A NOTE ON A PERIODIC REVIEW INVENTORY MODEL WITH UNCERTAIN DEMAND IN A RANDOM ENVIRONMENT. Hirotaka Matsumoto and Yoshio Tabata

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1 Scentae Mathematcae Japoncae Onlne, Vol. 9, (23), A NOTE ON A PERIODIC REVIEW INVENTORY MODEL WITH UNCERTAIN DEMAND IN A RANDOM ENVIRONMENT Hrotaka Matsumoto and Yosho Tabata Receved September 18, 23 Abstract. Ths paper s on the analyss of a sngle product, perodc revew nventory model, where the dstrbutons of demands vary wth the state of the envronment varable. The state of the envronment s assumed to follow a dscrete-tme Markov chan. The optmal nventory polcy to mnmze the total dscounted expected cost s derved va dynamc programmng. For the fnte-horzon model, we show that an envronmental-dependent base-stock polcy s optmal, and derve some characterstcs of the optmal polcy. Under addtonal condtons, we further derve the monotoncty of the optmal polcy. 1 Introducton The nventory control has long focused on managng certan specfc types of probablty n the demand for the products. But, on the other hand, consumer s lkng becomes varously n the real-lfe. The demand s fluctuated by the economc clmate, weather condton, trend of publc opnon, and so forth. Mere ncludng a purely random component n the demand process wll be mpossble to express such stuatons. So, n ths paper, under the assumpton that the envronmental process follows a dscretetme Markov chan, we model a sngle product nventory system of whch the dstrbutons of demands depend on envronmental fluctuatons, and dscuss the management polcy. We further nvestgate the effect of the envronmental fluctuatons on the optmal polcy. The man advantage of the Markov chan approach s that t provdes a natonal and flexble framework for formulatng varous changes descrbed above. The effect of a randomly changng envronment n nventory model receved only lmted attenton n the earler paper. Kalymon[12] studes a multple-perod nventory model n whch the unt cost of the product s determned by a Markov process, and the dstrbuton of demand n each perod depends on the current cost. Feldman[8] models the demand envronment as a contnuous-tme Markov chan. The demand s modulated by a compound Posson process where the parameters are determned by the state of the envronment. But he studes only the steady-state dstrbuton of the nventory poston. Song and Zpkn[18] present a contnuous-revew nventory model where the demand process s a Markov modulated Posson process, and they derve some basc characterstcs of the optmal polcy and algorthms for computng the optmal polcy. In recent artcles, Özekc and Parlar[15] develop an nfnte-horzon perodc-revew nventory model wth unrelable supplers where the demand, supply and cost parameters are nfluenced by a random envronment. Cheng and Seth[2] analyze the ont promoton-nventory management problem for a sngle tem n the context of Markov decson processes. The purpose of ths paper s to show that the envronmental-dependent base-stock polcy s optmal, and that the optmal polcy have the monotoncty for revew perods by analyzng fnte-horzon perodc-revew nventory model where the demand dstrbuton depend 2 Mathematcs Subect Classfcaton. 49J15; 6J1; 9B5; 9C39. Key words and phrases. Inventory; Dynamc programmng; Perodc revew models.

2 42 H. MATSUMOTO and Y.TABATA on a Markov envronmental process. In addton, we derve the monotoncty of the optmal polcy for the envronmental states by orderng them. In ths paper, we focus on the fnte-horzon analyss, snce t gves us concrete and realstc nsghts. Ths paper s organzed as follows; Secton 2 presents the formulaton of the general problem as a dynamc programmng model. Secton 3 provdes analyss for fnte-horzon problem. Secton 4 nvestgates the effect of the envronmental fluctuatons on the optmal polcy. The paper concludes wth some fnal remarks n Secton 5. 2 Assumpton and Notaton In ths secton, we ntroduce notatons and basc assumptons used throughout the paper: n : number of perods remanng n the fnte-horzon problem; E = 1, 2,...,R} : a fnte set of possble envronmental state; I n : the state of the envronment observed at the begnnng of perod n; I = I n ; n } : a Markov chan on E; : the transton probablty that the envronmental state changes from to n one perod,, E; D n : the total demand durng perod n; A (z) =P [D n z I n = ] : the condtonal dstrbuton functon of D n when I n = ; a (z) : the probablty densty functon correspondng to A (z); X n : the nventory level observed at the begnnng of perod n; Y n (, x n ) : the order-up-to level f the envronmental state s and the nventory level s x n at the begnnng of perod n; c : a unt orderng cost; c : a unt orderng cost n perod ; h : a unt holdng cost ncurred at the end of perod; p : a unt shortage cost ncurred at the end of perod; We assume A () =, a ( ) >. To motvate orderng, we assume that p>cas n standard models. Also, we assume that unsatsfed demands are fully backlogged. Remark 1 The basc assumpton of ths model s that the demand dstrbuton at any perod depends on the state of the envronment at the begnnng of that perod. Therefore, the decson maker observes both the nventory level and the envronmental state to decde on the optmal order quantty whch s delvered mmedately. Remark 2 The admssblty condton requres that Y n (, x n ) x n snce we do not allow for dsposng of any nventory wthout satsfyng demand. For any y n, t s noted that the nventory level X n s a Markov chan, where X = x n +[y n (, x n ) x n ] + D n, n. Fgure 1 llustrates the behavor of the nventory level. Now, let V n (x) be the mnmum expected total dscount cost of operatng for n-perod wth the state of the envronment and the ntal nventory level x, under the best orderng decson s used at perod n through perod 1. Then, a dynamc programmng equaton

3 INVENTORY MODEL IN A RANDOM ENVIRONMENT 421 y n y n 2 y n 3 y 2 y 1 D n D n 2 x = y D 2 D 1 x n D x n 3 x 2 x 1 x n 2 x n n 1 n Fgure 1: The behavor of the nventory level (DPE) for the problem can be gven by (1) (2) V (x) =, V n (x) = mn cy + L (y)+α } V (y z)da (z) cx, n>, E, where y s the nventory level after the order s delvered. L (y) =h y (y z)da (z)+p y (z y)da (z), n>, E s the expected one-perod holdng and shortage cost functon, and α s the dscount factor per perod. The frst and second dervatves of L (y) are L (y) =(h + p)a (y) p, L (y) =(h + p)a (y) >. To smplfy our analyss, by usng the relaton we change (1) and (2) to followng DPE. (x) =V n (x)+cx, n, (3) W (x) = c x, (x) = mn G n (y) },

4 422 H. MATSUMOTO and Y.TABATA where (4) G n (y) = cy(1 α)+cα zda (z)+l (y) +α W (y z)da (z). Furthermore, n ths paper, we assume that no acton s taken n perod. So, c =. Thus, W (x) =. The decson varable n ths model s y, so (4) whch s a functon of y plays a central role n gettng the optmal value y. We assume that all parameters and costs are nonnegatve, and that all relevant functons are dfferentable. 3 Fnte-Horzon Analyss In ths secton, we analyze the fnte-horzon problem for the model ntroduced n the last secton. When n = 2, from (3) and (4), (x) = mn G 2 (y) }, G 2 (y) = cy(1 α)+cα W 2 +α We obtan the frst two dervatves of G 2 (y) as follows: G 2 For n =1, R (y) = c(1 α)+l (y)+α G 2 (y) = L (y)+α zda (z)+l (y) W 1 (y z)da (z). W 1 (y z)da (z). W 1 (y z)da (z), Then, (x) = mn G 1 (y) }, W 1 G 1 (y) =cy(1 α)+cα G 1 (y) =c(1 α)+l (y), G 1 zda (z)+l (y), (y) =L (y) >. lm y G 1 (y) = c(1 α)+ h>, lm G 1 (y) =c(1 α) p<. y So, there exsts a unque S 1 such that G 1 (S1 ) =,.e., [ p c(1 α) S 1 = A 1 h + p ].

5 INVENTORY MODEL IN A RANDOM ENVIRONMENT 423 S 1 s nonnegatve and fnte because p c(1 α) > and p c(1 α) <h+ p. the optmal polcy for perod 1 s the envronmental-dependent base-stock polcy defned by Y 1 (, x) = and the optmal cost ncurred by ths polcy s Furthermore, for x>s 1, Then, S 1 (x S 1 ), x (x >S 1 ), W 1 (x) = G 1 (S 1 ) (x S1 ), G 1 (x) (x>s1 ). W 1 (x) =G 1 (x) >, W 1 (x) =G 1 (x) >. G 2 (y) >, lm y G 2 (y) =c(1 α 2 )+h(1 + α) >, lm G 2 (y) =c(1 α) p<. y So, there exsts a unque S 2 such that G 2 (S2 )=. Y2 S (, x) = 2 (x S 2), x (x >S 2), W 2 (x) = G 2 (S 2) (x S2 ), G 2 (x) (x>s2 ). Furthermore, for x>s 2, On the other hand, So, Moreover, So, W 2 (x) =G 2 R G 2 (y) G 1 (y) =α (x) >, W 2 (x) =G 2 (x) >. W 2 R G 2 (y) G1 (y) =α S 2 S 1, 1 (x) W (x). W 1 (y z)da (z). W 1 (y z)da (z). W 2 (x) W 1 (x) mn G 2 (y) G 1 (y) }.

6 424 H. MATSUMOTO and Y.TABATA For an n-perod problem, (x) = mn G n (y) }, G n (y) = cy(1 α)+cα +α G n (y) = c(1 α)+l (y)+α R G n (y) = L (y)+α zda (z)+l (y) W (y z)da (z), (y z)da (z), W (y z)da (z). To use nducton, we assume that the followng propertes hold for the ()-perod problem where the state of the envronment s E. Then, G lm (S y G lm y G W (x) = (x) = W (x) = W )=,G (y) >, n 2 (y) =c(1 α )+h (y) =c(1 α) p<, G k= α k >, (S ) (x S ), G (x) (x>s ), (x S ), G (x) (x>s, ), (x S ), G (x) (x>s, ), (x) 2 (x), (x) W n 2 (x). G n (y) >, lm y G n (y) =c(1 αn )+h k= α k >, lm So, there exsts a unque S n such that G n (Sn )=. Furthermore, for x>s n, W n G n y Yn S (, x) = n (x S n), x (x >S n), (x) = G n (S n) (x Sn ), G n (x) (x>sn ). (x) =G n (x) >, W n (x) =G n (x) >. (y) =c(1 α) p<.

7 INVENTORY MODEL IN A RANDOM ENVIRONMENT 425 G n (x) (x) S n Fgure 2: The behavor of (x) and Gn (x) The behavor of (x) and Gn (x) s shown graphcally n Fgure 2. On the other hand, So, G n (y) G (y) = α W (y z) W n 2 (y z) } da (z). Moreover, S n S, (x) (x). W n G n (y) G (y) = α W (y z) 2 (y z) } da (z). (x) W (x) mn G n (y) G (y) }. 4 The monotoncty of the optmal polcy for the envronmental states In ths secton, we nvestgate the effect of the envronmental fluctuatons on the optmal polcy. To smplfy our analyss, we ntroduce the concept of stochastc orderng, and set some addtonal assumptons. Defnton 1 Let F (x) =P (X x) and G(y) =P (Y y) denote cumulatve dstrbutons of the one-dmenson random varables, X and Y, respectvely. We defne that Y s stochastcally larger than X or that G s stochastcally larger than F, as follows: F (t) G(t), <t<. Then, t s represented by F SL G or X SL Y.

8 426 H. MATSUMOTO and Y.TABATA Theorem 1 The followng condton s necessary and suffcent to be X SL Y EΦ(X) EΦ(Y ) where Φ s any non-decreasng functon for whch expectatons exst. We assume that the followng relaton holds on the demand dstrbutons n respectve states. Assumpton 1 A 1 SL A 2 SL SL A R,.e., A 1 (y) A 2 (y) A R (y). We further make an assumpton on the transton probablty. Assumpton 2 The transton probablty s assumed to be totally postvely 2 (TP2). Now, when Assumpton 1 and 2 hold, we prove the followng (1) (5). For any states and (1 < R), (1) (x) (x), (2)G n (y) Gn (y), (3)W n (x) W n (x), (4)G n (y) G n (y), (5)S n S n. We use nducton to prove them. For n = 1, they are apparent from the last secton. For the (n 1)-perod problem where the state of the envronment s E, we assume that W (x) s non-ncreasng n. Then, snce R =r s non-decreasng n from Assumpton 2, we have W (y z)da (z) P (,) Snce W (x) s non-decreang n x, we obtan from Theorem 1 So, Moreover, W (y z)da (z) W (y z)da (z) P (,) G n (y) Gn (y). (x) (x). W (y z)da (z). W (y z)da (z). W (y z)da (z).

9 INVENTORY MODEL IN A RANDOM ENVIRONMENT 427 Next, for (n 1)-perod problem where the state of the envronment s E, we assume that (x) s non-ncreasng n. Then, from Assumpton 2, we have (y z)da (z) P (,) Snce (x) s non-decreasng n x, we obtan from Theorem 1 So, Moreover, (y z)da (z) (y z)da (z) P (,) G n (y) G n (y), S n S n. (x) W n (x). W n (y z)da (z). (y z)da (z). (y z)da (z). We further ntroduce a concept stronger than Defnton 1, and set a new assumpton. Defnton 2 Let X and Y be the one-dmenson random varables wth cumulatve dstrbutons F (x) =P (X x), G(y) =P (Y y), and probablty denstes f(x) =F (x), g(y) =G (y), respectvely. We defne that Y s larger than X n the sense of frst moment orderng or that g s larger than f n the sense of frst moment orderng, as follows. g(t) = f(t + C), t where C s an any postve constant. Then, t s represented by f FM g or X FM Y. Assumpton 3 a 1 FM a 2 FM FM a R. When Assumpton 2 and 3 hold, we prove that S n s constant n n. Frst, to smplfy our analyss, we ntroduce a new notaton. Let ν be a mean of the demand n an any state,.e., ν = zda (z),. Then, from Assumpton 3, a 1 (y ν 1 )=a 2 (y ν 2 )= = a R (y ν R ). Snce the demand should not be negatve, a 1 (y) =, y. (5) From S 1 = A 1 [ p c(1 α) h+p a (y) =, y ν ν 1, =1,...R. ], (6) S 1 1 ν 1 = S 1 2 ν 2 = = S 1 R ν R.

10 428 H. MATSUMOTO and Y.TABATA Now, for the ()-perod problem where the state of the envronment s E, we assume that S = S 1. Then, from the last secton, = (x S W ), (x) (x>s ). Snce S 1 S 2 S R from Assumpton 2, W = (x S (x) 1 = S1), 1 (x>s1 = S1 1). Moreover, from (5) and (6), = (y S 1 (y z)da (z) 1 + ν ν 1 = S 1), (y>s1 1 + ν ν 1 = S 1). S n = S 1. Therefore, when Assumpton 2 and 3 hold, S n s constant n n. 5 Concludng Remarks In ths paper, frst, we show the exstence of the envronmentaldependent optmal base-stock polcy and ts monotoncty for revew perods by analyzng fnte-horzon perodc-revew nventory model where the demand dstrbutons depend on a Markov envronment process. We further show that the optmal polcy have the monotoncty for the envronmental states by orderng them. When the cost-parameters depend on a Markov envronment process, we can prove the exstence of the envronmental-dependent optmal base-stock polcy. Also, n the presence of a fxed orderng cost, we can show the exstence of the envronmental-dependent optmal (s, S) polcy. Furthermore, we can derve the optmal polcy n the model that ncorporates varable capacty and stochastcally proportonal yeld. But, n above cases, t s dffcult to clarfy the effect of the envronmental fluctuatons on the optmal polcy. It s a future drecton to nvestgate the effect of the envronmental fluctuatons on the optmal polcy n more complex model. In our model, we analyze the nventory system that depends on the exogenous factors. So, t s also one of the possble extensons to present the nventory system that depends on the endogenous factors where the demand s nfluenced by the order quantty, prce dscount, marketng actvty, and so on. References [1] K. J. Arrow, S. Karln, H. Scarf, Studes n the Mathematcal Theory of Inventory and Producton, Stanford Unversty Press, Stanford, CA, [2] F. Cheng, S. P. Seth, A Perodc Revew Inventory Model wth Demand Influenced by Promoton Decsons, Management Scence, Vol. 45, No. 11, November 1999, [3] F. W. Carallo, R. Akella, T. M. Morton, A Perodc Revew Producton Plannng Model wth Uncertan Capacty and Uncertan Demand-Optmalty of Extended Myopc Polces, Management Scence, Vol. 4, No. 3, March 1994, [4] E. Çnlar, S. Özekc, Relablty of complex devces n random envronment, Probablty n the Engneerng and Informatonal Scence, 1 (1987),

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