Strong Stability in an (R, s, S) Inventory Model

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1 Strong Stability in an (R, s, S) Inventory Model Boualem RABTA and Djamil AÏSSANI L.A.M.O.S. Laboratory of Modelization and Optimization of Systems University of Bejaia, (Algeria) lamos bejaia@hotmail.com Abstract In this paper, we prove the applicability of the strong stability method to inventory models. Real life inventory problems are often very complicated and they are resolved only through approximations. Therefore, it is very important to justify these approximations and to estimate the resultant error. We study the strong stability in a periodic review inventory model with an (R, s, S) policy. After showing the strong v stability of the underlying Markov chain with respect to the perturbation of the demand distribution, we obtain quantitative stability estimates with an exact computation of constants. Keywords : Inventory control, (R, s, S) policy, Markov chain, Perturbation, Strong Stability, Quantitative Estimates. Introduction When modelling practical problems, one may often replace a real system by another one which is close to it in some sense but simpler in structure and/or components. This approximation is necessary because real systems are generally very complicated, so their analysis can not lead to analytical results or it leads to complicated results which are not Corresponding author : Boualem RABTA, L.A.M.O.S., Laboratory of Modelization and Optimization of Systems, University of Bejaia, (Algeria). Tel : (213) Fax : (213) brabta@yahoo.fr. 1

2 useful in practice. Also, all model parameters are imprecisely known because they are obtained by means of statistical methods. Such circumstances suggest to seek qualitative properties of the real system, i.e., the manner in which the system is affected by the changes in its parameters. These qualitative properties include invariance, monotonicity and stability (robustness). It s by means of qualitative properties that bounds can be obtained mathematically and approximations can be made rigorously [18]. On the other hand, in case of instability, a small perturbation may lead to at least a finite deviation [11]. The paper by F.W. Harris [8] was the first contribution to inventory research. Since, thousands of papers were written about inventory problems and this field still a wide area for research. for a recent contribution see for exemple [15]. Inventory models are the first stochastic models for which monotonicity properties were proved (see [12, 20]). In 1969, Boylan has proved a robustness theorem for inventory problems by showing that the solution of the optimal inventory equation depends continuously on its parameters including the demand distribution [5]. Recently, Chen and Zheng (1997) in [6] have shown that the inventory cost in an (R, s, S) model is relatively insensitive to the changes in D = S s. Also, Tee and Rossetti in [19] examine the robustness of a standard model of multi-echelon inventory systems by developing a simulation model to explore the model s ability to predict system performance for a two-echelon one-warehouse, multiple retailer system using (R,Q) inventory policies under conditions that violate the model s fundamental modeling assumptions. As many other stochastic systems, inventory systems can be entirely described by a Markov chain. We will say that the inventory system (or model) is stable if the underly- 2

3 ing Markov chain is stable. For the investigation of the stability of Markov chains many methods have been elaborated. Kalashnikov and Tsitsishvili [11] have proposed a method inspired from Liapunov s direct method for differential equations. Also, we can find in the literature the weak convergence method (Stoyan [18]), metric method (Zolotarev [22]), renewal method (Borovkov [3]), strong stability method (Aïssani and Kartashov [1]), uniform stability method (Ipsen and Meyer [10]),... Some of these methods allow us to obtain quantitative estimates in addition to the qualitative affirmation of the continuity. The strong stability method elaborated in the beginning of the 80 s is applicable to all operations research models which can be represented by a Markov chain. It has been applied to queueing systems (see for example [2]). According to this approach, we suppose that the perturbation is small with respect to a certain norm. Such a strict condition allow us to obtain better estimations on the characteristics of the perturbed chain. For definitions and results on this method we suggest [14]. In this paper, we apply the strong stability method to the investigation of the stability in an (R, s, S) periodic review inventory model and to the obtention of quantitative estimates. In section 1, we present the model, define the underlying Markov chain and study its transient and permanent regime. Notations are introduced in section 2. The main results of this paper are given in sections 3, 4 and 5. After perturbing the demand probabilities, we first prove the strong v stability of the considered Markov chain (section 3), estimate the deviation of its transition matrix (section 4) and then, we give an upper bound to the approximation error (section 5). A numerical example is given in section 6 to illustrate the use of the method. Definition and basic theorems of the strong stability method are given 3

4 in appendix. 1 The model Consider the following single-item, single-echelon, inventory problem. Inventory level is inspected every R time units. At the beginning of each period, the manager decides whether he should order and if yes, how much to order. Suppose that the outside supplier is perfectly reliable and that orders arrive immediately. During period n, n 1, total demand is a random variable ξ n. Assume that ξ n, n 1, are independent and identically distributed (i.i.d.) random variables with common probabilities given by a k = P (ξ 1 = k), k = 0, 1, 2,... So, a k is the probability to have a demand of k items during the period. For such inventory problems, (R, s, S) policy is proved to be optimal (see [17, 9, 21]). According to this control rule, if at review moment (t n = nr, n 1), the inventory level X n is below or equal to s (Order-Point) we order so much items to raise the inventory level to S (Order-Up-To level). Thus, at the beginning of n + 1 period, the size of the order is equal to Z n = S X n. Figure (1) illustrates the mechanism of this model. Note that for this model, R, s and S are decision variables which values should be optimally chosen through a suitable algorithm (see for example [21]). Assume that the initial inventory level s < X 0 S (if not, we raise it to S by ordering sufficient quantity). The inventory level X n+1 at the end of period n + 1 is given by : X n+1 = { (S ξ n+1 ) + if X n s, (X n ξ n+1 ) + if X n > s, 4

5 Figure 1: The inventory level in the (R, s, S) inventory system with zero lead time. where (A) + = max(a, 0). The random variable X n+1 depends only on X n and ξ n+1, where ξ n+1 is independent of n and the state of the system. Thus, X = {X n, n 0} is a homogeneous Markov chain with values in E = {0, 1,..., S}. 1.1 Transition probabilities Suppose that at date t n = nr, X n = i. 1. If 0 i s, then X n+1 = (S ξ n+1 ) +, and we have two cases : for j = 0, P (X n+1 = 0 X n = i) = P (S ξ n+1 0) = P (ξ n+1 S) = for 0 < j S, a k, k=s P (X n+1 = j X n = i) = P (S ξ n+1 = j) = P (ξ n+1 = S j) = a S j. 5

6 2. If s < i S, then X n+1 = (i ξ n+1 ) +, and we distinguish the following cases : for j = 0, P (X n+1 = 0 X n = i) = P (i ξ n+1 0) = P (ξ n+1 i) = a k, k=i for 0 < j i, P (X n+1 = j X n = i) = P (i ξ n+1 = j) = P (ξ n+1 = i j) = a i j, for j > i, P (X n+1 = j X n = i) = 0. So, we have a k if 0 i s and j = 0, k=s a S j if 0 i s and 1 j S, P ij = a k if s + 1 i S and j = 0, k=i a i j if s + 1 i S and 1 j i, 0 if s + 1 i S and j i + 1. Thus, the transition matrix of the Markov chain X is given by P = 0 1 s s + 1 S 0 1 s s + 1 S a k a S 1 a S s a S s 1 a 0 S a k a S 1 a S s a S s 1 a 0 S a k a S 1 a S s a S s 1 a 0 S a k a s a 1 a s a k a S 1 a S s a S s 1 a 0 S 6

7 The Markov chain X is irreducible and aperiodic. It has a unique stationary distribution π = (π 0, π 1,..., π S ) given by the solution of π.p = π π i = 1. Furthermore, this stationary distribution is independent from the initial inventory level X Stationary probabilities Consider the Markov chain Y = {Y n, n 1} given by Y n = {Items on-hand at the beginning of the period n (just after order Z n 1 delivery)}, with initial state Y 1 = X 0. So, we have X n = (Y n ξ n ) + and Y n+1 = { S if Y n ξ n s, Y n ξ n if Y n ξ n > s. Also, we define the Markov chain V = {V n, n 1} by V n = S Y n So, we have V n+1 = { 0 if V n + ξ n S s, V n + ξ n if V n + ξ n < S s. or equivalently V n+1 = (V n + ξ n ) 1I {Vn+ξ n<s s} (1) 7

8 V is a Markov chain with states in {0, 1,..., S s 1}. If we denote by V the random variable having the stationary distribution of the Markov chain V then V = (V + ξ 1 ) 1I {V +ξ 1 <S s} (2) This relation implies that for every 0 v S s 1 F V (v) = P (V v) = C + (F V F ξ )(v), (3) where C = 1 (F V F ξ )(S s 1) is a normalisation constant and F ξ (x) = P (ξ 1 x) is the cumulative function common to all ξ n, n 1. Since (3) is a renewal type equation, it follows (cf. [16]) that its uniquely determined solution is given by where H = n=1 F V (v) = C(1 + H(v)), F n ξ is the renewal function associated to F ξ. The constant C can be easily determined by the condition F V (S s 1) = 1, therefore we obtain that the unique invariant distribution of the Markov chain V is given by F V (v) = 1 + H(v) 1 + H(S s 1), 0 v S s 1. Now, we can deduce the stationary distribution of the Markov chain X from relation X = (S (V + ξ 1 )) + where X is the random variable having the stationary distribution of the Markov chain X. 8

9 2 Preliminary and notations In this section, we introduce necessary notations adapted to our case, i.e., the case of a finite discrete Markov chain. For a general framework see [14] and comments in the appendix. Let X = {X n, n 0} be a homogeneous discrete Markov chain with values in a finite space E = {0, 1,..N}, given by a transition matrix P. Assume that X admits a unique stationary vector π. For a transition matrix P and a function f : E R, we define P f(i) = j E P ij f(j), i E, and for a vector µ R N+1 we define µf = i E µ i f(i). Also, f µ is the matrix having the form f(i)µ i, i, j E. We introduce in R N+1 the special family of norms µ v = i E v(i) µ i, µ R N+1, (4) where v : E R + is a function bounded from bellow by a positive constant (not necessary finite). So, the induced norms on space of matrices of size (N + 1) (N + 1) will have the following form Q v = max i E (v(i)) 1 Q ij v(j), (5) j E 9

10 Also, for a function f : E R, f v = max i E (v(i)) 1 f(i). 3 Strong v stability of the (R, s, S) inventory model Denote by Σ the considered inventory model. Let X = {X n, n 0} be the Markov chain where X n is the on-hand inventory level at date t n = nr, n 1 and X 0 is the initial inventory level. Let Σ be another inventory model with the same structure but with demands ξ n having common probabilities a k = P (ξ 1 = k), k = 0, 1,... Let X = {X n, n 0} be the Markov chain where X n is the on-hand inventory level at date t n = nr, n 1 in Σ and X 0 = X 0 is the initial inventory level. Denote by P and Q the transition operators of the Markov chains X and X respectively. Definition 3.1. (cf. [1]) We say that the Markov chain X verifying P v < is strongly v stable, if every stochastic matrix Q in the neighborhood {Q : Q P v < ɛ} admits a unique stationary vector ν and : ν π v 0 when Q P v 0. Theorem 3.1. In the (R, s, S) inventory model with instant replenishments, the Markov chain X = {X n, n 0} is strongly v stable for a function v(k) = β k for all β > 1. Proof. To prove the strong v stability of the Markov chain X for a function v(k) = β k, β > 1, we check the conditions of Theorem A.1. 10

11 So, we choose the measurable function h(i) = and the measure α = (α 0, α 1,..., α S ), where { 1 if 0 i s, 0 if s < i S, α j = P 0j. Thus, we can easily verify that πh = S π i h(i) = s π i > 0, α1i = S P 0j = 1, αh = S P 0j h(j) = s P 0j = i=s s a i > 0. It is obvious that T ij = P ij h(i)α j = { 0 if 0 i s, P ij otherwise, is a nonnegative matrix. Now, we aim to show that there exists some constant ρ < 1 such that T v(k) ρv(k) for all k E. Compute T v(k) = S T kj v(j). If 0 k s, then T v(k) = 0. If s < k S, then we have T v(k) = k P kj β j = k a i + a k j β j = i=k j=1 k 1 a i + a i β k i i=k 11

12 = k 1 a i a i + β k a i β i = i=k + β k i=k a i i=k β + s+1 k 1 a i i=s+1 β s+1 + k 1 i=s+1 s a i β i βk = a i β i + a i i=s+1 β s+1 + s a i β i βk s a i β i βk. It suffices to take ρ = which is smaller then 1 for all β > 1. Now, we verify that P v < a i i=s+1 β s+1 + s a i β i (6) P v = max k {0,1,..,S} 1 β k P kj β j = max(a, B) where and A = max k {0,1,..,s} B = 1 β k P kj β j = max k {0,1,..,s} [ 1 a β k i + i=s ] a S j β j = j=1 S 1 S 1 S 1 ( = 1 a i + a i β S i = 1 + a i β S i 1 ) max k {s+1,..,s} 1 β k P kj β j = max k {s+1,..,s} [ 1 a β k i + i=k [ a i + i=s ] k a k j β j j=1 a S j β j [ 1 k 1 ( = max 1 + a k {s+1,..,s} β k i β k i 1 )] 1 k 1 ( max 1 + a k {s+1,..,s} β s+1 i β k i 1 )] [ 1 S 1 ( 1 + a β s+1 i β S i 1 )] S 1 ( 1 + a i β S i 1 ) = A. So, we have S 1 ( P v = 1 + a i β S i 1 ) <. 12 j=1

13 4 Deviation of the transition matrix The considered inventory system is strongly v stable. This means that its characteristics can approximate those of a similar inventory model having different demand distribution under the condition that this distribution is close (in some sense) to the ideal system demand distribution. To characterize this proximity we define the quantity W = S 1 a i a i + a i a i β S i. (7) i=s Before estimating numerically the deviation between stationary distributions of the chains X and X, we estimate the deviation of the transition matrix P of the chain X. So, we have Lemma 4.1. Let P (resp. Q) be the transition matrix of the Markov chain X (resp. X ). Then, P Q v W Proof. P Q v = max k {0,1,..,S} 1 β k P kj Q kj β j = max(c, D) where C = max k {0,1,..,s} 1 β k [ P kj Q kj β j 1 max a k {0,1,..,s} β k i a i + i=s ] a S j a S j β j j=1 = S 1 a i a i + a i a i β S i i=s 13

14 and D = max k {s+1,..,s} 1 β k = max k {s+1,..,s} [ P kj Q kj β j 1 max a k {s+1,..,s} β k i a i + i=k = max k {s+1,..,s} a i a i i=k + β k i=s+1 a i a i k 1 i=k + a β k i a i β i a i a i k 1 i=s+1 β s+1 + a i a i β i + ] k a k j a k j β j j=1 s a i a i β i s a i a i β i W. So, we obtain P Q v S 1 a i a i + a i a i β S i = W. i=s 5 Stability inequalities The purpose of this section is to obtain a numerical estimation of the deviation between stationary distributions of the Markov chains X and X. This can be done using Theorem A.2. First, estimate π v. Lemma 5.1. Let γ be the constant given by ( ) a i + S 1 a i β S i γ = i=s (1 ρ)(1 + H(S s 1)) (8) 14

15 where ρ is given by (6) and H = n=1 F n ξ distribution function F ξ of the random variable ξ 1. Then, is the renewal function associated to the cumulative π v γ Proof. According to Theorem A.2, π v (αv)(1 ρ) 1 (πh). Since αv = α(j)v(j) = P 0j β j = a i + a S j β j = i=s j=1 S 1 a i + a i β S i i=s and πh = π i h(i) = s π i = 1 π i = 1 P (X > s) = 1 P (S (V + ξ 1 ) > s) i=s+1 H(S s 1) = 1 P (V +ξ 1 S s 1) = 1 (F V F ξ )(S s 1) = H(S s 1) = H(S s 1), it follows that π v ( i=s a i + S 1 a i β S i ) (1 ρ)(1 + H(S s 1)) = γ. Now, we can prove the following result Theorem 5.1. Let π and ν be the stationary distributions of the Markov chains X and X respectively. Then, under the condition W < 1 ρ 1 + γ (9) 15

16 we have π ν v W γ (1 + γ) 1 ρ (1 + γ) W (10) where ρ is given by (6) and γ by (8). Proof. We have π v γ and 1I v = max k E 1 β k = 1. Imposing the condition W < 1 ρ 1 + γ and replacing constants in (12) by their values, we obtain π ν v W γ (1 + γ) 1 ρ (1 + γ) W. Remark 1. The quantitative estimates obtained by means of the strong stability method and given by the last result gives an upper bound of the deviation of the stationary distribution of the Markov chain X with respect to the perturbation of the transition matrix π ν C(P ) P Q (11) Some other stability methods can also be used to obtain such a bound. For the Absolutely stable chain defined in [10], the survey of Cho and Mayer [7] collects and compares several bounds of the form (11). Note that this method study the sensitivity of the individual stationary probabilities of a finite Markov chain. 16

17 6 Numerical example Consider the (R, s, S) inventory model with Poisson distributed demand. This can correspond to the case where clients arrive following a Poisson distribution and request all exactly one item. Now, we apply the above results to test the sensitivity of the model when perturbing the arriving rate λ to λ = λ + ε (for example, ε can be the error when estimating λ from real data). For model parameters we take R = 1, S = 6, s = 3, λ = 5. The transition matrix P of the Markov chain X is constructed and computations are made by means a computer program. In a first step, we vary β to see the impact of the choice of the norm. The perturbed parameter we take is λ = 5, 1 so ε = 0, 1. For each value of β, the program compute the deviation W of the transition matrix and check whether the condition (9) of Theorem 5.1 is satisfied. If yes, the program compute the upper bound of the approximation error from relation (10). The transition matrix of the chain X is given by 0, , , , , , , , , , , , , , , , , , , , , P = 0, , , , , , , , , , , , , , , , , , , , , , , , , , , , The stationary distribution of the chain X is given by π = (0, , 0, , 0, , 0, , 0, , 0, , 0, ) Table 1 gives the results of the computations and the error e = π ν v = f(β) is graphically represented in Figure 2. Observe that the condition (9) is satisfied only for 17

18 β W π ν v β W π ν v β W π ν v 1,11 0, , ,21 0, , ,31 0, , ,41 0, , ,51 0, , ,61 0, , ,71 0, , ,81 0, , ,91 0, , ,01 0, , ,11 0, , ,21 0, , Table 1: The deviation of the transition matrix and the error e = π ν v = f(β) for different values of β. Figure 2: The error e = π ν v = f(β). 1, 03 β 2, 24. So, the choice of the norm (β) is relevant. The minimum estimation of the error e = 0, 0489 is taken for β = 1, 29. Now, let us vary the perturbation ε for a fixed value of β. The error e = π ν v = g(ε) ε 0,01 0,02 0,03 0,04 0,1 0,2 0,3 0,4 0,5 e 0,0043 0,0087 0,0133 0,0179 0,0489 0,1150 0,2091 0,3537 0,6041 Table 2: The error e = π ν v = g(ε). is given in Table 2 for β = 1, 29. Naturally, the error is proportional to the perturbation 18

19 and the two quantities grow in the same time. Observe that the model have resisted to an important perturbation (0, 5 represents 10% of the value of λ). 7 Conclusion The Markov chain X denoting the on-hand inventory level at the end of periods in the (R, s, S) inventory system with instant replenishments, is strongly v stable with respect to the perturbation of the demand distribution, for a function v(k) = β k, β > 1. This means that its characteristics are close to those of another inventory model with the same structure but with demand distribution close to the ideal model demand distribution. The last result gives us a numerical estimation of the error due to the approximation. References [1] Aïssani, D. and N.V. Kartashov, 1983, Ergodicity and stability of Markov chains with respect to operator topology in the space of transition kernels, Dokl. Akad. Nauk. Ukr. SSR, ser. A 11, 3 5. [2] Aïssani, D. and N.V. Kartashov, 1984, Strong stability of the imbedded markov chain in an M/G/1 system, International Journal Theor. Probability and Math. Statist. 29, (American Mathematical Society), 1 5. [3] Borovkov, A.A., 1972, Processus probabilistes de la théorie des files d attente (Edition Navka, Moskou). 19

20 [4] Boualouche, L. and D. Aïssani, 2002, Performance evaluation of an SW Communication Protocol (Send and Wait), Proceedings of the MCQT 02 (First Madrid International Conference on Queueing Theory), Madrid (Spain), 18. [5] Boylan, E.S., 1969, Stability theorems for solutions to the optimal inventory equation, Journal of Applied Probability 6, [6] Chen, F. and Y.S. Zheng, 1997, Sensitivity analysis of an (s, S) inventory model, Operations Research Letters 21(1), [7] Cho, G. and C. Mayer, 2001, Comparaison of perturbation bounds for the stationary probabilities of a finite Markov chain, Linear Algebra and its Applications, 335, [8] Harris, F.W., 1913, How many parts to make at once, Factory (The magazine of Management), 10(2), [9] Iglehart, D., 1963, Dynamic programming and stationary analysis of inventory problems, multistage inventory models and techniques, in : H. Scarf, D. Gilford, and M. Shelly, editors, Multistage Inventory Models and Techniques, (Stanford University Press) [10] Ipsen, I. and C. Meyer, 1994, Uniform stability of Markov chains, SIAM Journal on Matrix Analysis and Applications, 15(4),

21 [11] Kalashnikov, V.V. and G.Sh. Tsitsishvili, 1971, On the stability of queueing systems with respect to disturbances of their distribution functions, Queueing theory and reliability, [12] Karlin, S., 1960, Dynamic inventory policy with varing stochastic demands, Management Sciences 6, [13] Kartashov, N.V., 1981, Strong Stability of Markov Chains, VNISSI, Vsesayouzni Seminar on Stability Problems for stochastic Models, Moscow, (See also : 1986, J. Soviet Mat. 34, ). [14] Kartashov, N.V., 1996, Strong Stable Markov Chains (VSP, Utrecht, TbiMC Scientific Publishers). [15] Mohebbi, E., 2004, A replenishment model for the supply-uncertainty problem, International Journal of Production Economics, 87(1), [16] Ross, S.M., 1970, Applied probability with optimization applications (Holden-Day, San Francisco). [17] Scarf, H., 1960, The optimality of (S, s) policies in the dynamic inventory problem, in : K.J. Arrow, S. Karlin, and P. Suppes, editors, Mathematical methods in the social sciences, (Stanford University Press, Stanford, Calif.) [18] Stoyan, D., 1983, Comparaison Methods for Queues and Other Stochastic Models, English translation, D.J. Daley, Editor ( J. Wiley and Sons, New York). 21

22 [19] Y.S. Tee and M.D. Rossetti, 2002, A robustness study of a multi-echelon inventory model via simulation, International Journal of Production Economics, 80(3), [20] Veinott, A., 1965, Optimal policy in a dynamic, single product, non-stationary inventory model with several demand classes, Operations Research 13, [21] Veinott, A., and H. Wagner, 1965, Computing optimal (s, S) policies, Management Sciences 11, [22] Zolotarev, V.M., 1975, On the continuity of stochastic sequenses generated by recurent processes, Theory of Probability and Its Applications, XX(4), A Strong stability criterion Conserve notations of section 2. The following theorem (cf. [1]) gives sufficient conditions for the strong stability of a Harris recurrent Markov chain. Theorem A.1. The Harris recurrent Markov chain X verifying P v < is strongly v-stable, if the following conditions are satisfied 1. α R N+1 +, h : E R + such that : πh > 0, α1i = 1, αh > 0, 2. T = P h α is a nonnegative matrix, 3. ρ < 1 such that, T v(x) ρ v(x), x E. where 1I is the function identically equal to 1. 22

23 One important particularity of the strong stability method is the possibility to obtain quantitative estimates. The following theorem (cf. [13]) allows us to obtain a numerical estimation of the deviation of the stationary distribution of the strongly stable Markov chain X. Theorem A.2. Under conditions of theorem (A.1) and for verifying the condition v < C 1 (1 ρ), we have : ν π v v π v C (1 ρ C v ) 1, (12) where C = 1 + 1I v π v and π v (αv)(1 ρ) 1 (πh) 23