Filtering of an Inventory Model with a Multinomial Thinning Operator

In this paper a multivariate discrete-time, discrete-state stochastic inventory model for perishable items is discussed. This model draws on earlier works by the authors and the fractional thinning operator of Steutel and van Harn. Items in stock are assumed to belong to one of M possible categories...

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Datum:	2007
Hauptverfasser:	Lakhdar Aggoun, Lakdere Benkherouf
Format:	Artikel
Sprache:	English
Veröffentlicht:	Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України 2007
Schriftenreihe:	Электронное моделирование
Schlagworte:	Математические методы и модели
Online Zugang:	http://dspace.nbuv.gov.ua/handle/123456789/101621
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Zitieren:	Filtering of an Inventory Model with a Multinomial Thinning Operator / Lakhdar Aggoun, Lakdere Benkherouf // Электронное моделирование. — 2007. — Т. 29, № 1. — С. 3-18. — Бібліогр.: 13 назв. — англ.

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spelling	irk-123456789-1016212016-06-06T03:02:36Z Filtering of an Inventory Model with a Multinomial Thinning Operator Lakhdar Aggoun Lakdere Benkherouf Математические методы и модели In this paper a multivariate discrete-time, discrete-state stochastic inventory model for perishable items is discussed. This model draws on earlier works by the authors and the fractional thinning operator of Steutel and van Harn. Items in stock are assumed to belong to one of M possible categories (representing qualities). At each time t items in the stock may stay in the same class, move to one of theÌ 1 classes or perish. The movement between classes is assumed to be regulated by a multinomial thining operator (to be defined below) which is dependent on some vector-valued parameter process. Recursive estimates for the parameter process are proposed for three possible scenarios. Рассмотрена стохастическая модель управления запасами с многими случайными переменными, дискретная во времени и пространстве, для скоропортящихся товаров. Модель построена на основе предыдущих работ авторов с использованием дробного оператора разрежения Стентела и Ван Харна. Предполагается, что товары на складе относятся к одной из М возможных категорий качества. В каждый момент времени t товары на складе могут оставаться в одном и том же классе, переходить в один из М &#2 1 классов или портиться. Предполагается также, что перемещение между классами регулируется мультиномиальным оператором разрежения, который зависит от некоторого процесса с векторно-оцениваемыми параметрами. Для трех возможных сценариев предложены рекурсивные оценки параметров процесса. Розглянуто стохастичну модель управління запасами з багатьма випадковими змінними, дискретну у часі та просторі, для товарів, що швидко псуються. Модель базується на попередніх роботах авторів з використанням дробового оператора розрідження Стентела та Ван Харна. Прийнято припущення про те, що товари на складі належать до одної з М можливих категорій якості. У кожну мить часу t товари на складі можуть залишатись в одному і тому ж класі, переходити в один із М &#2 . класів, або псуватись. Припускається також, що пересування поміж класами регулюється мультиноміальним оператором розрідження, який залежить від деякого процесу з параметрами, що векторно оцінюються. Для трьох можливих сценаріїв запропоновано рекурсивні оцінки параметрів процесу. 2007 Article Filtering of an Inventory Model with a Multinomial Thinning Operator / Lakhdar Aggoun, Lakdere Benkherouf // Электронное моделирование. — 2007. — Т. 29, № 1. — С. 3-18. — Бібліогр.: 13 назв. — англ. 0204-3572 http://dspace.nbuv.gov.ua/handle/123456789/101621 en Электронное моделирование Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Математические методы и модели Математические методы и модели
spellingShingle	Математические методы и модели Математические методы и модели Lakhdar Aggoun Lakdere Benkherouf Filtering of an Inventory Model with a Multinomial Thinning Operator Электронное моделирование
description	In this paper a multivariate discrete-time, discrete-state stochastic inventory model for perishable items is discussed. This model draws on earlier works by the authors and the fractional thinning operator of Steutel and van Harn. Items in stock are assumed to belong to one of M possible categories (representing qualities). At each time t items in the stock may stay in the same class, move to one of theÌ 1 classes or perish. The movement between classes is assumed to be regulated by a multinomial thining operator (to be defined below) which is dependent on some vector-valued parameter process. Recursive estimates for the parameter process are proposed for three possible scenarios.
format	Article
author	Lakhdar Aggoun Lakdere Benkherouf
author_facet	Lakhdar Aggoun Lakdere Benkherouf
author_sort	Lakhdar Aggoun
title	Filtering of an Inventory Model with a Multinomial Thinning Operator
title_short	Filtering of an Inventory Model with a Multinomial Thinning Operator
title_full	Filtering of an Inventory Model with a Multinomial Thinning Operator
title_fullStr	Filtering of an Inventory Model with a Multinomial Thinning Operator
title_full_unstemmed	Filtering of an Inventory Model with a Multinomial Thinning Operator
title_sort	filtering of an inventory model with a multinomial thinning operator
publisher	Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України
publishDate	2007
topic_facet	Математические методы и модели
url	http://dspace.nbuv.gov.ua/handle/123456789/101621
citation_txt	Filtering of an Inventory Model with a Multinomial Thinning Operator / Lakhdar Aggoun, Lakdere Benkherouf // Электронное моделирование. — 2007. — Т. 29, № 1. — С. 3-18. — Бібліогр.: 13 назв. — англ.
series	Электронное моделирование
work_keys_str_mv	AT lakhdaraggoun filteringofaninventorymodelwithamultinomialthinningoperator AT lakderebenkherouf filteringofaninventorymodelwithamultinomialthinningoperator
first_indexed	2025-07-07T11:09:51Z
last_indexed	2025-07-07T11:09:51Z
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fulltext	Lakhdar Aggoun Department of Mathematics and Statistics, Sultan Qaboos University (P.O.Box 36, Al-Khod 123, Sultanate of Oman, E-mail: laggoun@squ.edu.om), Lakdere Benkherouf Department of Statistics and Operations Research College of Science, Kuwait University (P.O.Box 5969, Safat 13060, Kuwait, E-mail: lakdereb@kuc01.kuniv.edu.kw) Filtering of an Inventory Model with a Multinomial Thinning Operator (Recommended by Prof. E. Dshalalow) In this paper a multivariate discrete-time, discrete-state stochastic inventory model for perishable items is discussed. This model draws on earlier works by the authors and the fractional thinning operator of Steutel and van Harn. Items in stock are assumed to belong to one of M possible cate- gories (representing qualities). At each time t items in the stock may stay in the same class, move to one of the Ì�1 classes or perish. The movement between classes is assumed to be regulated by a multinomial thining operator (to be defined below) which is dependent on some vector-valued parameter process. Recursive estimates for the parameter process are proposed for three possible scenarios. Ðàññìîòðåíà ñòîõàñòè÷åñêàÿ ìîäåëü óïðàâëåíèÿ çàïàñàìè ñ ìíîãèìè ñëó÷àéíûìè ïåðå- ìåííûìè, äèñêðåòíàÿ âî âðåìåíè è ïðîñòðàíñòâå, äëÿ ñêîðîïîðòÿùèõñÿ òîâàðîâ. Ìîäåëü ïîñòðîåíà íà îñíîâå ïðåäûäóùèõ ðàáîò àâòîðîâ ñ èñïîëüçîâàíèåì äðîáíîãî îïåðàòîðà ðàçðåæåíèÿ Ñòåíòåëà è Âàí Õàðíà. Ïðåäïîëàãàåòñÿ, ÷òî òîâàðû íà ñêëàäå îòíîñÿòñÿ ê îäíîé èç Ì âîçìîæíûõ êàòåãîðèé êà÷åñòâà. Â êàæäûé ìîìåíò âðåìåíè t òîâàðû íà ñêëàäå ìîãóò îñòàâàòüñÿ â îäíîì è òîì æå êëàññå, ïåðåõîäèòü â îäèí èç Ì�1 êëàññîâ èëè ïîðòèòüñÿ. Ïðåäïîëàãàåòñÿ òàêæå, ÷òî ïåðåìåùåíèå ìåæäó êëàññàìè ðåãóëèðóåòñÿ ìóëüòè- íîìèàëüíûì îïåðàòîðîì ðàçðåæåíèÿ, êîòîðûé çàâèñèò îò íåêîòîðîãî ïðîöåññà ñ âåêòîð- íî-îöåíèâàåìûìè ïàðàìåòðàìè. Äëÿ òðåõ âîçìîæíûõ ñöåíàðèåâ ïðåäëîæåíû ðåêóðñèâíûå îöåíêè ïàðàìåòðîâ ïðîöåññà. K e y w o r d s: partially observed inventory model, multinomial thinning operator, optimal filtering. 1. Introduction. Deterioration (perishability) of items while in stock is a real fact. Food, electronic components, pharmaceuticals, and drugs are just a few ex- amples of such items: see [1—3]. In this paper we consider a multivariate dis- crete state, discrete time stochastic inventory model for perishable items, where ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2007. Ò. 29. ¹ 1 3 �� items are assumed to belong to M +1 possible categories (representing qualities). Categories are assumed to be ordered so that Category 1 houses the best quality and quality M houses the pre-perished quality and the perished items are housed in Category M + 1. At each time t, t = 1, … items in Category i, i = 1,..., M, that have not been sold either stay in the same class, or move to a lower class. The move- ment between classes is regulated by some multinomial thinning operator to be de- fined below. As a matter of fact the proposed model builds on an earlier work by the authors where the Binomial thinning operator « ° » is used: see [4, 5] . To Binomial thinning operator is defined as follows. For any nonnegative integer-valued random variable X and ��[ , ]0 1 , let � � X Y j j X � � � 1 , where Y1, Y2, ... is a sequence of of i. i. d. random variables independent of X, such that P(Yj = 1) = 1 – P (Yj = 0) = �. We assume that � � X �0 if X � 0. The operator « ° » was used by [6, 7] to examine integer-valued time series and to model count data. Here we shall assume that the inventory consists of a single item. Let X n i , i = = 1, ..., M + 1, n = 1, ... be the level of stock of the item in category i, at time n. We also assume that within period n, an item of quality i either keeps its quality with probability� i i or move to any of the M – i lower qualities i+1, i+ 2, ..., M with probabilities � � �i i i i M i � �1 2, ,..., , or perish with probability �� i i � �1 � � � �� i i M i M i 2 1... . The inventory dynamics now take the form X X U Vn n n n 1 1 1 1 1 1�� , X X X Vn n n n 2 2 1 1 1 2 2 1 2 2�� , � (1) X X X X Vn M M n M n M M n M n M�� 1 1 1 2 1 2 1� � �� ... , X X X Xn M M n M n M M n M� � � � � � �� 1 1 1 1 1 1 2 1 2 1 1�� ... , where U is a Z�-valued process representing the replenishment process which is assumed to be predictable with respect to the filtration generated by the inven- tory process, and Vn n n n MV V V� � ( , , ,..., , )1 2 0 (2) is a Z� M -valued random variable with distribution� n representing the demand at each epoch n. Lakhdar Aggoun, Lakdere Benkherouf 4 ISSN 0204–3572. Electronic Modeling. 2007. V. 29. ¹ 1 Now we introduce the multinomial operator . This operator can be seen as a natural generalization of the binomial thinning. Let X be an non-negative, inte- ger-valued random variable. For a given value of X, suppose that a random ex- periment consists of classifying each of X objects into one of M + 1 categories with probabilities � � � �1 2 1, ,..., ,M M � such that � � � �1 2 1 1� � � � ��... M M . Write S X M i i M X� � � � � � � � � � � � � �� ( , ..., ) :1 1 1 1 , where � �1 1,..., M � are nonnegative integers. Let �� ( , ,..., )1 2 1M . (3) Then �� X I SX � �� ( )Y , where Y = (Y 1, ..., Y M +1)� , Y i is the (random) num- ber of objects that result in class i. Then with ei denoting the M-dimensional standard unit vector with 1 in the i-th position and zeroes elsewhere, the dynamics in (1) take the form X X U Vn i n i i M n ne� � �� 1 1 , , (4) where X n n n n M X X X � � � � � � � � � � � � �� 1 2 1 � , Un nU � � � � � � � � � � � � � 0 0 � , �� 1 1 1 2 1 1 1 1 � � � � � � � � � � � � � � � � �� M M , �� 2 2 2 2 1 2 0 � � � � � � � � � � � � � � � � � M M , ..., �� M M M M M � � � � � � � � � � � � � � � � 0 0 1 � , Vn is given in (2). A generalization of the operator [8] was proposed in [9]. In both articles this operator or its generalization gave rise to some new integer-valued time se- ries where their properties were examined. The present paper take a different ap- proach and focusses on estimating dynamically the parameters� � , ..., M . To do that we shall consider three possible scenarios. Initially, we shall adopt a Bayesian point of view and assume that parameters � � , ..., M have some given prior density. Based on observing the level of stock of the items, recursive esti- mate for the posterior density is proposed. This is done in the next section. In the second scenario, it is assumed that the parameters� � , ..., M are no longer static in time but dynamic and changes according to some Markovian rule. Also, based Filtering of an Inventory Model with a Multinomial Thinning Operator ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2007. Ò. 29. ¹ 1 5 on observing the inventory history recursive estimates are proposed in section 3. The final scenario builds on the fact that the assumption of full observation of the level of stock is not always valid as transaction errors, spoilage, product quality and yield render full observation of the level of stock difficult. In this paper, we shall consider the zero balance walk proposed by [10, 11], where at each period demand is only observed when the inventory level drops to zero. The paper concludes with the analysis of the zero-balance walk model and some general remarks. 2. Recursive parameter estimation. In this section we derive recursive es- timates for the parameters � �� , ..., .M We suppose that each �� i takes values in a measurable space (�i i i, ,� � ). The values of � �� , ..., M are unknown and, in this section, we suppose they are constant. Write Fn for the complete history generated by the observed inventory Xk, k = 0, 1, . . ., n, and Gn for the complete history generated by the inventory and the parameters � �� , ..., .M To make computations easy: see [12] we shall work under a reference prob- ability measure P where the process X is a sequence of i.i.d. random variables with probability distribution �. Set � 0 1� , and for k �1 � � � � � k n i n i i M n n n e � � � ! "" # $ %%� � � X U X X 1 1 , ( ) , &n k k n � � '� 0 . It can be shown that the process {&n } is a martingale with respect to the filtration Gn . Therefore, we can relate P and P by setting dP dP n nG � & . We shall call the probability measure P the «real world» measure. Using similar arguments to those used in [12], we can also show that under the «real world» measure P that the dynamics in (4) hold where V Xn i n i i M e� �� 1 1 , � �U Xn n , and under P, Vn has distribution � n . Now, from observing the inventory level, we are interested in computing E I di i i M n( )� (� (� � � � � � � � ' 1 F . A generalized version of Bayes Theorem: see [12] gives: E I d E I d i i i M n n i i i M n ( ) ( ) � ( � ( � ( � ( � � � � � � � � � � � � � �' ' 1 1F F& ) * � � � E n n& F . Lakhdar Aggoun, Lakdere Benkherouf 6 ISSN 0204–3572. Electronic Modeling. 2007. V. 29. ¹ 1 The numerator of the above expression represents a unnormalized condi- tional expectation. Write E I d q dn i i i M n n M& ( ) ( , ..., ) (� ( ( ( (� ( ( ( � (� � � � � � �� ' 1 1 1F 1 ) ... ( ).d M M� (( The normalizing denominator E n n[ ]& F is given by � �1 1 1 1 + + , ... ( , ..., ) ( ) ... ( ) M q d dn M M M( ( � ( � (( ( ( ( . The next theorem provides a recursion for qn M( , ..., )( (( (1 . Theorem 1. Suppose h M( , ..., )( (( (1 is the prior density for � �� , ..., .M Then q hM M 0 1 1( , ..., ) ( , ..., ) ,( ( ( (( ( ( (� and the updated estimates are given recur- sively by qn M( , ..., )( (( (1 � � � � ! "" # $ %% � � � �� i M n i i M n n n n i i n i X 1 1 1 1 � �� S X U X X � � ( ) � � �i i i i M i i M � �� ! "" # $ %%+' 1 11 ... + � � � �( ) ( ) ...( ) ( ,..., )( ( ( ( (( (i i i i M i Mi i i i M i f� � � 1 1 11 1 qn M �1 1( ,..., ) .( (( ( Here S X M i n i M n X � � � � �� 1 1 1 1 1 2 1 1 1 1 1 1 1 1 { ( , , ..., ) : }�� , S X M i n i M n X � � � � �� 1 2 2 2 2 3 2 1 2 2 1 2 2 1 0{ ( , , , ..., ) :�� }, … S X M M M M M M M M M n M n M X � � � � � �� 1 0 0 1 1 1{ ( , , ..., , ) : }�� , X M Xn i i i M i n i i i M � � � � ! "" # $ %% �1 1 2 1 1 1 2 1� � � � � �... ! ! !... i ! , (( ( ( ( 1 1 1 2 1 1 1 � � � � � � � � � � � � � � � M , (( ( ( 2 2 2 1 2 0 � � � � � � � � � � � � � � � M , ..., (( ( ( M M M M M � � � � � � � � � � � � � � � � 0 0 1 � , Filtering of an Inventory Model with a Multinomial Thinning Operator ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2007. Ò. 29. ¹ 1 7 �� 1 1 1 2 1 1 1 � � � � � � � � � � � � � �M , �� 2 2 2 1 2 0 � � � � � � � � � � � � � �M , ..., �� M M M M M � � � � � � � � � � � � � � � � 0 0 1 . P r o o f. Let f be a test function then by definition: E fn M n[ ( ,..., )& � �� F � � + + , � �1 1 1 1 1 ... ( , ..., ) ( , ..., ) ( ) . M f q dn M n M( ( ( ( (( ( ( ( � ( .. ( )d M M� (( . However E fn M n[ ( ,..., )& � �� F � � � � ! "" # $ %% � � �� Å e fn n i i M n i n n n & 1 1 1� � � � � � X U X X , ( ) ( ,..., )�� M nF � � � � � � � � � � � � � � � � ! "" # $ %% + + � � , � Å e f M n i i M n i n n n� �1 1 1 ... , ( ) � ( � ( X U X X ( ,..., )( (( ( M � � � � � � � � � � � � , q d dn M M M n� �+ �1 1 1 1 1( ,..., ) ( ) ... ( )( ( ( (( ( � ( � ( F � � � ! "" # $ �� + + � � �� , � �� i n i Mi M n i i M n n Å S X U X 1 11 1 � �... � %% � � � � � � � � ' � ( ) ( ) X Y n i i i M I �� 1 , f q d dM n M M M( ,..., ) ( ,..., ) ( ) ... )( ( ( ( ( (( ( ( ( � ( � - (1 1 1 1 1 � Fn� � �� 1 � � � ! "" # $ % �� + + � � �� , � �� i n i Mi M n i i M n n S X U X 1 11 1 � �... � % ! "" # $ %%+ � � �� ' � ( ) ...X n n i i i i i M i i M X 1 1 11 � � � Lakhdar Aggoun, Lakdere Benkherouf 8 ISSN 0204–3572. Electronic Modeling. 2007. V. 29. ¹ 1 + � � � �( ) ( ) ...( ) ( ,..., )( ( ( ( (( (i i i i M i Mi i i i M i f� � � 1 1 11 1 + + �q d dn M M M 1 1 1 1( ,..., ) ( ) ... ( )( ( ( (( ( � ( � ( . Since f is an arbitrary test function the result follows. 3. A finite state case with Markovian dynamics. In this section we assume that each group of parameters �� 1 1 1 1 1� �( ,..., )M , �� 2 2 2 1 2� �( ,..., )M , ..., �� M � � �( , )� �M M M M 1 is a set of dependent finite-state Markov chains. For the sake of simplicity we suppose that the Markov chains in group i have state spaces equal to Ki. More precisely we have for n � 0: � �1 1 11 1 11 1 1 1 1 11 ( ) { ,..., },..., ( ) { ,...( )n p p n pK M M� �� , }( )p M K 1 1 1 � , � �2 2 22 1 22 1 2 2 1 12 ( ) { ,..., },..., ( ) { ,...( )n p p n pK M M� �� , }( )p M K 2 1 2 � , ... � �M M MM MM K M M M M Mn p p n p p M ( ) { , ..., }, ( ) { , ...,( ) (� �� 1 1 1 1 M K M �1) } . Without any loss of generality, we identify the state space of each Markov chain in group i, with the set of standard unit vectors R K i . Write Fn for the complete filtration generated by the observed inventory X, and Gn for the complete filtration generated by the inventory and the processes �� 1 1 1 1 1� �( , ..., )M , �� 2 2 2 1 2� �( , ..., )M , ..., �� M M M M M� �( , )1 . Now we define the probability transitions of the above processes. With . denoting the tensor product of two vectors we assume the following: P n e M M s n. � .� �� 1 1 1 1 1 1 1� ( ) G � .� �� . � � � � � �P n e n n M M s M � � � �1 1 1 1 1 1 1 1 1 11� � �( ) ( ), ..., ( � � �� 1) , P n e M M s n.� �� . � �� 2 1 2 2 1 2 1� ( ) G � .� �� . � � � � � �P n e n n M M s M � � � �2 1 2 2 1 2 2 2 1 21� � �( ) ( ), ..., ( � � �� 1) , . . . P n n e eM M M M s M s M nM M � �( ) ( ) ].� �� . � �� 1 11 G Filtering of an Inventory Model with a Multinomial Thinning Operator ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2007. Ò. 29. ¹ 1 9 � . � .� �� P n n e e nM M M M s M s M M M M M M M � � � �( ) ( ) ( ),...,1 11 1 ( )n� � �� 1 . Write P n e n e M M s r M.� �� . � � � � � � � � � � �1 1 1 1 1 1 1 1 11 1 � � �( ) ( ) ,..., 1 1 11 1 ( )n erM � � � �� a s s r rM M1 1 1 1 1 1 1 1 1 , ..., ; , ..., , P n e n e M M s r M.� �� . � � � � � � � � � � �2 1 2 2 1 2 2 2 21 2 � � �( ) ( ) ,..., 1 2 21 1 ( )n erM � � � �� a s s r rM M2 2 1 2 2 2 1 2 2 , ..., ; , ..., , . . . P n n e e n eM M M M s M s M M M r M M M M M M � � � �( ) ( ) ( ) ,.� �� . � �� 1 11 1 ( )n er M M � � � �� 1 1 � � � a s s r r M M M M M M M M M, ..., ; , ...,1 1 ; A a s s r rM M 1 1 1 1 1 1 1 1 1 1� � � { } , ..., ; , ..., , s s r r K M M1 1 1 1 1 1 1 1 1 2 1,..., , ..., , , ...,,� � � , A a s s r rM M 2 2 2 2 1 2 2 2 1 2� � � { } , ..., ; , ..., , s s r r K M M2 2 1 2 2 2 1 2 1 2 2,..., , ..., , , ...,,� � � , ... A aM s s r r M M M M M M M M M� � � { } , ..., ; , ...,1 1 , s s r r K M M M M M M M M M, ..., , , , , ...,� � � 1 1 1 2 . We have the following representations [12, 13]: . � . � � � � � � i M i i M i nn A n W 1 1 1 1 1 1 1 11� �( ) ( ) , . � . � � � � � � i M i i M i nn A n W 2 1 2 2 2 1 2 21� �( ) ( ) , ... (5) ..., � � � �M M M M M M M M M n Mn n A n n W( ) ( ) ( ) ( ). � � . � �� 1 11 1 . Here W1 is a martingale increment process with respect to the filtration generated by the processes � �1 1 1 1 2,..., ,M W� is a martingale increment process with re- spect to the filtration generated by the processes � �2 2 1 2,..., ,M MW� is a martin- Lakhdar Aggoun, Lakdere Benkherouf 10 ISSN 0204–3572. Electronic Modeling. 2007. V. 29. ¹ 1 gale increment process with respect to the filtration generated by the processes � �M M M M, �1. Since . � .� �� / � � � � � � � � �1 1 1 1 1 1 1 1 11 11M M s sn e n p� �( ) ( ) , ...,� M M sn p M � �� 1 1 1 1 1 1 ( ) ( ) ( ) , . � .� �� / � � � � � � � � �2 1 2 2 1 2 2 2 22 22M M s sn e e n p� �( ) ( ) , ..., ( ) ( ) ( ) � M M sn p M � �� 1 2 2 1 2 1 , … � � � �M M M M s M s M M M MM s Mn n e e n p M M MM ( ) ( ) ( ) ,. � .� �� / �� 1 1 � �� 1 1 1M M M sn p M M ( ) ( ) ( ) . Therefore ��n n s s M s e p p p M 1 1 1 11 12 1 1 11 12 1 1 � � � � � � � � � � � � � � X , ( ) ( ) � � � � � + � � �� s s K n M e 11 1 1 1 1 1 1 ,..., ( ) ,X + � �� I n p n ps M M s M [ ( ) , ..., ( ) ]( ) ( ) � �1 1 11 1 1 1 1 11 1 1 , ��n n s M s s e p p M 2 1 2 22 2 1 0 22 2 1 22 � � � � � � � � � � � � � � � � X , ( ) ( ) � ,..., ( ) , s K n M e 2 1 2 1 1 2 � � �� +X + � �� I n p n ps M M s M [ ( ) , ..., ( ) ]( ) ( ) � �2 2 22 1 2 2 1 22 2 1 , ��n M n M MM s M M s e p p MM M M � � � � � � � � � � � � � � � � � � X 1 1 0 0 1 , ( ) ( ) � � � + � � �� s s K n M MM M M M e ,..., ( ) , 1 1 1X + � �� I n p n pM M MM s M M M M sMM M M [ ( ) , ( ) ]( ) ( ) � � 1 1 1 , and the dynamics in (4) take the form X X U Vn n i i M n i n ne� � � � �� 1 1 , . (6) Filtering of an Inventory Model with a Multinomial Thinning Operator ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2007. Ò. 29. ¹ 1 11 Write p 1 11 12 1 1 11 12 1 1 � � � � � � � � � � � � � � �� p p p s s M s M � ( ) ( ) , p 2 22 2 1 0 22 2 1 � � � � � � � � � � � � � � � p p s M s M � ( ) ( ) , ..., p M MM s M M s p p MM M M � � � � � � � � � � � � � � � � �� 0 0 1 1 � ( ) ( ) We shall adopt a similar approach to that used in the previous section and work under a reference probability measure P where the process X is a sequence of i.i.d. random variables with probability distribution �. Set � 0 1� , and &n k k n � � '� 0 , with � � � � � k n k i i M k i k k k e � � � ! "" # $ %% � �� 1 1X U X X , ( ) , k = 1, ... . It can be shown that the process {&n } is a martingale with respect to the filtration Gn . Therefore, we can relate P and P by setting dP dP n n� & . The probability measure P is the «real world» measure. Using similar argu- ments to those used in [12], we can also show that under the «real world» mea- sure P that the dynamics in (6) hold. We shall be interested in computing E n e n e M M s M M s. . . . � � � � � � � � � � � � � �� 1 1 1 1 1 1 2 1 2 2 1 � �( ), ( ), 2 1 1 ... ( ) ( ), .� �M M M M s M s M nn n e e M M . .� �� F (7) A generalized version of Bayes Theorem: see [12] shows that equation (7) is equal to: E n e nn M M s M M [ ( ), ( ),& . . . . � � � � � � � � � � � � � ��1 1 1 1 1 1 2 1 2 2 1 � � e n n e e E s M M M M s M s M n n n M M� 2 1 1 ... ( ) ( ), ] [ ] . � �. .� � F F& The numerator of the above expression represents a unnormalized condi- tional expectation. Let this expression be denoted by qn (s1, s 2, ..., s M), where s 1 1 1 1 1� �( ,..., )s sM , s 2 2 2 1 2� �( ,..., )s sM , ..., s M M M M Ms s� �( , )1 . The next theorem gives a recursion for qn (s1, s 2, ..., s M). Lakhdar Aggoun, Lakdere Benkherouf 12 ISSN 0204–3572. Electronic Modeling. 2007. V. 29. ¹ 1 Theorem 2. Suppose p0 is the probability distribution of � �� 1, ..., .M Then for n �1the updated estimates are given recursively by qn M( , , ..., )s s s 1 2 � � � � �� r r K r r K r rM M M M M M 1 1 1 1 1 2 2 1 2 2 11 1, ..., , ..., , ... 1 1 1 1 1 1 1 1 1 1 2 2 1 2 2 K s s r r s s r M M M M a a� � � �, ..., ; , ..., , ..., ; 2 1 2 2 , ..., ... rM� ... , ..., ; , ..., a s s r r M M M M M M M M M � �1 1 i M n i i m n n n n i i n i X � � � �� ! "" # $ %% 1 1 1 1 �� S X U X X � � ( ) � � �i i i i M i i m � �� ! "" # $ %%+' 1 11 ... + � � � � � ( ) ( ) ...(( ) ( ) ( ) ( p p pii s i i s i M sii i i i i i i i M � � 1 1 1 1 1) ) ( ,..., ) �M i qn M� � 1 1 1 r r . Here r 1 1 1 1 1� �( ,..., )r rM , r 2 2 2 1 2� �( ,..., )r rM , ..., r M M M M Mr r� �( , )1 . P r o o f. First note that E n e nn M M s M M & . .� �� . . � � � � � � �� 1 1 1 1 1 1 2 1 2 2 � �( ), ( ), � �. . 1 2 1e n n es M M M M s M M� ... ( ) ( ),� � . � �� . . . � � � � � � � � e E n es M n n M M s M M 1 1 1 1 1 1 1 1 2 F [ ( ),& � � � �� 1 2 2 1 2� � � � ( ), ...n e M s. � � ... ( ) ( ),� �M M M M s M s Mn n e e M M . .� �1 1 . Using the representations in (5) this is � � n i i M n i n n n n ep X U X X � �� ! "" # $ %% �1 1 , ( ) ]F � . � .� �� .� � � � � � � E A n e An i M i M s i M i& 1 1 1 1 1 1 1 1 2 2 1 21� �( ), � � ( ), ...n e M s� . � � 1 2 1 2 � � ... ( ) ( ),A n n e eM M M M M s M s M M M � �� . � .� � 1 11 1 , � � n i i M n i n n n n ep X U X X � �� ! "" # $ %% � � � � � � �1 1 , ( ) F Filtering of an Inventory Model with a Multinomial Thinning Operator ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2007. Ò. 29. ¹ 1 13 � � � �� r r K r r K r rM M M M M M 1 1 1 1 1 2 2 1 2 2 11 1, ..., , ..., , ... 1 1 1 1 1 1 1 1 1 1 2 2 1 2 2 K s s r r s s r M M M M a a� � � �, ..., ; , ..., , ..., ; 2 1 2 2 , ..., ... rM� ... , ..., ; , ..., a s s r r M M M M M M M M M � �1 1 E n e nn i M i M r i M i[ ( ), ( ),& � � � � � � � . � . . � .1 1 1 1 1 1 1 2 1 21 1� � � � � �� 2 1 2 M re ... ... ( ) ( ),� �M M M M r M r Mn n e e M M � . � .� � 1 11 1 , � � n i i M n i n n n n ep X U X X � � � � � � ! "" # $ %% �1 1 1 , ( ) ]F � � � �� r r K r r K r rM M M M M M 1 1 1 1 1 2 2 1 2 2 11 1, ..., , ..., , ... 1 1 1 1 1 1 1 1 1 1 2 2 1 2 2 K s s r r s s r M M M M a a� � � �, ..., ; , ..., , ..., ; 2 1 2 2 , ..., ... rM� ... , ..., ; , ..., a s s r rM M M M M M M� �1 1 2 2 i M n i i m n n n n i i n i X X � � � �� ! "" # $ %% 1 1 1 1 �� S X U X � � ( ) � � �i i i i M i i m � �� ! "" # $ %%+' 1 11 ... + � � � � � ( ) ( ) ...(( ) ( ) ( ) ( p p pii s i i s i M sii i i i i i i i M � � 1 1 1 1 1) ) ( ,..., ) �M i qn M� � 1 1 1 r r . Here r 1 1 1 1 1� �( , ..., )r rM , r 2 2 2 1 2� �( , ..., )r rM , ..., r M M M M Mr r� �( , )1 . 4. A partially observed inventory. In this section we assume that the in- ventory level is not observed at all time. However, the management observes the event when the inventory falls to zero and cannot observe the inventory when it is positive. To study such partial observations of the inventory levels, we intro- duce a signal (message) random variable Z I Xn i n i� � ( )0 , n = 0, 1, 2, ... . The processes Z i, i = 1, ..., M are discrete-time Markov Chains with the state space the set {0,1} where 1 means an empty inventory and 0 means a nonempty one. Write Gn k k i i k k Z i M k n� � �0 ��{ , , , , ..., , , , }X U V1 , Fn k i k k Z i M U V k n� � �0{ , ,..., , , , }1 . We shall suppose that: P Z m P Z m Z X U Vn i n n i n i n i n n [ ] [ , , , ]� � �� 1 1 1 1 1 . Lakhdar Aggoun, Lakdere Benkherouf 14 ISSN 0204–3572. Electronic Modeling. 2007. V. 29. ¹ 1 Write a X U V P Z m Z X U Vm n n n n i n i n n n, ( , , ) [ , , ,� �� 1 1 1 1 1 1 1 ] . We shall work under a reference probability measure P where the the pro- cess X is a sequence of i.i.d. random variables with probability distribution �, and the processes Z i are i.i.d. random variables uniformly distributed on the set {0, 1}. Set � 0 1� , and for k � 1, � � � � � k n i i M k i k k k e � � � ! "" # $ %% +� �� 1 1X U X X , ( ) + � � � � � �' ' � i M m m i k k k I Z m Z a k i k 1 0 1 1 1 12 1 � � , , , ( , ( ( , , ))X U V i ��) , and &n k k n � � '� 0 . It can be shown that the process {&n } is a martingale with re- spect to the filtration Gn . Therefore, we can relate P and P by setting dP dP n nG � & . Under the «real world» measure P the dynamics in (6) hold. We wish to find a re- cursion for E I I dn n i i i M n& ( ) ( )X x� � � � � � � � � ' � (� ( 1 F . Let this expression be denoted by qn M( , , ..., )s s s 1 2 , where s 1 1 1 1 1� �( , ..., )s sM , s 2 2 2 1 2� �( , ..., )s sM , ..., s M M M M Ms s� �( , )1 . Theorem 3. Suppose h M( , ,..., )x ( (( (1 is the prior density for X0, �� , ... ..., �� M . Then q x h xM M 0 1 1( , , ..., ) ( , , ..., )( ( ( (( ( ( (� , and the updated estimates are given recursively by qn M( , ,..., )x ( (( (1 � z i M m m i n n I Z m Z a n i n i � ' ' � � � � � �� 1 0 1 1 12 1 � � , , , ( , ( ( , , ))z U V �) + + � � ! "" # $ %% � � � � � � � i M n i i M n i i i i Mi zi1 1 1� � � � � � S U x z � ... �� ! " # $ %+' 11 i i M + � � � � �( ) ( ) ...( ) ( , , ...( ( ( ((i i i i M i n i i i i M i q� � � 1 1 1 11 1 z , )(( M . For the notation see Theorem 1. Filtering of an Inventory Model with a Multinomial Thinning Operator ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2007. Ò. 29. ¹ 1 15 P r o o f. Let f be a test function then by definition: E I fn n M n[ ( ) ( ,..., )& X x� �� F � + + , � �1 1 1 1 1 ... ( , ..., ) ( , , ..., ) ( ) M f q dM n M( ( ( (( ( ( ( � ((x ... ( )d M M� (( . However E I fn n M n[ ( ) ( , ..., )& X x� �� F � � � � ! "" # $ %% � � � � �� E I e n n n i i M n i n ( ) , ( ) X x X U x x & 1 1 1� � � �� + + � � � � � �' ' � i M m m i n n n I Z m Z a n i n 1 0 1 1 1 12 1 � � , , , ( , ( ( , , ))X U V i f M n � � � � ��) ( , ..., )� �� F � � � ! "" # $ %% � � � +� � ��E en n i i M n i n& 1 1 1� �� X U x, + � � � � � �' ' � i M m m i n n n I Z m Z a n i n 1 0 1 1 1 12 1 � � , , , ( , ( ( , , ))X U V i f M n � � � � ��) ( , ..., )� �� F �� ' ' � � � � � � z z U V i M m m i n n I Z m Z a n i n i 1 0 1 1 12 1 � � , , , ( , ( ( , , )) � +�) + � � � ! "" # $ %%� � � �E I e fn n n i i M i n[ ( ) , ( ,X z z U x1 1 1 & � � �� ..., ) ]�� M nF � �� ' ' � � � � � � z z U V i M m m i n n I Z m Z a n i n i 1 0 1 1 12 1 � � , , , ( , ( ( , , )) � +�) + � � � � � � ! "" # $ %% + + � , �E e f M n i i M i n � �1 1 1 ... , ( ,� ( (( (z U x ..., )(( M + + � �� q d dn M M M n1 1 1 1 1( , , ..., ) ( ) ... ( )z ( ( ( (( ( � ( � ( F �� ' ' � � � � � � z z U V i M m m i n n I Z m Z a n i n i 1 0 1 1 12 1 � � , , , ( , ( ( , , )) � +�) Lakhdar Aggoun, Lakdere Benkherouf 16 ISSN 0204–3572. Electronic Modeling. 2007. V. 29. ¹ 1 + � � ! "" # $ %% � � + + � � � , � � i M n i i M n i n i M1 1 1 1�� S X U x � �... � z � � �i i i i M i i M � �� ! " # $ %+' 1 11 ... + � � � �( ) ( ) ...( ) ( , ..., )( ( ( ( (( (i i i i M i Mi i i i M i f� � � 1 1 11 1 qn M � +1 1( , , ..., )z ( (( ( + d d M M� ( � (( (1 1( ) ... ( ) . Since f is an arbitrary test function the result follows. In this paper we proposed a multivariate discrete-time, discrete state sto- chastic inventory model for perishable items. The proposed model is based on the multinomial thinning used in integer-valued time series analysis and for modeling count data. The present paper was concerned with estimating the vec- tor valued parameter process of the multinomial thinning where recursive esti- mators were proposed from the Bayesian point of view, the dynamic view and fi- nally the partial observed case. Ðîçãëÿíóòî ñòîõàñòè÷íó ìîäåëü óïðàâë³ííÿ çàïàñàìè ç áàãàòüìà âèïàäêîâèìè çì³ííèìè, äèñêðåòíó ó ÷àñ³ òà ïðîñòîð³, äëÿ òîâàð³â, ùî øâèäêî ïñóþòüñÿ. Ìîäåëü áàçóºòüñÿ íà ïîïåðåäí³õ ðîáîòàõ àâòîð³â ç âèêîðèñòàííÿì äðîáîâîãî îïåðàòîðà ðîçð³äæåííÿ Ñòåíòåëà òà Âàí Õàðíà. Ïðèéíÿòî ïðèïóùåííÿ ïðî òå, ùî òîâàðè íà ñêëàä³ íàëåæàòü äî îäíî¿ ç Ì ìîæëèâèõ êàòåãîð³é ÿêîñò³. Ó êîæíó ìèòü ÷àñó t òîâàðè íà ñêëàä³ ìîæóòü çàëèøàòèñü â îäíîìó ³ òîìó æ êëàñ³, ïåðåõîäèòè â îäèí ³ç Ì�� êëàñ³â, àáî ïñóâàòèñü. Ïðèïóñêàºòüñÿ òàêîæ, ùî ïåðåñóâàííÿ ïîì³æ êëàñàìè ðåãóëþºòüñÿ ìóëüòèíîì³àëüíèì îïåðàòîðîì ðîç- ð³äæåííÿ, ÿêèé çàëåæèòü â³ä äåÿêîãî ïðîöåñó ç ïàðàìåòðàìè, ùî âåêòîðíî îö³íþþòüñÿ. Äëÿ òðüîõ ìîæëèâèõ ñöåíàð³¿â çàïðîïîíîâàíî ðåêóðñèâí³ îö³íêè ïàðàìåòð³â ïðîöåñó. 1. Goyal S. K, Giri B. C. Recent trends in modeling of deteriorating inventory//European J. of Operational Research. — 2001. — 79. — Ð. 123—137. 2. Nahmias S. Perishable inventory theory: A review// Operations Research. — 1982. — 30. — Ð. 680—707. 3. Raafat F. Survey on continuously deteriorating inventory models// J. of the Operational Re- search Society. —1991. — 42. — Ð. 27—37. 4. Aggoun L., Benkherouf L. Filtering and predicting the cost of hidden perished items in an in- ventory model//J. of Applied Mathematics and Stochastic Analysis. — 2002. — 15. — Ð. 251—261. 5. Aggoun L., Benkherouf L., Tadj L. A Hidden Markov Model for an Inventory System with Perishable Items //J. of Applied Mathematics and Stochastic Analysis. — 1997. — 10, 4. — P. 423—430. 6. McKenzie E. Some simple models for discrete variate time series// Water Res Bull. — 1985. — 21. — Ð. 645—650. 7. Al-Osh M. N. and Alzaid A. A. First order integer-valued autoregressive (INAR(1)) process// J. Time Series Anal. — 1987. — 8. — Ð. 261—275. 8. McKenzie E. Some ARMA models for dependent sequences of Poisson counts// Advances in Applied Probability. — 1988. — 20. — Ð. 822—835. 9. Aly A. A, Bouzar N. On some integer-valued autoregressive moving average models// J. of Multivariate Analysis. — 1994. — 50. — Ð. 132—151. Filtering of an Inventory Model with a Multinomial Thinning Operator ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2007. Ò. 29. ¹ 1 17 10. Bensoussan Alain, Metin Cakanyildirim, Suresh P. Sethi Partially Observed Inventory Sys- tems: The case of Zero Balance Walk/Working paper SOM 200548. School of Management, University of Texas at Dallas. — 2005. 11. Bensoussan Alain, Metin Cakanyildirim, Suresh P. Sethi On the optimal control of partially observed inventory systems. Comptes Rendus de l’Academie des Sciences, 2005. 12. Aggoun L., Elliott R. J. Measure Theory and Filtering: Introduction with Applications.— Cambridge Series In Statistical and Probabilistic Mathematics. — 2004. 13. Elliot R. J., L. Aggoun, J. B. Moore Hidden Markov Models: Estimation and Control//Appli- cations of Mathematics. — 1995. — No. 29. Ïîñòóïèëà 21.03.06 Lakhdar Aggoun, Lakdere Benkherouf 18 ISSN 0204–3572. Electronic Modeling. 2007. V. 29. ¹ 1

Filtering of an Inventory Model with a Multinomial Thinning Operator

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