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Information and Management Sciences Volume 18, Number 4, pp. 317-334, 2007

A Stochastic Perishable Inventory System with Random Supply Quantity Paul Manuel

A. Shophia Lawrence

Kuwait University

Madurai Kamaraj University

Kuwait

India G. Arivarignan Madurai Kamaraj University India

Abstract This paper considers a continuous review perishable inventory system with demands arrive according to a Markovian arrival process (M AP ). We model, in this paper, the situation in which not all the ordered items are usable and the supply may contain a fraction of defective items. The number of usable items is a random quantity. We consider a modified (s, S) policy which allows a finite number of pending order to be placed. We assume full backlogging of demands that occurred during stock out periods and that the recent backlogged demand may renege the system after an exponentially distributed amount of time. The limiting distribution of the inventory level is derived and shown to have matrix geometric form. The measures of system performance in the steady state are derived.

Keywords: Stochastic Inventory System, Perishable Items, Positive Lead Time, Modified (s, S) Policy, Matrix Geometric Solution. 1. Introduction Inventory systems with stochastic input and output processes have been attracting the researchers from the mid twentieth century. Hadley and Whitin [4] used probabilistic methods to analyse such systems. Sivazlian [14] has used the methods of renewal processes to analyse continuous review (s, S) inventory systems (CRIS). The work of Srinivasan [15] provided a general framework for the analysis of CRIS with arbitrarily distributed inter-demand time points and random lead times. Since then many researchers have contributed to the analyses of CRIS (see Kalpakam and Arivarignan [5, 6, 8], Kalpakam and Sapna [9], Arivarignan [1] and Elango and Arivarignan [3]). Received August 2006; Revised October 2006; Accepted July 2007. Supported by University Grants Commision, INDIA, reserch award 32-170/2006 (SR).

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Weiss [18] extended the notion of perishable inventories to the realm of continuous review system by obtaining (0, S) policy as the optimal ordering policy for a model with Poisson demand, fixed life time for items and instantaneous supply of ordered items. Arivarignan and Kalpakam [6] have extended this work to include exponential life time for items. Schmidt and Nahmias [13] considered (S − 1, S) ordering policy for a perishable inventory system. Kalpakam and Sapna [9] considered a Markovian inventory system with exponential lifetime for items. Arivarignan [1] showed that in the case of perishable items, renewal demands, and instantaneous supply of ordered items, (−r, S) policy (which places an order at the r-th demand after the depletion of stock) is optimal and has provided optimal decision rules for r and S. Kalpakam and Arivarignan [6] introduced a modified (s, S) ordering policy which allows more than one pending orders at a time but fixes an upper bound for it. This policy allowed full backlogging of demands. They have also assumed that the number of non defective items in a supply is a random variable. This modified (s, S) policy has been adopted by Liu and Yang [10] for a perishable inventory model. In this work, we extend the work of Kalpakam and Arivarignan [7] by including perishable items and by incorporating a general class of processes for the sequence of demand points, namely, a Markovian arrival process(M AP ). This process includes a large collection of renewal and non renewal processes [2]. The plan of the manuscript is as follows. The section 2 describes the model assumptions. The section 3 presents the steady state distribution of the inventory level and expresses it in a matrix-geometric form. Finally the measures of system performance in the steady state are calculated.

2. Problem Formulation Consider an inventory system which can stock a maximum of S perishable items. The items are removed from the stock as and when either items perish (each item has exponential life time) or demands occur. We assume that the demand time points form a Markovian arrival process (MAP). The ordering policy is as follows: An order for Q (0 < Q < S) items is placed whenever Q items are removed from the stock and a maximum of m (> S/Q) orders can be pending. This condition ensures that orders for Q items will be placed until all the available stock is exhausted. The demands that

A Stochastic Perishable Inventory System with Random Supply Quantity

319

occurred during the stock out periods are fully backlogged and the recent backlogged demands may leave the system. In this paper, we consider the situation in which not all supplied items in the lot are in good usable condition and the number of non defective items in the supply of the order is a random quantity. The operating policy is a modified (s, S) ordering policy according to which an order for fixed quantity Q (= S − s) is placed, 1. whenever the net inventory level (on hand minus backorders) drops to any one of the reorder levels, viz., S − Q, S − 2Q, . . . , S − mQ, 2. at the time of replenishment, if the supply is not sufficient enough to take the net inventory level above the preceding reorder level. S

?

S−Q

? ?

?

? ?

S − 2Q

?

?

?

? ?

6? ?

? ? ?

S − 3Q

?

R

?

R

?

S

?

R

?

?

6

?

S&R

R- Reorder placed, S- Supply received, S&R- Supply received and reorder placed. The M AP in continuous time is described as follows. Consider an irreducible Markov chain with infinitesimal generator D = (dij ) on the state space EN = {1, 2, . . . , N }. At the end of a sojourn time in state i which is distributed as exponential with parameter λi ≥ −dii , the chain enters a state j with probability pij (1), if the transition corresponds to an arrival, ∀j and pij (0), if there is no arrival, j 6= i

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Note that the Markov chain can go from state i to state i only through an arrival. Also we have

N X

pij (1) +

j=1

We define the matrices

N X

pij (0) = 1,

for all i = 1, 2, . . . , N.

j=1,j6=i

Dk = ((dij (k)))i,j∈EN ,

k = 0, 1 so that

dij (1) = λi pij (1) and dij (0) =

(

−λi ,

j=i

λi pij (0),

j 6= i.

By assuming D0 to be a nonsingular matrix, it is ensured that the inter arrival times are finite with probability one and the arrival process does not terminate. Hence, we see that D0 is a stable matrix. The generator D is then given by D = D0 + D1 . Thus the M AP is described by the matrices D0 and D1 with D0 governing the transitions corresponding to no demand and D1 governing those corresponding to a demand. This point process is a special class of semi-Markov process. For use in sequel, let Ir denote an identity matrix of dimension r and er (k) denote the k-th column of Ir . When there is no need to emphasize the dimension of these vectors we will suppress the suffix. Thus, e will denote a column vector of 1’s of appropriate dimension. The notation ⊗ will stand for the Kronecker product of two matrices. Thus, if A is a matrix of order m × n and if B is a matrix of order p × q, then A ⊗ B will denote a matrix of order mp × nq, whose (i, j)th block matrix is given by aij B. Let υ be the stationary probability vector of the Markov process with generator D. That is υ is the unique (positive) probability vector satisfying υD = 0,

υe = 1.

Let η be the initial probability vector of the underlying Markov chain which governs the M AP . Then, by choosing η appropriately we can model the time origin to be (a) an arbitrary demand point; (b) the end of an interval during which there are atleast k demands; (c) the point at which the system is in specific state. The most interesting case is the one where we get the stationary version of the MAP by having η = υ. The constant λ = υD1 e, referred to as the Fundamental rate, gives the expected number of arrivals per unit of time of the M AP under steady state.

A Stochastic Perishable Inventory System with Random Supply Quantity

321

3. Analysis Let I(t) denote the net inventory level and J(t) denote the phase of the demand process at time t. Then the process {(I(t), J(t)), t ≥ 0} has the state space F = {S, S − 1, S −2, . . .}×EN . Let Z(t) be the number of pending orders at time t which takes values 0, 1, 2, . . . , m. The demand is for single item and the occurrence time points of demands form a M AP . The life time of each item has exponential distribution with parameter µ(> 0). A reorder will be for a fixed quantity of Q items and the maximum number of orders that can be pending at any time is fixed as m with S − mQ ≤ 0. The policy for placing orders is as follows: A reorder will be placed (i) at a demand epoch t when I(t) > I(t+) = S − kQ, k = 1, 2, . . . , m and (ii) at a replenishment epoch t when S − (l + 1)Q < I(t) < I(t+) ≤ S − lQ, l = 1, 2, . . . , m − 1 and I(t) < I(t+) ≤ S − mQ Note that Z(t) is not a random variable and is uniquely determined by the inventory level. For instance Z(t) = i,

if S − (i + 1)Q < L(t) ≤ S − iQ i = 0, 1, . . . , m

= m,

if L(t) < S − mQ

For the input process, the rate of replenishment is assumed to depend not only on the number of outstanding orders but also on the quantity of non defective items in the supply. We assume the existence of constants γij > 0, i = 1, 2, . . . , m, j = 1, 2, . . . , Q such that P [a replenishment for j-items in (t, t + ∆t)|Z(t) = i] = γij ∆t + o(∆t). P [more than one replenishment in (t, t + ∆t)|Z(t) = i] = o(∆t), i = 1, 2, . . . , m,

j = 1, 2, . . . , Q.

Then we have P [a replenishment in (t, t + ∆t)|Z(t) = i] = γi1 ∆t + o(∆t) and P [no replenishment in (t, t + ∆t)|Z(t) = i] = 1 − γi1 ∆t + o(∆t) where γi1 =

Q P

j=1

γij . We also write γij =

Q P

k=j

γik , and gij =

j P

k=1

γik .

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We assume that during the stock out periods, the recent backlogged demands may leave the system after a random time which is distributed as negative exponential with parameter α (> 0). From the assumptions made on the input and output processes, it can be shown that the stochastic process {(I(t), J(t)), t ≥ 0} on the state space F is a Markov Process. Defining q = ((q, 1), . . . , (q, N )) with q = (S − qQ, S − qQ − 1, . . . , S − (q + 1)Q + 1), q = 0, 1, 2, . . .. The infinitesimal generator A can be conveniently expressed as a blockpartitioned matrix with blocks of size QN × QN , as follows, 

where

A0 B0 0 0 · · ·

  C1    0   ..  . A=   0   0    0  .. . 

    A0 =     

0 0 0 .. .

0

0

0

D0 − µS I D1 + µS I 0

0

0

0



 0    A2 B2 · · · 0 0 0 0 0   .. ..  . . ··· 0 0 0 0 0    0 0 · · · Cm−1 Am−1 Bm−1 0 0   0 0 ··· 0 Cm Am Bm 0    0 0 ··· 0 0 Cm Am Bm   .. .. .. .. .. .. .. . . ··· . . . . .

A1 B1 0 · · · C2 .. .

0

0

··· 0

0

D0 − µS−1 I D1 + µS−1 I · · · 0

0

D0 − µS−2 I · · · 0 .. . . · · · ..

0 .. .

0 .. .

0 .. .

0

0

0

0

0

         

· · · 0 D0 − µS−Q+1 I  D0 − βi0 I D1 + µS−iQ I · · · 0    (γi1 + αk )I D0 − βi1 I · · ·  0   Ai =   , i = 1, 2, . . . , m − 1 .. .. ..   . . · · · .   γi(Q−1) I γi(Q−2) I · · · D0 − βi(Q−1) I 

Bi = eQ (Q)eQ (1)T ⊗ [D1 + µS−(i+1)Q+1 I], i = 0, 1, . . . , m − 1 

   Am =   

1 + α )I D0 − (γm k

(γm1 + αk )I .. . γm(Q−1) I

D1

···

0

1 + α )I · · · D0 − (γm k .. . ···

0 .. .

γm(Q−2) I

· · · D0 − (γi1 + αk )I

      

A Stochastic Perishable Inventory System with Random Supply Quantity

323

Bm = eQ (Q)eQ (1)T ⊗ D1 

where

γiQ I γi(Q−1) I · · · (γi1 + αk )I

   Ci =   

0 .. .

γiQ I .. .

··· ···

γi2 I .. .

0

0

···

γiQ I



    , i = 1, 2, . . . , m  

βij = γi1 + αk + µS−iQ−j , µk =

(

αk =

(

kµ,

if k > 0

0,

if k ≤ 0

0,

S − mQ < k ≤ S

α,

k ≤ S − mQ ≤ 0

0≤j ≤Q−1

and m is the smallest integer such that S − mQ ≤ 0. In other words m is the smallest integer greater than or equal to S/Q.

4. The Steady-State Analysis Before we consider the steady state distribution of the inventory level process, we first obtain the necessary condition for the stability of the process. Define A∗ = Bm +Am +Cm which is given by  D0 − βm I + γmQ I D1 + γm(Q−1) I   (γ + α )I D0 − βm I + γmQ I m1 k    γm2 I (γm1 + αk )I   . .. ..  .  γm(Q−1) I + D1

γm(Q−2) I

···

(γm1 + αk )I

···

γm2 I

···

γm3 I .. .

···

· · · D0 − βm I + γmQ I



    .    

As this matrix is a generator of a continuous time Markov chain, its steady state prob-

ability vector Π = (π1 , π2 , . . . , πQ ) with πi = (π(i,1) , π(i,2) , . . . , π(i,M ) ) satisfying ΠA∗ = 0, and Πe = 1 is given by Π = 1/Q(eT ⊗ υ)

(1)

where υ is the steady-state probability vector of D. This can be verified by noting that A∗ is a circulant matrix,

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Lemma 1. The stability condition of the inventory system under study is given by λ