Perishable inventory system with service interruptions, retrial demands and negative customers

Applied Mathematics and Computation 262 (2015) 102–110

Contents lists available at ScienceDirect

Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Perishable inventory system with service interruptions, retrial demands and negative customers Vijaya Laxmi P.∗, Soujanya M.L. Department of Applied Mathematics, Andhra University, Visakhapatnam 530 003, India

a r t i c l e

i n f o

Keywords: Continuous review (s, S) policy Perishable inventory Retrial demands Service interruptions Negative customers Matrix analytic method

a b s t r a c t In this paper, we consider a continuous review perishable inventory system wherein demands arrive according to the Poisson process, each demanding exactly one unit of inventory item and the life time of each item is assumed to be exponential. The operating policy is (s, S) policy, i.e., whenever the inventory level drops to s, an order for Q(=S − s) items is placed. The ordered items are received after a random time which is distributed as exponential. The service may be interrupted according to the Poisson process in which case it restarts after an exponentially distributed time. The demands that occur during the server breakdown period or stock-out period may turn out to be ordinary or a negative demand and then they enter into the orbit of infinite size. These orbiting demands send out a signal to compete for their demand which is distributed as exponential. The matrix analytic method is used for the steady state distribution of the model. Various performance measures and cost analysis are shown with numerical results. © 2015 Elsevier Inc. All rights reserved.

1. Introduction Perishable items are those items, which have a fixed or specified life time after which they are considered unsuitable for utilization. The classical inventory theory did not take into account these perishable items. However, the analysis of perishable inventory systems is important because in real life products like milk, blood, drug, food, vegetables, photographic film and some chemicals do have fixed life times after which they will perish. Weiss [17] introduced the notion of perishable inventories with a continuous review (0, S) policy, Poisson demand and instantaneous supply of ordered items. The author assumed that the items fail after a fixed lifetime. Kalpakam and Arivarignan [4] studied a similar model with (s, S) ordering policy in which items have exponential life times. For recent reviews on perishable inventory systems, see [2,5,9,13]. The delay in service caused by server interruption is a common phenomenon in almost all practical situations. Such interruptions may be caused due to breakdown of a machine, for example, an electronic computer, or the arrival of a high-priority customer that may bring interruption in the service of lower-priority customers. Krishnamoorthy et al. [6] first introduced the concept of service interruptions in an inventory model. They assumed that there is no bound on the number of interruptions that can occur in the middle of a single service and also that an order is instantaneously processed. Later, they extended the above model with positive lead time in [7] and recently in [8] they considered an (s, S) production inventory system with positive service time and the production processes are subject to multiple interruptions.

∗

Corresponding author. Tel.: +919989578075. E-mail address: [email protected], [email protected] (Vijaya Laxmi P.).

http://dx.doi.org/10.1016/j.amc.2015.04.013 0096-3003/© 2015 Elsevier Inc. All rights reserved.

Vijaya Laxmi P., Soujanya M.L. / Applied Mathematics and Computation 262 (2015) 102–110

103

The above models consider that the demands that occur during stock-out period or during service interruption period are either backlogged or lost. In the later case, it is assumed that the demands that occur in such situations enter into an orbit of infinite size and retry for their demand after a random time. Artalejo et al. [1] first used the concept of retrial demands in inventory models. They have assumed that an arriving demand at the time of stock-out leaves the service area temporally and repeat its request after some random time. Sivakumar [16] considered a continuous review perishable inventory system with a finite number of homogeneous sources of demands. Periyasamy [14] extended [16] with multiple server vacations. In all the above models, the authors have considered that the arrival of a customer to the service station should join the system unless it is full. However in some applications an arriving customer may turn out to be a positive or a negative customer. Positive customer joins the system whereas negative customer always removes some positive customers without joining the system. Negative customers can be thought of viruses or commands that delete some transaction in computer network or a database. The concept of queues with negative arrivals was first introduced by Gelenbe [3]. Sivakumar and Arivarignan [15] introduced the concept of negative customers in perishable inventories with finite waiting hall. Manual et al. [10,11] considered a continuous review perishable (s, S) inventory system with two types of customers, ordinary and negative, arrived according to a Markovian arrival process (MAP). In [11], they assumed demands that occur during the stock-out period either enter a pool which has a finite capacity or are lost whereas in [10] such demands are considered as lost. To the best of our knowledge, there is no literature which takes negative arrivals, retrial demands and service interruptions into consideration in perishable inventory systems. However, in real life there is a chance to form such perishable inventory systems. For example, in a medical store, the drugs can be considered as perishable items. At the time of billing, the service may be interrupted due to a fault in the computer or in the printer. If the drugs required by a customer are not available immediately then he may retry for that drug on the next day (based on two assumptions that the rare drug may not be available nearby or the customer is a regular buyer from the medical shop). On the other hand, the customer may opt for another drug store. Such type of demands are considered as negative customers. Motivated by the above example in the present paper, we consider an (s, S) policy perishable inventory model with exponential life time for an inventory and service interruption due to server breakdown. Whenever the server is down, it is immediately sent for repair. The broken down time follows Poisson distribution and repair time is exponentially distributed. Primary demands occur according to Poisson distribution. During the service interruption or stock-out period, any arriving primary demands turn to be positive with probability p or negative with probability q = 1 − p and then they enter the orbit of infinite size. These orbiting demands compete for their demands after a random time which is assumed to be exponentially distributed. Using the matrix analytic method, we obtain the steady state distribution of the model. Various performance measures of the model such as expected reorder rate, expected inventory level, expected number of demands in an orbit, etc., are presented. Cost analysis is also carried out using the direct search method. The rest of the paper is organized as follows. Section 2 and 3 present the description and analysis of the model, respectively. Various performance measures are obtained in Section 4. Section 5 is devoted to the cost analysis which is illustrated by means of numerical examples in Section 6 followed by conclusion in Section 7. 2. Description of the model Let us consider that the demand arrive according to Poisson process with rate λ(>0) and demand only single unit at a time. Life time of each item has exponential distribution with rate γ (>0). Inventory is replenished according to (s, S) policy, the replenishment time being exponentially distributed with parameter η(>0). The service for a demand may be interrupted due to server breakdown which follows Poisson distribution with parameter δ 1 . An interrupted service, after repair, resumes from where it was stopped. The repair time follows exponential distribution with parameter δ 2 . We assume that the demands that occur during the stock-out period or during service interruption period may turn out to be an ordinary customer with probability p or a negative customer with probability q(=1 − p) and then they enter the orbit of infinite size. These orbiting customers compete for their demands according to an exponential distribution with parameter θ (>0). We also assume that no inventory is lost due to service interruption and any order placed in server breakdown state is canceled. Some notations:

⎧ ⎨pλ + η, i = 0; ci = η + iγ , i = 1, 2, . . . , s; ⎩ i = s + 1, . . . , S. iγ , ⎧ i = 0; ⎨pλ + η + qλ, di = λ + θ + δ1 + η + iγ , i = 1, 2, . . . , s; ⎩ λ + θ + δ1 + iγ , i = s + 1, . . . , S. fi = λ + iγ ,

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

hi = θ + iγ + λ,

i = 1, . . . , S.

g0 = −(δ2 + qλ + pλ). e : a column vector of appropriate dimension containing all ones. In : an identity matrix of order n.

104

Vijaya Laxmi P., Soujanya M.L. / Applied Mathematics and Computation 262 (2015) 102–110

Let N(t) be the number of demands in the orbit, L(t) be the inventory level at time t and

⎧ ⎨0, if the server is idle, ζ (t) = 1, if the server is busy, ⎩ 2, if the server is broke down,  be the state of the server. Then  = {(N(t), ζ (t), L(t))} is a Markov chain with state space E = {(0, 0, k) | 0 ≤ k ≤ S} {(i, 0, 0) |  i ≥ 1} {(i, j, k) | i ≥ 1, j = 1, 2, 1 ≤ k ≤ S}} which is partitioned into level ¯i defined as 0¯ = {(0, 0, 0), . . . , (0, 0, S)} and ¯i = {(i, 0, 0), (i, 1, 1), . . . , (i, 1, S), (i, 2, 1), . . . , (i, 2, S)}, for i  1. This makes the Markov chain a level independent Quasi-Birth-Death (QBD) process. The infinitesimal generator of the process  is

⎛

B0 ⎜B2 ⎜ ⎜ T =⎜0 ⎜0 ⎝ .. .

B1 A1 A2 0 .. .

0 A0 A1 A2 .. .

0 0 A0 A1 .. .

0 0 0 A0 .. .

⎞ ... . . .⎟ ⎟ . . .⎟ , ⎟ . . .⎟ ⎠ .. .

where the sub-matrices A0 , A1 , A2 are of order (2S + 1) × (2S + 1), B0 is of order (S + 1) × (S + 1), B1 is of order (S + 1) × (2S + 1) and B2 is (2S + 1) × (S + 1) matrix. These sub-matrices are expressed as shown below:

⎞ pλ . . . 0

0 ⎜ .. .. ⎟ , A = B1 , D = (0 pλI ), B = C0 , A = C0 B1 = ⎝ ... , C1 = (0 qλIS ), ⎠ S 0 0 2 2 . . D0 C1 0 qλIS 0 ... 0 ⎛ ⎞ qλ 0 . . . 0 0

⎜ θ 0 . . . 0 0⎟ E 0 E1 0 ⎜ ⎟ C0 = ⎜ . , E1 = , E2 = (0 δ2 IS ), E3 = g0 IS , ⎟ , A1 = . . . . .. .. ⎠ .. .. E2 E3 δ1 IS ⎝ .. 0 0 ... θ 0 ⎛ ⎛ ⎞ ⎞ 0 ... 0 ... 0 0 0 ... 0 ... 0 0 −d0 −co ⎜ f0 ⎜ γ −d1 . . . 0 ... 0 0 ⎟ −c1 . . . 0 ... 0 0 ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ . ⎜ . .. .. .. .. .. .. ⎟ .. .. .. .. .. .. ⎟ ⎜ .. ⎜ ⎟ ⎟ . . . . . . . . . . . . . . ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ 0 . . . −d(s−1) . . . 0 . . . −c(s−1) . . . η 0 ⎟ η 0 ⎟ ⎜ 0 ⎜ 0 E0 = ⎜ . ⎟ , B0 = ⎜ ... 0 η ⎟ ... 0 η ⎟ 0 ... fs 0 ... sγ ⎜ 0 ⎜ 0 ⎟ ⎜ ⎜ ⎟ ⎟ .. ⎟ .. ⎟ .. .. .. .. .. .. .. .. .. .. ⎜ .. ⎜ .. ⎜ . ⎜ . . ⎟ . ⎟ . . . . . . . . . . ⎜ ⎜ ⎟ ⎟ ⎝ 0 ⎝ 0 0 ... 0 . . . −d(S−1) 0 ... 0 . . . −c(S−1) 0 ⎠ 0 ⎠ −dS −cS 0 0 ... 0 ... fS 0 0 ... 0 ... Sγ ⎛

3. Analysis of the model In this section, using the matrix analytic method, we perform the steady state analysis. First we determine the stability condition under which the irreducible Markov chain is positive recurrent which guarantees the existence of steady state solution. 3.1. Stability condition To discuss the stability condition of the process under study, we consider the generator matrix A = A0 + A1 + A2 , which is given by

⎛

A=

E 3 E1 E2 E4

,

E4 = −δ2 IS ,

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ E3 = ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

−η h0 .. . 0 0 .. . 0 0

0 −d1 .. . 0 0 .. . 0 0

... ... .. . ... ... .. . ... ...

0 0 .. . −d(s−1) hs .. . 0 0

... ... .. . ... ... .. . ... ...

0 0 .. .

η 0 .. . −d(S−1) hS

0 0 .. . 0

η

.. . 0 −dS

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

Let φ = (φ (0, 0), φ (1, 1), . . . , φ (1, S), φ (2, 1), . . . , φ (2, S)) denote the steady state probability vector of the generator A, i.e., φ A = 0, φ e = 1.

Vijaya Laxmi P., Soujanya M.L. / Applied Mathematics and Computation 262 (2015) 102–110

105

Lemma 3.1. The steady state probability vector φ corresponding to the generator A is given by

φ(2, k) =

δ1 φ(1, k), δ2

(1)

φ(1, k) = φ(0, 0)rk , k = 1, . . . , S, where

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

η , ψ(k, k) ηχ (1, k − 1) , ψ(1, k) rk = ηχ (1, s)ψ(s + 1, k − 1) , ⎪ ⎪ ⎪ ψ(1, k) ⎪ ⎪   ⎪ k ⎪  ⎪ ηχ (1, s)ψ(s + 1, k − 1) η ⎪ ⎪ − r 1 + , ⎪ i−Q ⎩ ψ(1, k) ψ(k, k) i=Q+1

(2)

k = 1; k = 2, . . . , s + 1; k = s + 2, . . . , Q; k = Q + 1, . . . , S.

With χ(1, k) = sk=1 (di − δ1 ), ψ(j, k) = ki=j (hi ), 1 ≤ k ≤ S, and φ (0, 0) can be obtained by solving ηφ (1, s) − hS φ (1, S) = 0 and φ e = 1. Proof. The equation φ A = 0 yields the following set of equations.

−ηφ(0, 0) + h1 φ(1, 1) = 0,

(3)

−dk φ(1, k) + hk+1 φ(1, k + 1) + δ2 φ(2, k) = 0; 1 ≤ k ≤ s,

(4)

−dk φ(1, k) + hk+1 φ(1, k + 1) + δ2 φ(2, k) = 0; s + 1 ≤ k ≤ Q − 1,

(5)

ηφ(0, 0) − dQ φ(1, Q ) + hQ+1 φ(1, Q + 1) + δ2 φ(2, Q ) = 0,

(6)

ηφ(1, k − Q ) − dk φ(1, k) + hk+1 φ(1, k + 1) + δ2 φ(2, k) = 0; Q + 1 ≤ k ≤ S − 1,

(7)

ηφ(1, s) − dS φ(1, S) + δ2 φ(2, S) = 0,

(8)

δ1 φ(1, k) − δ2 φ(2, k) = 0; 1 ≤ k ≤ S.

(9)

Clearly Eq. (1) reduces from Eq. (9). To get Eq. (2), use Eq. (1) in the system of Eqs. (3)–(8) and solve them recursively. Now, from Eqs. (1), (2) and (8) we get φ (0, 0)(ηrs − hS rS ) = 0. Using this and the normalizing condition φ e = 1 we get the value of φ (0, 0).  The following result gives the stability condition. Lemma 3.2. The stability condition of the system under study is given by



p<

S S  θ rk 1 − rk λ k=1 k=1

−1

+ q.

Proof. From the well known result of Neuts [12] on the positive recurrence of A, we have φ A0 e < φ A2 e. Simplification of this yields the stated result.  3.2. Computation of steady state vector Let π = (π 0 , π 1 , π 2 , . . . ) be the steady state vector, where π i is the probability of i demands in the system, i  0. Further, these probabilities are partitioned as π 0 = (π 0 (0, 0), . . . , π 0 (0, S)) and π i = (π i (0, 0), π i (1, 1), . . . , π i (1, S), π i (2, 1), . . . , π i (2, S)), i  1, where π i (j, k) indicates the probability of i demands in the system, j(=0, 1, 2) indicates the state of the server and k is the inventory level. Since  is a level independent QBD process, its steady state probability vector is given by π i = π 0 Ri , i  1 (see Neuts [12]), where R is the minimal non-negative solution of the matrix-quadratic equation R2 A2 + RA1 + A0 = 0, which can be obtained by using the following iterative procedure. Computational algorithm for R: Step 1: Step 2: Step 3: Step 4:

Take k = 1. Take U = A1 and calculate G = (I − U)−1 A2 . Increment k by 1. Replace U = A1 + A0 G and G = (I − U)−1 A2 .

106

Vijaya Laxmi P., Soujanya M.L. / Applied Mathematics and Computation 262 (2015) 102–110

Step 5: Repeat the Step-3 and Step-4 until ||e − Ge|| < ε , where ε is a stopping tolerance. Step 6: Calculate R = A0 (I − U)−1 . For finding the boundary vector π 0 we have from π T = 0,

π1 = −π0 B1 (A1 + RA2 )−1 = π0 W, where

W = −B1 (A1 + RA2 )−1 . Further, π 0 B0 + π 1 B2 = 0, i.e., π 0 (B0 + WB2 ) = 0. First we take π 0 as the steady state vector of the generator matrix B0 + WB2 . Then π i , for i  1 can be found using the formulae π 1 = π 0 W, π i = π 1 Ri − 1 , i  2. Finally, the steady state probability distribution of the system under study is obtained by dividing each π i with the normalizing constant

(π0 + π1 + π2 + . . .)e = π0 (1 + W (I − R)−1 )e. 4. System performance measures In this section, we derive some performance measures of the system using which the total expected cost per unit time can be constructed.  S (i) The probability of the busy server is given by Pβ = ∞ i=1 j=1 πi (1, j).  S (ii) The probability for the breakdown server is given by Pα = ∞ i=1 j=1 πi (2, j). (iii) The probability that the server is idle is given by Pγ = 1 − Pβ − Pα . (iv) Expected reorder rate: Let EOR denote the expected reorder rate. We note that the reorder will occur either at the time of service completion or failure (perish) of an item in stock, which bring the inventory level to s. This leads to

EOR =

∞ 

[λ + θ + (s + 1)γ ]πi (1, s + 1) + (s + 1)γ π0 (0, s + 1).

i=1

(v) Expected inventory level: Since π i (j, k), j = 0, 1, 2, denotes the steady state probability vector for kth inventory level, the expected inventory level (EIL ) is given by

EIL =

S 

jπ0 (0, j) +

j=0

∞  S 

j(πi (1, j) + πi (2, j)).

i=1 j=1

(vi) Expected rate of arrivals of negative customers: The expected arrival rate of negative customers (ENC ) at steady state is given by

ENC =

∞ 

qλπi (0, 0) +

i=1

∞  S 

qλπi (2, j).

i=1 j=1

(vii) Expected number of demands in the orbit: We note that the primary demands enter into the orbit whenever the service is interrupted or during stock-out period. Hence, the expected number of demands (EOC ) in the orbit is given by

EOC =

∞  S 

iπi (2, j) +

∞ 

i=1 j=1

iπi (0, 0).

i=1

(viii) Expected replenishment rate: The expected replenishment rate (ERR ) is given by

ERR =

∞  s 

ηπi (0, j) +

i=0 j=0

∞  s 

ηπi (1, j).

i=1 j=1

(ix) Successful rate of retrials: Let ERC denote the successful rate of retrials. We note that the orbiting demands are retry for their service only when the server is in busy state. Hence, it is given by

ERC =

∞  S 

θ πi (1, j).

i=1 j=1

(x) Expected interruption rate: The expected interruption rate (EIN ) is given by

EIN =

∞  S  i=1 j=1

δ1 πi (1, j).

Vijaya Laxmi P., Soujanya M.L. / Applied Mathematics and Computation 262 (2015) 102–110

107

(xi) Expected repair rate: The expected repair rate (ERE ) is given by

ERE =

∞  S 

δ2 πi (2, j).

i=1 j=1

(xii) Expected failure rate: The expected failure rate (EFR ) is given by

EFR =

∞  S 

jγ πi (1, j).

i=1 j=1

5. Cost analysis We develop a total expected cost function per unit time with an objective to determine the optimum values of s and S so that the total cost is minimized. Let CS = Setup cost per order, CO = Waiting cost of a demand in the orbit per unit time, CH = Holding cost per unit item per unit time, CN = Loss per unit time due to arrival of a negative customer, CF = Failure cost per unit item per unit time, and CI = Cost incurred per unit time due to interrupted service. Based on the definitions of each cost element listed above and various performance measures of the model, the total expected cost function per unit time is defined as

TC (s, S) = CS EOR + CO EOC + CH EIL + CN ENC + CF EFR + CI EIN . Substituting EOR , EIL , EOC , EFR , ENC and EIN in the above equation, we get

TC (s, S) = CS

 ∞

(λ + θ )πi (1, s + 1) +

i=1

+ CH

(s + 1)γ πi (1, s + 1) + CO

 ∞  S

i=0

 S

jπ0 (0, j) +

j=0

+ CF



∞ 

∞  S  i=1 j=1

∞  S 

i=1 j=1



j(πi (1, j) + πi (2, j)) + CN

i=1 j=1

jγ πi (1, j) + CI

∞  S 

iπi (2, j) +

 ∞ i=1

qλπi (0, 0) +

∞ 

 iπi (0, 0)

i=1 ∞  S 

 qλπi (2, j)

i=1 j=1

δ1 πi (1, j).

i=1 j=1

Since the computation of the π  s are recursive, it is quite difficult to find the optimality of the cost function using analytical methods. Hence, we present the following numerical examples to demonstrate the computability of the results derived in our work and to illustrate the effect of the parameters of the main performance characteristics. 6. Numerical analysis This section presents the numerical results in the form of tables and graphs. We have used the direct search procedure to find the optimal values of s and S and to discuss the effect of parameters on expected demands in the orbit, expected inventory level and successful rate of retrials. The parameters of the model are assumed as λ = 1.5, p = 0.5, δ 1 = 1, δ 2 = 1.2, η = 0.8, γ = 1.4, θ = 3, S = 16 and s = 3 unless they are considered as variables in the tables and graphs. The cost values are fixed as CS = 5, CH = 1, CO = 15, CF = 25, CN = 10 and CI = 20. In tables, the upper entries in each cell correspond to S∗ and s∗ and the lower entry gives the optimum cost (TC(s∗ , S∗ )). Table 1 gives the optimum values S∗ and s∗ that minimize the total expected cost function TC(s, S) for different values of the retrial rate (θ ) and the failure rate (γ ). It is observed that as θ increases the optimum cost decreases and it increases with the increase of γ which agrees with reality. It is also observed that if γ increases, the maximum inventory level (S) decreases for fixed θ as expected in practice. The optimum values S∗ , s∗ and TC(s∗ , S∗ ) for different values of arrival rate (λ) and the interruption rate (δ 1 ) are presented in Table 2. It is observed that as λ and δ 1 increases simultaneously the optimum cost is also increasing. It happens because the arriving demand enters into the orbit and the server spends more time in interruption state whenever λ and δ 1 increases simultaneously and therefore all the performance measures involved in the cost function increase. Fig. 1 shows the effect of retrial rate (θ ), interruption rate (δ 1 ) and the repair rate (δ 2 ) on the successful rate of retrials (ERC ). It can be observed that with the increase of θ and δ 2 , ERC increases. This is because the orbital demand retry for their service whenever the server is in busy state. Further, one may also observe that ERC decreases with the increase of δ 1 , since in this case most of the time the sever spends in server beak down state.

108

Vijaya Laxmi P., Soujanya M.L. / Applied Mathematics and Computation 262 (2015) 102–110 Table 1 The optimal values s∗ , S∗ and TC(s∗ , S∗ ) for different θ and γ .

γ ↓θ →

2.1

2.4

2.7

3.0

3.3

3.6

1.2

(1,5) 15.5491 (1,5) 17.7215 (2,4) 19.5183 (2,4) 19.9390 (2,4) 21.2424 (2,4) 22.4448

(1,5) 14.8343 (1,5) 16.8875 (1,5) 18.8352 (2,4) 19.0623 (2,4) 20.3083 (2,4) 21.1254

(1,5) 14.2391 (1,5) 16.1888 (1,5) 18.0414 (2,4) 18.3136 (2,4) 19.5070 (2,4) 20.6139

(2,7) 13.4298 (1,5) 15.5952 (1,5) 17.3640 (2,4) 17.6672 (2,4) 18.8121 (2,4) 19.8763

(2,7) 13.0171 (1,5) 15.0848 (1,5) 16.7792 (1,5) 17.3905 (2,4) 18.2041 (2,4) 19.2288

(2,7) 12.6616 (1,5) 14.6413 (1,5) 16.2694 (1,5) 16.8187 (2,4) 17.6678 (2,4) 18.6559

1.4 1.6 1.8 2.0 2.2

Table 2 The optimal values s∗ , S∗ and TC(s∗ , S∗ ) for different λ and δ 1 .

δ 1 ↓λ →

0.5

1.0

1.5

2.0

2.5

0.5

(4,6) 6.1384 (3,5) 7.1513 (1,5) 9.2284 (1,5) 9.5895 (1,5) 9.9391

(1,5) 11.9701 (1,5) 12.5504 (1,5) 13.0967 (1,5) 13.6111 (1,5) 14.0954

(1,5) 14.9100 (1,5) 15.5952 (1,5) 16.2273 (1,5) 16.8112 (2,7) 16.9371

(1,5) 17.5442 (1,5) 18.2865 (2,7) 18.4122 (2,7) 18.9455 (2,7) 19.4338

(2,7) 19.3535 (2,7) 20.0251 (2,7) 20.6307 (2,7) 21.1786 (3,9) 21.3440

1.0 1.5 2.0 2.5

0.09

Successful rate of retrials (ERC)

0.085 0.08 0.075

ERC = 0.08514 at δ1 or δ2 = 1.06 E = 0.08073 RC at θ or δ = 1.87

0.07

1

0.065 θ versus ERC δ1 versus ERC

0.06

δ versus E 2

0.055 0.05

1

RC

1.5

θ or δ or δ 1

2

2.5

2

Fig. 1. Effect of θ , δ 1 and δ 2 on the successful rate of retrials.

The two figures in Fig. 2 plot the effect of retrial rate (θ ), replenishment rate (η), failure rate (γ ), interruption rate (δ 1 ) and the repair rate (δ 2 ) on the expected demands in the orbit (EOC ). In this paper, we assume that the replenishment for an order takes place and an item is perished whenever the server is in busy state or in idle state. But, the demand enter into the orbit whenever the server is on interruption or the inventory is stock-out. Thus if δ 2 or η or θ increases, the server spends in busy state and then EOC decreases. But in the case δ 1 as it increases, the sever spends more time in interruption state. Then EOC increases. We can clearly observe these cases in Fig. 2. We can also observe that EOC increases with the increase in γ . This is because as γ increases the life time of inventory decreases and hence the inventory is stock-out.

Vijaya Laxmi P., Soujanya M.L. / Applied Mathematics and Computation 262 (2015) 102–110

109

0.32

) EOC = 0.3390

0.45

at θ or γ = 1.78

0.4 0.35 0.3 0.25

0.15

0.3

OC

0.5

0.2

0.31

Expected demands in the orbit (E

Expected demands in the orbit (E

OC

)

0.55

θ versus E

OC

EOC = 0.1680

η versus EOC

at η or γ = 1.08

γ versus E

OC

0.28

0.5

1

1.5

2

θ or η or γ

δ1 versus EOC

0.27

δ2 versus EOC

0.26 0.25 0.24 0.23

0.1 0

EOC = 0.2648 at δ1 or δ2 = 1.0385

0.29

0.22 0.6

2.5

0.8

1

1.2

δ or δ 1

1.4

1.6

1.8

2

2

Fig. 2. Effect of various parameters on expected demands in the orbit. 4.21 EIL = 4.2017 at θ or δ1 = 1.62 IL

Expected inventory level (E )

IL

Expected inventory level (E )

4.2 4.19

θ versus E

IL

δ1 versus EIL

4.18

δ versus E 2

4.17

IL

E = 4.1631 at IL δ or δ = 1.13

4.16

1

2

4.15 4.14 4.13 0.5

1

1.5

2

2.5

θ or δ or δ 1

3

3.5

4

4.5

η versus EIL

5.5

γ versus E

IL

5 E = 4.794 at IL

η or γ = 1.08 4.5

4

0.6

0.8

2

1

η or γ

1.2

1.4

1.6

Fig. 3. Effect of various parameters on expected inventory level.

As in Fig. 2, we observe the effect of same parameters on the expected inventory level (EIL ) in Fig. 3. We note that the inventory level decreases if the server is in busy period or inventory is perished and it increases if the server is on interruption or an order is replenished. As we discussed in Fig. 2, the server spends more time in busy period if δ 2 , η or θ increases and he spends more time in interruption state if δ 1 increases. Hence, EIL decreases with the increase in δ 2 , θ or γ and it increases with the increase in δ 1 or η. 7. Conclusion In this paper, we have analyzed a perishable inventory model with service interruptions, retrial demands and negative customers. We have derived the joint probability distribution of the inventory model at steady state through the matrix analytic method. Various performance measures of the system are discussed and also cost analysis is carried out to obtain the optimum value of s and S. The technique used in this paper can be adopted for more complex models like perishable inventory models with retrial MAP arrivals, negative customers and service interruptions, which point to future investigation. Further, the other order policies like (s, q), (t, S), (t, q) may be considered in future research work. Acknowledgment The authors are thankful to the referees for their valuable comments and suggestions which have helped in improving the quality of the presentation of this paper.

110

Vijaya Laxmi P., Soujanya M.L. / Applied Mathematics and Computation 262 (2015) 102–110

References [1] J.R. Artalejo, A. Krishnamoorthy, M.S. Lopez-Herrero, Numerical analysis of (s, S) inventory systems with repeated attempts, Ann. Oper. Res. 141 (2006) 67–83. [2] O. Baron, O. Berman, David Perry, Continuous review inventory models for perishable items ordered in batches, Math. Methods Oper. Res. 72 (2010) 217–247. [3] E. Gelenbe, Random neural networks with negative and positive signals and product form solution, Neural Comput. 1 (1989) 502–510. [4] S. Kalpakam, G. Arivarignan, A continuous review perishable inventory model, J. Stat. 19 (1988) 389–398. [5] C. Kouki, E. Sahin, Z. Jemai, Y. Dallery, Periodic review inventory policy for perishables with random lifetime, in: Proceedings of the 8th International Conference of Modelling and Simulation – MOSIM’10, Tunisia, 2010. [6] A. Krishnamoorthy, S.S. Nair, V.C. Narayanan, An inventory model with server interruptions, in: Proceedings of the 5th International Conference on Queueing Theory and Network Applications, July 24–26, Beijing, China, 2010. [7] A. Krishnamoorthy, S.S. Nair, V.C. Narayanan, An inventory model with positive service time and server interruptions, Calcutta Statist. Assoc. Bull., 2012. [8] A. Krishnamoorthy, S.S. Nair, V.C. Narayanan, Production inventory system with positive service time and interruptions, Int. J. Syst. Sci. (2013), doi:10.1080/00207721.2013.837538. [9] A.S. Lawrence, B. Sivakumar, G. Arivarignan, A perishable inventory system with service facility and finite source, Appl. Math. Model. 37 (2013) 4771–4786. [10] P. Manual, B. Sivakumar, G. Arivarignan, A perishable inventory system with service facilities, MAP arrivals and PH-service time, J. Syst. Sci. Syst. Eng. 16 (2007) 62–73. [11] P. Manual, B. Sivakumar, G. Arivarignan, Perishable inventory systems with postponed demands and negative customers, J. Appl. Math. Decis. Sci. 2007 (2007) 12 Article ID 94850. [12] M.F. Neuts, Matrix-Geometric Solutions in Stochastic Models – An Algorithmic Approach, John Hopkins Universal Press, Baltimore, 1981. [13] S. Nahmias, Perishable Inventory Systems, International Series in Operations Research and Management, 160, Springer, 2011. [14] C. Periyasamy, A finite source perishable inventory system with retrial demands and multiple server vacation, Int. J. Eng. Res. Technol. 2 (2013) 3802–3815. [15] B. Sivakumar, G. Arivarignan, A perishable inventory system with service facilities and negative customers, Adv. Model. Optim. 7 (2005) 193–210. [16] B. Sivakumar, A perishable inventory system with retrial demands and a finite population, J. Comput. Appl. Math. 224 (2009) 29–38. [17] H.J. Weiss, Optimal ordering policies for continuous review perishable inventory models, J. Oper. Res. Soc. Am. 28 (1980) 365–374.