A Finite source Perishable Inventory system with Retrial demands and Multiple server vacation

A Finite source Perishable Inventory system with Retrial demands and Multiple server vacation

A Finite source Perishable Inventory system with Retrial demands and Multiple server vacation

C. Periyasamy Department of Mathematics,

Madurai Kamaraj University Constituent College, Sattur – 626 203, INDIA.

Abstract

In this article, we consider a continuous review perishable inventory system with a finite number of homogeneous sources of demands. The in- ventory is replenished according to a state dependent (s, S) ordering policy and the lead times are assumed to follow an exponential distribution. The life time of each item is assumed to be exponential. The server goes for a vacation of an exponentially distributed duration whenever the inventory level reaches zero. If the server finds empty stock when he returns to the system, he continues his vacation. The demands that occur during stock out period and/or during the server vacation period enter into the orbit. These orbiting demands send out signal to compete for their demand and the retrial times are distributed as exponential. The joint probability dis- tribution of the inventory level and the number of demands in the orbit are obtained in the steady state case. Various system performance measures are derived and the results are illustrated numerically.

Keywords : Continuous review Inventory System, Random life time,

State dependent (s, S) policy, Finite population, Retrial demands Multiple vacation

1. Introduction

   In most of the inventory models considered in the literature, the demanded items are directly delivered from the stock (if available). The demands occurring during the stock-out period are either lost (lost sales) or satisfied only after the arrival of ordered items (backlogging). The often quoted review articles Nahmias

   1. and Raafat [10] and Goyal and Giri [5] provide excellent summaries of many of these modelling efforts.

      However, in Queueing systems with server vacations have been widely stud- ied in different contexts in the literature. Continuous review inventory system with server vacation has been received little attention in the literature. Daniel and Ramanarayanan [3] have first introduced the concept of server vacation in

      inventory with two servers. In [4], they have studied an inventory system in which the server takes a rest when the level of the inventory is zero.

      The concept of retrial demands in inventory was introduced by Artalejo et al. [2]. They have assumed Poisson demand, exponential lead time and exponential retrial time. In that work, the authors proceeded with an algorithmic analysis of the system. Ushakumari [14] considered a retrial inventory system with clas- sical retrial policy. Krishnamoorthy and Jose [6] analysed three different retrial inventory with positive service time and positive lead-Time. Sivakumar [12] has considered a retrial inventory system with multiple server vacation and in[11] he has considered a perishable inventory system with retrial demands.

      In this paper, we address a continuous review perishable inventory system with a finite number of homogeneous sources of demands. The operating policy is a state dependent ordering policy. According to this policy the placement of order occurs in the following situations:

      1. When the on hand inventory level reaches the prefixed level s, he places an order for Q(= S s) items.

      2. When the server returns to the system (the following situations may arise because of the nature of the item),

         • If the on hand inventory level is i which is less than or equal to s, he place an order for Q items and terminates his vacation.

         • If the inventory level is zero and the ordered items are not pending, he place an order for Q units and continues his vacation.

   The server terminates his vacation only when he finds the positive inventory level. During the vacation period, any arriving primary demands enter the orbit. These orbiting demands compete for their demands after a random time. The inter-retrial times follows exponential distribution.

   The rest of the paper is organized as follows. In Section 2, we describe the problem and in the next section analyse the mathematical model of the problem under study. The steady-state analysis of the model is presented in section 4 and some key system performance measures are derived in Section 5. In the last section, we perform sensitivity analyses on the total expected cost rate in terms of numerical illustrations.

2. Problem formulation

   We consider an inventory system with a maximum stock of S units and the items are distributed by the server to the demands. The items are perishable in nature. The lifetime of each item is exponential with parameter (> 0). The demands are originated from the population of finite size N . The demand time points form a Quasi-random distribution with parameter , demand only single unit at a time. The operating policy is the state dependent (s, S) ordering policy. The replenishment of stock occur after some random time. The lead

   time is exponentially distributed with mean rate µ(> 0). When the on-hand inventory level zero, the server goes for vacation. The duration of the server vacation is an exponential random variable with parameter (> 0). Due to the perishable nature of the items, during the server vacation period the stock replenished and the items may perish. The situation makes to place an order. According to our ordering policy, when the server returns to the system, if the on hand inventory level is i(> 0) which is less than or equal to the prefixed level s, he place an order for Q items and he terminates his vacation or the inventory level zero and the ordered items are not pending, he place an order for Q units and continues his vacation. Server terminates his vacation only when he finds the positive stock. Demands that occur during stock-out period and/or during server vacation periods enters into the orbit. These orbiting demands compete for their demands according to an exponential distribution with parameter (> 0). We consider the classical policy where each demands in the orbit conducts his own attempts to obtain service independently from the other demands present in the orbit. We can then assume that the probability of a retrial during the time interval (t, t+dt), given that j demands were in orbit at time t, is jdt + o(dt). Each source is either free or in the orbit at any time. We also assume that the inter-demand times between the primary demands, lead times, lifetime of each items, retrial demand times and server vacation time are mutually independent random variables.

   Notation :

   Aij : element/sub-matrix at ith row, jth column of the matrix A. e : a column vector of appropriate dimension containing all ones. I : an identity matrix of appropriate dimension.

3. Analysis

   Let X(t), Y (t), Z(t) and Z(t), respectively, denote the inventory level, number of demands in the orbit, server status(0-is on vacation & 1-is available for provide item) and the status of the ordered item(0-received & 1-not received) at time

   t. From the assumption made on the input and output processes, it may be verified that the stochastic process {X(t), Y (t), Z(t), Z(t) : t 0} is a Markov process with state space E, which is defined as, Here

   E	=	E1 E2 E3 E4,
   E1	=	{(i, j, k, l) | i = s + 1, s + 2, . . . , S, j= 0, 1, 2, . . . , N, k = 0, 1, l = 0}
   E2	=	{(i, j, k, l) | i = 1, 2, . . . , s, j = 0, 1, 2, . . . , N, k = 0, l = 0}
   E3	=	{(i, j, k, l) | i = 1, 2, . . . , s, j = 0, 1, 2, . . . , N, k = 1, l = 1}
   E4	=	{(i, j, k, l) | i = 0, j = 0, 1, 2, . . . , N, k = 0, l = 0, 1}

   The values taken by these random variables are listed in the following table.

   i	j	k	l
   	0
   	1		0
   0	2	0

   … 1

   N

   1 0 0 0

   2 1

   … … 1 1

   s N

   s + 1 0 0

   s + 2 1 0

   …

   …

   1

   S N

   The state space of the stochastic process {X(t), Y (t), Z(t), Z(t)|t 0} is the collection of all quadruples (i) = (i, j, k, l) where each entry is selected from each column as we move from left to right; we may cross vertical lines but not horizontal ones. These quadruples can be ordered in the lexicographic order in each box separated by the horizontal lines. Define the following sets:

   (i)	=	(<< i, 0 >>, << i, 1 >>, . . . , << i, N >>),
   << i, k >>	=	(< i, k, 0 >, < i, k, 1 >),
   < i, k, 0 >	=	((i, k, 0, 0)),
   < i, k, 1 >	=	((i, k, 1, 1)) for i = 1, 2, . . . , s,
   < i, k, 1 >	=	((i, k, 1, 0)) for i = s + 1, s + 2, . . . , S.

   Then the state space of the process can be ordered as {(0), (1), (2), . . . , (S)}, where (0) = ((0, j, 0, 0), (0, j, 0, l)) for j = 0, 1, . . . , N,

   Then the infinitesimal generator P can be conveniently expressed in block partitioned matrix with entries,

   (0) (1) (2) · · · (s) · · · (Q) (Q + 1) · · · (S)

   (0) A C

   0 0

   (1) B1 A1 C1

   (1) B1 A1 C1

   (1) B1 A1 C1

   (1) B1 A1 C1

   … . . . . . . . . .

   (s) Bs As C1

   .

   .

   .

   .

   P = .

   . . .

   . . .

   (Q) BQ AQ

   (Q + 1)

   …

   …

   BQ+1 AQ+1

   . . . . . .

   . . . . . .

   (S) BS AS

   For i = 0, 1, 2, . . . , S,

   0 1 2 · · · N 1 N

   0 Di0 Ei0

   1 Di1 Ei1

   1 Di1 Ei1

   1 Di1 Ei1

   1 Di1 Ei1

   2

   .

   .

   Ai = .

   .

   .

   Di2 Ei2

   . . .

   . . .

   N 1

   DiN 1 EiN 1

   N DiN

   For i = 1, 2, . . . , S,

   0 1 2 · · · N 1 N

   0 Fi0

   1 Gi1 Fi1

   1 Gi1 Fi1

   1 Gi1 Fi1

   1 Gi1 Fi1

   2

   .

   .

   Bi = .

   .

   .

   Gi2 Fi2

   . . .

   . . .

   N 1

   GiN 1 FiN 1

   N GiN FiN

   For i = 0, 1, . . . , s,

   0 1 2 . . . N

   0 Hi0

   1 Hi1

   1 Hi1

   1 Hi1

   1 Hi1

   Ci =

   2

   …

   Hi2

   . . .

   N HiN

   The dimension of the main matrices are defined in Table 1 and the dimension of the sub-matrices are explicitly from the structure of the matrices. The sub- matrices are defined as, For j = 0, 1, 2, . . . , N,

   0 D

   0 D

   0

   0

   !

   !

   ij

   ij

   00

   i = 0

   0 1

   D =

   0 D00

   D01

   i = 1, 2, . . . , Q

   i = 1, 2, . . . , Q

   1 0 D11

   1 0 D11

   ij ij ij

   ij

   1

   ij

   ij

   1 D11 !

   i = Q + 1, Q + 2, . . . , S

   0 E

   0 E

   0

   0

   !

   !

   00

   0j

   0j

   Eij =

   i = 0

   0 1

   0 1

   ij

   ij

   0 E00 0

   1

   1

   0 0 i = 1, 2, . . . , Q

   0

   0 F

   0 F

   00

   1j

   1 F

   1 F

   10 i = 1

   1j

   0 1

   0 F 00 0

   1 0 F

   1 0 F

   ij

   11

   i = 2, 3, . . . , Q

   Fij = ij

   1 0 F

   1 0 F

   !

   !

   0 1

   11

   ij

   i = Q + 1

   1 F !

   1 F !

   1

   11

   ij i = Q + 2, Q + 3, . . . , S.

   0

   0

   0 0

   1 G10

   i = 1

   0 0

   1 G10

   i = 1

   1j

   0 0 1

   0 0

   ij

   ij

   Gij =

   1 0 G11 i = 2, 3, . . . , Q

   1 0 G

   1 0 G

   !

   !

   0 1

   11

   ij i = Q + 1

   1 G !

   1 G !

   1

   11

   ij i = Q + 2, Q + 3, . . . , S.

   0 1

   0 1

   !

   0 H

   0

   0 H

   0

   00

   ij

   Hij =

   i = 0

   0 1

   0 0 0

   1 0 H11

   i = 1

   ij

   Define j = (N j), ij = (N j) + i

   1j

   1j

   0 1 0 1

   0j

   0j

   D00 = 1 ( j µ

   F 10 = 1 ( 1j

   For i = 1, 2, . . . , s

   0

   00

   For i = 2, 3, . . . , s + 1, 0

   00

   Dij = 0

   ij

   ij

   D01 = 0

   ( ij

   0

   0

   (

   Fij = 0

   ij

   ij

   F 11 = 1

   ( i

   1

   1

   ( ij

   0 For i = s + 2, s + 3, . . . , S,

   11 0

   0

   0

   Fij = 0 ij

   0

   Fij = 0 ij

   0

   (

   (

   Dij = 0 ( ij j 11

   00

   00

   (

   (

   Dij = 0 ( ij 00

   For i = s + 1, s + 2, . . . , S

   1

   01

   Fij = 0 i

   0 1

   10

   Dij = 0

   (

   G1j = 1 (

   j

   1 For i = 2, 3, . . . , s + 1,

   11 1

   (

   (

   Dij = 1 ( ij µ 11

   0 1

   0j

   0j

   1 j

   1 j

   E00 = 0 ( j \

   Gij = 1 j

   For i = s + 2, s + 3, . . . , S,

   0

   11

   11

   For i = 1, 2, . . . , S

   0

   Gij = 0

   ( j

   0

   0

   ij

   ij

   E00 = 0

   ( j

   H00 = 0 ( 0 \

   0 0j 1 µ

   H0j = 1 µ

   H0j = 1 µ

   (

   (

   00 0

   F1j = 0

   ( 11

   Matrix Dimension Matrix Dimension

   A0 (2(N + 1), 2(N + 1)) C1 (2(N + 1), N + 1)

   Ai, i = 1, 2, . . . , Q (2(N + 1), 2(N + 1)) Bi, i = 1, 2, . . . , Q (2(N + 1), 2(N + 1))

   Ai, i = Q + 1, . . . , S (N + 1, N + 1) BQ+1 (N + 1, 2(N + 1))

   C0 (2(N + 1), 2(N + 1)) Bi, i = Q + 2, . . . , S (N + 1, N + 1)

   Table 1: Dimension of the matrices

4. Steady state analysis

   It can be seen from the structure of P that the homogeneous Markov process

   {(X(t), Y (t), Z(t), Z(t)), t 0} on the finite state space E is irreducible. Define

   ((i, j, k, l; i1 , j1 , k1, l1 ), t) = P r (X(t), Y (t), Z(t), Z (t)) = (i, j, k, l)|(X(0), Y (0) Z(0), Z (0)) = (i1, j1 , k1, l1 )Z

   Hence the limiting distribution

   (i,j,k,l) = lim ((i, j, k, l; i1, j1, k1, l1), t) exists.

   t

   Let = (<<<0>>>, <<<1>>>, . . . , <<<S>>>) where,

   <<<i>>> = (<<i,0>>, <<i,1>>, . . . , <<i,N>>), i = 0, 1, . . . , S

   <<i,j>> = (<i,j,0>), i = 0, j = 0, 1, . . . , N,

   <i,j,0> = ((i,j,0,0) , (i,j,0,1)), i = 0, j = 0, 1, . . . , N,

   <<i,j>> = (<i,j,0>, <i,j,1>), i = 1, 2, . . . , Q, j = 0, 1, . . . , N,

   <i,j,0> = ((i,j,0,0)), i = 1, 2, . . . , Q, j = 0, 1, . . . , N,

   (i,j,1,1)

   <i,j,1> = ( (

   ), i = 1, 2, . . . , s, j = 0, 1, . . . , N,

   ((i,j,1,0)), i = s + 1, s + 2, . . . , Q, j = 0, 1, . . . , N

   <<i,j>> = (<i,j,1>), i = Q + 1, Q + 2, . . . , S, j = 0, 1, . . . , N,

   <i,j,1> = ((i,j,1,0)), i = Q + 1, Q + 2, . . . , S, j = 0, 1, . . . , N.

   The limiting distribution can be computed by using

   P = 0 and e = 1. (4.1)

   The first equation of the above yields the following set of equations :

   Q>>>C = 0, i = Q, (4.2)

   Q>>>C = 0, i = Q, (4.2)

   <<<i+1>>>Bi+1 + <<<i>>>Ai = 0, i = 0, 1, . . . , Q 1, <<<i+1>>>Bi+1 + <<<i>>>Ai + <<<i 0

   Q>>>C = 0, i = Q + 1, . . . , S 1,

   Q>>>C = 0, i = Q + 1, . . . , S 1,

   <<<i+1>>>Bi+1 + <<<i>>>Ai + <<<i 1

   1

   1

   <<<i>>>Ai + <<<iQ>>>C = 0, i = S.

   The equations (except (4.2)) can be recursively solved to get

   <<<i>>> = <<<Q>>>i, i = 0, 1, . . . , S,

   where

   (1)QiBQ

   1

   A

   A

   Q1

   BQ1

   · · · B

   i+1

   A1, i = 0, 1, . . . , Q 1,

   i

   i

   I, i = Q,

   i =

   P h B

   P h B

   Si

   (1)2Qi+1

   A1 B

   · · · B

   A1 \ C A1

   j=0

   Sj

   Sj

   Q Q1

   B

   B

   Q1

   s+1j

   i+1

   i+1

   sj 1 Sj

   × B

   Sj

   A1

   Sj1

   B

   Sj1

   · · · B

   i+1

   A1 \i , i = Q + 1, . . . , S,

   i

   × B

   Sj

   A1

   Sj1

   B

   Sj1

   · · · B

   i+1

   A1 \i , i = Q + 1, . . . , S,

   i

   and <<<Q>>> can be obtained by solving

   s1

   <<<Q>>>

   (1)Q X h(B

   A1 B

   · · · B

   A1 C A1

   j=0

   Q Q1

   Q1

   s+1j

   sj 1 Sj

   A

   A

   × (B

   Sj

   1

   Sj1

   BSj1

   · · · B

   Q+2

   1

   A

   A

   Q+1

   i B

   Q+1

   + AQ

   +(1)QB

   A1 B

   · · · B A1 C i = 0,

   Q

   Q

   Q1

   Q1

   Q1

   Q1

   1

   1

   0

   0

   0

   0

   and

   " X (

   " X (

   Q1

   <<<Q>>> (1)QiBQ

   i=0

   1

   A

   A

   Q1

   BQ1

   · · · B

   i+1

   A1 + I

   i

   i

   S Si

   Q

   Q1

   Q1

   s+1j

   js

   1

   Sj

   Q

   Q1

   Q1

   s+1j

   js

   1

   Sj

   + X (1)2Qi+1 X h(B

   i=Q+1

   j=0

   i=Q+1

   j=0

   A1 B

   · · · B

   A1 C A1

   i=Q+1

   j=0

   i=Q+1

   j=0

   × (B

   Sj

   1

   A

   A

   Sj1

   BSj1

   · · · B

   i+1

   A1 i i e = 1.

   i

   i

5. System performance measures

   In this section , we derive some system performance measures in the steady-state case.

   1. Expected inventory level

      Let i denote the expected inventory level in the steady-state. Then i is given by

      S

      X

      X

      i = i<<<i>>>e

      i=1

   2. Expected reorder rate

      Let r denote the expected reorder level in the steady-state. Then r is given by

      N s

      r = X "[(s + 1) + (N k) + k] <s+1,k,1>e + X <i,k,0> e#

      k=0

      i=0

      k=0

      i=0

      k=0

      i=0

      k=0

      i=0

   3. Expected perishable rate

      Let p denote the expected perishable rate in the steady-state. Then p is given by

      S

      X

      X

      i = i<<<i>>>e

      i=1

   4. Expected number of demands in the orbit

      Let o denote the expected number of demands in the orbit in the steady-state. Then o is given by

      N S

      X X

      X X

      o = k<<i,k>>e

      k=1 i=0

   5. The blocking probability

      Let B denote the probability that the demands is blocked and B is given by

      Q N 1

      X X

      X X

      B = <i,k,0>e + <<<0>>>e

      i=1 k=0

   6. The overall rate of retrial

      The overall rate of trials at which the orbiting demands request his demand is given by

      S N

      X X

      X X

      SR = k<<i,k>>e

      i=1 k=1

   7. The successful rate of retrial

      The rate at which the orbiting demands successfully receive his demands is given by

      S N

      X X

      X X

      SR = k<i,k,1>e

      i=1 k=1

   8. The fraction of time the server is on vacation

      The fraction of time the server is on vacation is given by

      N Q

      SV = X "X <i,k,0>e + (0,k,0,0)#

      k=0

      i=1

      k=0

      i=1

      k=0

      i=1

      k=0

      i=1

   9. Fraction of successful rate of retrials

      The fraction of successful rate of retrials is given by

6. Cost analysis

   FR

   SR

   =

   OR

   The long-run expected cost rate for this model is defined to be

   T C(S, s) = chi + csr + cpp + cw o.

   where,

   cs : Setup cost per order

   ch : The inventory carrying cost per unit item per unit time

   cw : Waiting cost of a demand in the orbit per unit time

   cp : The cost per unit failure

   Due to the complex form of the limiting distribution, it is difficult to discuss the properties of the cost function analytically. Hence, a detailed computational study of the cost function is carried out.

   1. Numerical Examples

      We have studied the effect of varying the cost and the other system parameters on the optimal values and the results agreed with what one would expect.

      Example 1. In this example, we study the impact of the setup cost cs, holding cost ch, perishable cost cp and the waiting cost cw on the the total expected cost rate T C(s, S). Towards this end, we first fix the parameter values as = 7, = 0.2, = 0.5, = 4 and µ = 5. We observe the following from Table 1.

      • The optimal cost increases when ch, cs, cw and cp increase.

        Example 2. Here, we study the impact of the primary demand rate , the lead time rate µ, the retrial demand rate and the vacation time on the total expected cost rate T C(s, S). We first fix the cost values as ch = 0.02, cs = 25, cp = 2 and cw = 8 We observe the following from Figure 1 to 4.

      • The optimal expected cost rate increases when increases.

      • As is to be expected, µ increases the total expected cost rate decreases.

      • The total expected cost rate increases when increases and decreases when

and increases.

Table 2: Effect of cost parameters on total expected cost rate

References

1. Artalejo, J. R., (1998), Retrial queues with a finite number of sources, Journal of the Korean Mathematical Society, 35, 503 – 525.

2. Artalejo, J. R., Krishnamoorthy, A. and Lopez-Herrero, M. J., (2006), Nu- merical analysis of (s, S) inventory systems with repeated attempts, Annals of Operations Research, 141, 67 – 83.

3. Daniel, J. K. and Ramanarayanan, R., (1987), An inventory system with two servers and rest periods, Cahiers du C.E.R.O,Universite Libre De Brux- elles, 29, 95 – 100.

201.51

201.5

201.49

201.48

TC(s,S)

TC(s,S)

201.47

201.46

201.45

201.44

201.43

201.42

201.41

Vs on TC(s,S)

=2

=4

=6

=2

=4

=6

5 10 15 20

201.7

201.65

201.6

TC(s,S)

TC(s,S)

201.55

201.5

201.45

201.4

vs on TC(s,S)

=12

=8

=4

=12

=8

=4

5 10 15 20

Figure 1: vs variation on

T C(s, S)

Figure 2: vs µ variation on

T C(s, S)

207

206

205

TC(s,S)

TC(s,S)

204

203

202

201

vs onTC(s,S)

=0.6

=0.4

=0.2

=0.6

=0.4

=0.2

5 10 15 20

206

205

204

TC(s,S)

TC(s,S)

203

202

201

200

vs on TC(s,S)

=0.2

=0.4

=0.6

=0.2

=0.4

=0.6

5 10 15 20

Figure 3: vs variation on

T C(s, S)

Figure 4: vs variation on

T C(s, S)

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2. Goyal, S. K. and Giri, B. C., (2001), Recent trends in modeling of deterio- rating inventory, European Journal of Operational Research, 34(1), 1-16.

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