 œ! 5 œ! 6 œ" 6

M M R

<œ! 4< Ð6 M Y Ñ ÐM Ñ

" Q

8 8 8" 8"

" Q " Q

" Q

" "

8 " 8 8

" "

8 8

^G _G

‚#^Q #Š ‹ Š ‹^Q ÐJ Ñ Ð"  J Ñ

3œ# 3œ"

Ð6 M 56 M Ñ

ÐM Ñ 5

M M

6 3 6 3 M 6

< <

G G

3 3" 3

8 3 8" 3" 8

3 3

8 8

3 8 3

3

8" 3 3

3

3 8"

‚ Ð"  J Ñ ÐJ Ñ₄^{3 5} ₄^{3 M 5};_8"^< Ð- ^Q5 : Ñ

3œ" 3 3 8

3 3 3

8 ) ,

with normalized form : Ð-Ñ œ⁴₈ ^{; Ð-Ñ48}_{; Ð5Ñ}.

5 6

68

Also, the marginal distribution of the cost is : Ð-Ñ œ₈ ^R_4œ": Ð-Ñ⁴₈ .

In view of (1.4), (2.1) and (2.2) and for any test function , expression (2.8) is

Proof: 1

equal to

” œ •

R Q

<œ" 4< U 8" < 8" 3œ" 3 3 3 8 8" 8 8

4 8

E I Ø\ ß / Ù1 -  Ð"  J Ñ ‰ M ): A - ±l

œ ^R á á ^R E

<œ" <œ!

M M

5 œ! 5 œ! 6 œ" 6

M M

4< Ð6 M Y Ñ

ÐM Ñ

" Q

8 8 8" 8"

" Q " Q

" Q

" "

8 " 8 8

" "

8 8

G G

‚#^Q #Š ‹ Š ‹^Q ÐJ Ñ Ð"  J Ñ

3œ# 3œ"

Ð6 M 56 M Ñ

ÐM Ñ 5

M M

6 3 6 3 M 6

< <

G G

3 3" 3

8 3 8" 3" 8

3 3

8 8

83 3

3

8" 3 3

3

3 8"

‚ Ð"  J Ñ ÐJ Ñ₄^{3 5} ₄^{3 M 5}I ÒØ\_{U 8"}ß / Ù1Ð-_{< 8"}^Q5_{3 3 8}: Ñ ± _8"Ó

3œ"

3 3 3

8 ) l

œ ^R á á ^R E

<œ" <œ!

M M

5 œ! 5 œ! 6 œ" 6

M M

4< Ð6 M Y Ñ

ÐM Ñ

" Q

8 8 8" 8"

" Q " Q

" Q

" "

8 " 8 8

" "

8 8

G G

‚#^Q #Š ‹ Š ‹^Q ÐJ Ñ Ð"  J Ñ

3œ# 3œ"

Ð6 M 6 M Ñ

ÐM Ñ 5

M M

6 3 6 3 M 6

< <

G G

3 3" 3

8 3 8" 3" 8

3 3

8 8

83 3

3

8" 3 3

3

3 8"

‚ Ð"  J Ñ ÐJ Ñ₄^{3 5} ₄^{3 M 5} 1Ð?  5 : Ñ ± ;_8"^< Ð?Ñ

?

Q 3œ" 3 3 8

3 3 3

8 )

œ ^R á á ^R E

<œ" <œ!

M M

5 œ! 5 œ! 6 œ" 6

M M

4< Ð6 M Y Ñ

ÐM Ñ

" Q

8 8 8" 8"

" Q " Q

" Q

" "

8 " 8 8

" "

8 8

G G

‚#^Q #Š ‹ Š ‹^Q ÐJ Ñ Ð"  J Ñ

3œ# 3œ"

Ð6 M 6 M Ñ

ÐM Ñ 5

M M

6 3 6 3 M 6

< <

G G

3 3" 3

8 3 8" 3" 8

3 3

8 8

3 8 3

3

8" 3 3

3

3 8"

‚ Ð"  J Ñ ÐJ Ñ₄^{3 5} ₄^{3 M 5} 1Ð-Ñ;_8œ"^< Ð-  5 : Ñ

-

Q 3œ" 3 3 8

3 3 3

8 ) .

This holds for all test functions . So the recursion for 1 ;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 . That is, we wish to predict the behavior of 8 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

I ÒØ\U 8"ß / Ù1Ð-7 8"Ñ 8"± 8Ó œ 1Ð-Ñ Ð-ÑÞ Ð#Þ*Ñ

-

78ß8"

A l 3

Then, we have

Theorem 2: The value of 3_8ß8"⁷ Ð † Ñ is given by the recursion:

3_8ß8"⁷ G^"

2  " 2  " 5 œ! 5 œ! M œ! M œ!

2 2 M M R

4œ" 74 8" " " 8"

Ð-Ñ œ á á á E Ð6  2  ? Ñ

" Q " Q " Q

" Q " Q

8 8

‚#^Q Ð6  M  6  2 Ñ#Š ‹ Š ‹^Q ÐJ Ñ Ð"  J Ñ

3œ# 3œ"

3 3" 3 6 3 M 6

8" 3 8 3" 3 2 M 4 4

5 6

G ³

3 3

3

8 3 3 3

8

‚ Ð"  J Ñ ÐJ Ñ₇^{3 2} ₇^{3 2 5}; Ð- ₈^{4 Q}5 T Ñß

3œ" 3 3 8"

3 3 3 )

where ; Ð † Ñ₈⁴ is given recursively in Theorem ."

Proof: The left-hand side of (2.9) is equal to

4 74 U 8 4 8 3œ" 3 8 8 8" 8

Q 3 3

7 8"

E I ÒØ\ ß / Ù1Ð-  Ð"  J Ñ ‰ M ): ÑA - ±l ÓÞ

Note that M_8"³ , 3 œ "ß á ß Q are known at time . So to get the result, use the fact that under8 probability measure , they are sequences of independent random variables with probabilityU distribution G³_8", 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 ÖJ ß E ß₄³ _=< G³Ð.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 ÒH  HÓ œ !³₈ 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 Ð ß ß T ÑH Y . We shall denote by T T) to emphasize the dependence of on .T )

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 based on the information) 8 gathered in the -field generated by the inventory levels and the prices.5

Let

) G

s œ J ß E ßšs³4 s=< s³Ð.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 at time , define the martingale) s) 8

>8

8 Q R

5œ"3œ"6œ"4œ"

P Js "Js

J3

MÐ^ œ"Ñ MÐ^ œ!Ñ

"J

œ# # # #Œ⁸ Œ

3 3

4 4

4 3

3 3

56 5ß6

4

‚#^R Š ‹ # #^{Q H}Œ Ð$Þ"Ñ

<=œ" 3œ"

Es

E Ð.Ñ

MÐ\ œ/ ß\ œ/ Ñ

.œ!

s Ð.Ñ MÐH œ.Ñ

=<

=<

5" < 5 = 3

3 3 5

G .

G

It can be shown that I Ò_T)>₈Ó œ " so that we can set

.T ¹

.T^s)₎ 8

Z8 œ> Þ Ð$Þ#Ñ

Let

[₈œl₈” ÖH ß 3 œ "ß á ß Q ß 5 Ÿ 8×ß5 ³₅ and

IÒ † ÓÀ œ I Ò † ÓÞs _T

s)

Then it follows from (3.1) and (3.2) that

IÒs og>8 ±[8Ó œ Is MÐ^ œ "Ñ ±[8 logJs

R R 8

4œ" 3œ" 5œ" 6œ"

P 3

5ß6

3

– — 4

5

4œ" 3œ"^{R R}Is–5œ" 6œ"⁸ MÐ^ œ !Ñ ± — Ð"  J Ñs

P 3

5ß6 8

3

log 4

5

[

4œ" 3œ"^{R R} Is–5œ" 6œ"⁸ Ø\ ß / ÙØ\ ß / Ù ± — Es

P

5" < 5 = 8 =<

log

5

[

^{Q H 8} MÐH œ .Ñ sÐ.Ñ  VÐ Ñß Ð$Þ$Ñ

3œ" .œ! 5œ"

58

logG3 )

where VÐ Ñ) is an expression which does not depend on s)Ð8Ñ. Write

^ À œ₈^{3 8} MÐ^³ œ "Ñß

5œ" 6œ"

P 5ß6

5

^₈^3"À œ ⁸ MÐ^³ œ !Ñß

5œ" 6œ"

P

5ß6

5

N À œ₈^{=< 8} Ø\ ß / ÙØ\ ß / Ù

5œ" 5" < 5 =

0^α₈Ð^ ÑÀ œ I Ò^ Ø\ ß / Ù₈³ _{U 8}³ _{5 α} A₈±[₈Óß

0_{8 8}^{3 R} 0_{8 8}³ _{U 8}³A₈ [₈

œ"

Ð^ ÑÀ œ Ð^ Ñ œ I Ò^ ± Ó

α α

; À œ / ÒØ\ ß / Ù^α_{8 U 5 α} A₈±[₈Ó

; ÐN \ ÑÀ œ I ÒN8 =< 8 U =< 8\ ±8 8Ó

8 8 A [

; ÐN ÑÀ œ Ø; ÐN \ Ñß Ð"ß "ß á ß "ÑÙÞ8 =< 8 =< 8

8 8

Maximizing the conditional log-likelihood in (3.3) with respect to , we obtain the followings) result.

Theorem 2:

Gs Ð.Ñ œ^{3 MÐH œ.Ñ}₈ ß Ð$Þ%Ñ

8

5œ" 3

5

J Ð8Ñ œs³₄ ^{IÒ^ ± Ó} œ ^{Ð^ Ñ} Ð$Þ&Ñ

IÒ^ ± ÓIÒ^ ± Ó 3 Ð^ Ñ Ð^ Ñ

3 3

8 8 8 8

3 3" 3 3"

8 8 8 8 8 8 8 8

[ 0

[ [ 0

E œs⁸_=< ^{; ÐN Ñ} ß Ð$Þ'Ñ

; ÐN Ñ

8 =<

8 R

=œ" 8 =<

8

and the unnormalized conditional probability distribution of 0₈Ð^ Ñ₈³ ; 0₈Ð^₈^3"Ñ and ; ÐN Ñ_{8 8}^=<

are given by the following recursions:

08 3 0 3

8 8 8

R œ"

Ð^ Ñ œ Ð^ Ñ

α

α ,

0^{α α} 0 - "

"

"

"α

8 83 8 83 R 3

œ" 8" 8" 8

Ð^ Ñ œ M ;  Ð^ Ñ Ð ÑE ß Ð$Þ(Ñ

where

- "8 ÐJ ‰M M ? Ñ

ÐM Ñ ÐM Ñ

Q 3œ#

ÐJ ‰M Ð"J щM M Ñ

Ð Ñ œ ^G _G ^G _G

" " " "

8 8" 8 8 8" 8"

" " " 3

8 8 8 8

3 3 3 3" 3" 3

8 8

" # " "

; ÐN \ Ñ œ_{8 8}^=< ₈ ^R Ø;_8"ÐN^=< ß \_8"Ñß / ÙE_{/ 8}Ð Ñ  E / ;_{<= =} ^< ₈Ð<Ñß

œ" 8" 8"

" " "- " -

;_8"^< œ I ÒØ\U 8"ß / Ù< A8"±[8"ÓÞ

Proof: Straightforward algebra gives (3.4)-(3.6). To derive (3.7), write

^ œ ^₈³ _8"³ ^P MÐ^³ œ "Ñ œ ^_8"³  M_8Þ³

6œ" 8ß6

8

Hence,

0₈^αÐ^ Ñ œ I Ò^ Ø\ ß / Ù₈³ _{U 8}³ _{8 α} A₈±[₈Ó

Ð$Þ)Ñ œ I Ò^_{U 8"}³ ØE\_8"ß / Ù_α A_{8" 8}- ±[₈Ó  M I ÒØ\ ß / Ù₈³ _{U 8 α} A₈±[₈ÓÞ

The second expression in (3.8) is M ;_{8 8}^{3 α}. The scalar product in the first expression in (3.8) is equal to _"Ø\_8"ß / ÙE_{" "α}, which leads to

0^α - " A [ ^α

" "α "

8 83 R 3 8 83

œ" 8 U 8" 8" 8" 8"

Ð^ Ñ œ E Ð ÑI ÒØ\ ß / Ù^ ± Ó  M ;

œ ^R E Ð Ñ Ð^ Ñ  M ;

œ" 8 8" 3 3

8" 8 8

" "α- " 0" α

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.

References

[1] Aggoun, L., Benkherouf, L., and Tadj, L., A hidden Markov model for an inventory system with perishable items, J. of Applied Math. and Stoch. Anal. 10:4 (1997), 423- 430.

[2] Aggoun, L., Benkherouf, L., and Tadj, L., A stochastic jump inventory model with deteriorating items, Stoch. Anal. and Appl., to appear.

[3] Aggoun, L., Benkherouf, L., and Tadj, L., Optimal adaptive estimators for partially observed numbers of defective items in inventory models, Math. and Comp. Modeling 29 (1999), 83-93.

[4] Aggoun, L., Benkherouf, L., and Tadj, L., A stochastic inventory model with perishable and aging items, J. of Appl. Math. and Stoch. Anal. 12:1 (1999), 23-29.

[5] Baum, L.E. and Petrie, T., Statistical inference for probabilistic functions of finite state Markov chains, Annals of the Inst. of Stat. Math. 37 (1966), 1554-1563.

[6] Elliot, R.J., Aggoun, L., and Moore, J.B., Hidden Markov Models: Estimation and Control, Springer-Verlag, New York 1995.

[7] Nahmias, S., Perishable inventory theory: A review, Oper. Res. 30 (1982), 680-707.

[8] Rafaat, F., Survey of literature on continuously deteriorating inventory models, J. of Oper. Res. Soc. 42 (1991), 27-37.