# A Perishable Inventory System with Postponed Demands and Multiple Server Vacations

Authors: R. Jayaraman; B. Sivakumar; G. Arivarignan
Journal: Modelling and Simulation in Engineering
 (2012)

---

## Abstract

A mathematical modelling of a continuous review stochastic inventory
system with a single server is carried out in this work. We assume
that demand time points form a Poisson process. The life time of each
item is assumed to have exponential distribution. We assume(
s
,
S
)
ordering policy to replenish stock with random lead time. The server
goes for a vacation of an exponentially distributed duration at the time
of stock depletion and may take subsequent vacation depending on the
stock position. The customer who arrives during the stock-out period
or during the server vacation is offered a choice of joining a pool which
is of finite capacity or leaving the system. The demands in the pool
are selected one by one by the server only when the inventory level
is above s, with interval time between any two successive selections
distributed as exponential with parameter depending on the number
of customers in the pool. The joint probability distribution of the
inventory level and the number of customers in the pool is obtained
in the steady-state case. Various system performance measures in the
steady state are derived, and the long-run total expected cost rate is
calculated.


---

## Body

## 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 case) or satisfied only after the arrival of ordered items (backlogging). In the latter case, it is assumed that either all (full backlogging) or any prefixed number of demands (partial backlogging) that occurred during the stock-out period are satisfied. The often quoted review articles Nahmias [1] and Raafat [2] and the recent review articles N. H. Shah and Y. K. Shah [3] and Goyal and Giri [4] provide excellent summaries of many of these modelling efforts. For some recent references, see Chakravarthy and Daniel [5], Yadavalli et al. [6], and Kalpakam and Shanthi [7, 8].In the case of continuous review perishable inventory models with random life times for the items, most of the models assume instantaneous supply of order (Lian and Liu [9], Liu and Shi [10], and Kalpakam and Arivarignan [11], and Gürler and Özkaya [12]). When a server is assumed to process the demands in continuous review inventory systems, the concept of server vacation was also taken up in some works. Daniel and Ramanarayanan [13] introduced the concept of server vacation in an inventory model with two servers. In [14], they studied an (
s
,
S
) inventory system in which the server takes a rest when the level of the inventory is zero. The demands that occurred during stock-out period were lost. The interoccurrence times between two successive demands, the lead times, and the rest times are assumed to follow general distributions, which are mutually independent. Using renewal and convolution techniques, they obtained the steady-state transition probabilities. However, server vacations in Queueing systems have been widely studied in different contexts in the literature. We refer the readers to Doshi [15, 16], Takagi [17, 18], and Tian and Zhang [19] for details on various vacations systems and for additional references.In the context of inventory system, any arriving demands during the server vacation either enter a pool or leave the system. These pooled customers are selected at a later time, preferably after the stock is replenished. This type of inventory problems is called inventory with postponed demands.The concept of postponed demands in the inventory model has been introduced by Berman et al. [20]. They assumed that both demand and service rates are deterministic. Krishnamoorthy and Islam [21] considered an inventory system with Poisson demand, exponential lead time and the pooled customers are selected according to an exponentially distributed time lag. Sivakumar and Arivarignan [22] considered an inventory model in which the demand occurs according to a Markovian arrival process, lead time has a phase type distribution, life time for the items in the stock has exponential distribution, and the pooled customers are selected exponentially. Manuel et al. [23] dealt with an inventory system in which the positive and negative demands occur according to an independent Markovian arrival processes, the lead time of the reorder, the life time of the items, the interselection time of customers from the pool and the reneging time points of the customers in the pool are independent, and each has the exponential distribution. In all the above models, the authors assumed that the pool size is finite. Sivakumar and Arivarignan [24] considered an inventory system with independent Markovian arrival processes for positive and negative demands, exponential lead time for the reorders, exponential life times for each item in the stock, and the pool size is infinite.In earlier works listed above, the server is assumed to be available even during the stock-out periods. In this work, we have introduced the concept of server vacation. It may be noted that this model also includes situations in which the server may not be availing vacation, but the server may attend some other work during the stock-out period and may return at random times to the inventory system. If the stock is still not replenished, the server goes for another vacation again or to attend the other work. Employees who are assigned extra work on additional remuneration basis can also be thought of such servers.In this paper, we consider a continuous review perishable inventory system. The unit demands occur according to a Poisson process. The(
s
,
S
) ordering policy is implemented to replenish stock. When the inventory is completely depleted, the server is assumed to go on a vacation of random duration. At the end of vacation, if the stock is not replenished, the server again starts his/her vacation. The demands that occurred during stock-out period are offered postponement of their demand after the stock is replenished. Those demands that accept the postponement joins a pool of finite capacity. We assume that fraction p of demands join the pool and the rest leave the system. The pooled demands are selected once the stock is replenished, and the server resumes service. But the selection from the pool will be stopped if the inventory falls below the reorder level, s.The rest of the paper is organized as follows. In Section2, we describe the mathematical model of the problem under study. The steady-state analysis of the model is presented in Section 3, and some key system performance measures are derived in Section 4. In Section 5, we calculate the total expected cost rate, and, in the final 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 ofS unit items. The items are assumed to have identical and independent exponential life-time distributions with parameter γ
(
>
0
). When an item perishes, it is removed from the stock. Another process which removes items from the stock is the demand process. The demand is for single item, and the demand occurrences form a Poisson process with rate λ
(
>
0
). A server processes the unit demands. The stock is replenished by the server by implementing (
s
,
S
) ordering policy, where the lead time is assumed to have independent exponential distribution with parameter μ
(
>
0
). If the stock is depleted before the supply of reorder, the server goes on a vacation duration which has exponential distribution with parameter β
(
>
0
). At the end of a vacation period, if the stock is still empty, the server takes another vacation. The demands that occur during stock-out period are offered postponement at a later time. It is assumed that the demands accept this postponement or leave the system according to a Bernoulli distribution with probability p. The demand which accepts an offer of postponement may join a pool of finite capacity fixed as N.The demands from the pool are selected if the server is on service and the inventory level is aboves such that the interval time between successive selections is exponentially distributed with parameter α
n
(
>
0
) where n is the number of customers in the pool.Notation.
e: a column vector of appropriate dimension containing all ones.
0: a zero matrix of appropriate dimension.
0
(
i
×
j
): a zero matrix with i rows and j columns.


## 3. Analysis

LetX
(
t
)
and
L
(
t
), respectively, denote the number of demands in the pool and the on-hand inventory level at time t. We define a binary variable Y
(
t
) (which takes 0
and
1) to denote the status of the server, respectively, “on vacation’’ and “on service.’’ Similarly, the binary variable Z
(
t
) will indicate the status of the reorder to replenish the stock: Z
(
t
)
=
0
or
1, respectively, if   “no order is pending’’ or “an order is pending.’’The values taken by these random variables are listed in the following.(1)We consider the stochastic process{
X
(
t
)
,
Y
(
t
)
,
Z
(
t
)
,
L
(
t
)
,
t
≥
0
}. Since the components carry sufficient information on the sample path of the process and from the assumptions made on the input and output processes, this process can easily be seen to be a Markov process.The state space of the stochastic process{
X
(
t
)
,
Y
(
t
)
,
Z
(
t
)
,
L
(
t
)
,
t
≥
0
} is the collection of all quadruples 〈
i
1
〉
=
(
i
1
,
i
2
,
i
3
,
i
4
) 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.We shall write the transition rate matrix in a block-partitioned form wherei
1th row (block) corresponds to the collection of quadruples 〈
i
1
〉 (those with leftmost entry i
1), i
1
=
0,1
,
2
,
…
,
N. It may be recalled that the pool size (i
1) may increase or decrease by one unit (except the end states) when a demand joins or leaves the pool. Hence, the transition rate matrix is a tridiagonal one.We also note that when the number of demands in the pool isi
(
i
=
0,1
,
2
,
…
,
N
-
1
), a new demand joining the pool will shift the current set of quadruple 〈
i
〉 to 〈
i
+
1
〉 and the transition rate does not depend upon i. Therefore, all the entries in the upper diagonal of the rate matrix are equal to a constant block matrix, which we shall denote by B
0. The lower diagonal entries correspond to the transition from states of 〈
i
〉
,
i
=
1,2
,
…
,
N, to states 〈
i
-
1
〉. This is due to the selection of a demand from the pool, and the rate of selection of any one of the demands is α
i. Hence, these transition rates depend on row indices. Therefore, the entries of the lower diagonal depend on the row index. We shall denote them by C
i. In the case of diagonal entries, again, the transition rate may depend upon the pool size (row index), if it involves selection of a demand from the pool. Hence, the entries here also depend upon the respective row index. We shall denote them by A
i.Hence, the infinitesimal generator matrix can be written as(2)
0
1
2
3
⋯
N
-
2
N
-
1
N
0
1
2
⋮
N
-
2
N
-
1
N
(
A
0
B
0
0
0
⋯
⋯
0
0
C
1
A
1
B
0
0
⋯
⋯
0
0
0
C
2
A
2
B
0
⋯
⋯
0
0
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋯
⋯
⋯
⋯
⋯
A
N
-
2
B
0
⋯
⋯
⋯
⋯
⋯
⋯
C
N
-
1
A
N
-
1
B
0
⋯
⋯
⋯
⋯
⋯
0
C
N
A
N
)
.
We shall determine the submatrix B
0, the elements of which correspond to the transitions from 〈
i
〉
to
〈
i
+
1
〉
,
i
=
0,1
,
2
,
…
,
N
-
1. This corresponds to the case that one demand has joined the pool. In a Markov process, at most one event can occur in a small interval. Hence, for the transitions from (
i
1
,
i
2
,
i
3
,
i
4
)
to
(
i
1
+
1
,
i
2
′
,
i
3
′
,
i
4
′
), we have 0 when i
2
′
≠
i
2
,
i
3
′
≠
i
3
,
i
4
′
≠
i
4 and p
λ when they are equal and i
2
=
0. The matrix B
0 is given by(3)Next, we consider the submatrixC
i, the elements of which give the transitions from 〈
i
〉 to 〈
i
-
1
〉. This corresponds to the event that a demand from the pool is selected, and this indicates that the server is “on service’’ and that “ no order is pending.’’ Therefore, possible value is i
2
′
=
i
2
=
1. As a result of a selection from the pool, an item from the inventory is used to meet this demand. Hence, L
(
t
) moves from i
4 to i
4
-
1
,
if
i
4
=
s
+
2
,
s
+
3
,
…
,
S (with i
3
′
=
0) and from s
+
1 to s (with i
3
′
=
1). Hence, the submatrix C
i
,
i
=
1,2
,
…
,
N, is given by(4)where(5)
C
0
(
i
)
=
(
0
0
⋯
0
0
α
i
0
⋯
0
0
0
α
i
⋯
0
0
⋮
⋮
⋮
⋮
⋮
0
0
⋯
α
i
0
)
,
C
1
(
i
)
=
(
0
0
⋯
0
α
i
0
0
⋯
0
0
0
0
⋯
0
0
⋮
⋮
⋮
⋮
⋮
0
0
⋯
0
0
)
.Let us consider the submatrixA
i
,
i
=
1,2
,
…
,
N. In this case, there is no addition to nor selection from the pool. Firstly, we note the transitions(i)
from(
i
1
,
0,1
,
i
4
) to (
i
1
,
1,0
,
i
4
)
or
(
i
1
,
1,1
,
i
4
),
(ii)
from(
i
1
,
1,0
,
i
4
) to (
i
1
,
0,0
,
i
4
)
or
(
i
1
,
0,1
,
i
4
),
(iii)
from(
i
1
,
1,1
,
i
4
) to (
i
1
,
0,0
,
i
4
),as these involve more than one occurrence.Secondly, we observe that, only in the case of diagonal entriesa
(
(
i
1
,
1,0
,
i
4
)
,
(
i
1
,
1,0
,
i
4
)
) (i.e., the intensity of passage), the rate α
i (the rate of selection from the pool) will appear.Hence, the block-partitioned submatrixA
i will have the following structure:(6)The following cases are pertaining to the event that the server is on vacation before and after the transition.(i)
In the case of column vectora
01, the reorder status changes from “no order is pending’’ to “an order is pending.’’ Thus, i
4
=
0 and, after transition, i
4
′
=
0. No transition is possible from other states i
4
=
1,2
,
…
,
Q. Thus, we obtain
(7)
a
01
=
(
β
,
0
,
…
,
0
)
′
.
(ii)
The row vectora
10 considers the change of reorder status (from “order is pending’’ to “no order is pending’’). Thus, a supply of Q items is received. This changes i
2
=
0
,
i
3
=
0
,
i
4
=
0 to i
2
′
=
0
,
i
3
′
=
1
,
i
4
′
=
Q. Hence,
(8)
a
10
=
(
0,0
,
…
,
0
,
μ
)
.
(iii)
The constanta
11 is a diagonal entry, and, considering all transitions from (
i
1
,
0,1
,
0
) to other states, we get
(9)
a
11
=
{
-
(
p
λ
+
μ
)
if
i
1
=
1,2
,
…
,
N
-
1
,
-
μ
if
i
1
=
N
,
or simply a
11
=
-
(
p
λ
δ
̅
i
1
N
+
μ
), where δ
̅
i
1
N
=
1
-
δ
i
1
N.
(iv)
The nondiagonal entry ofA
00, namely,a
(
(
i
1
,
0,0
,
i
4
)
,
(
i
1
,
0,0
,
i
4
′
)
) is equal to i
4
γ for i
4
′
=
i
4
-
1 and zero for other nondiagonal elements. For the diagonal entry (considering the transitions to other states), we get
(10)
[
A
00
]
i
4
,
i
4
=
{
-
(
p
λ
+
β
+
i
4
γ
)
if
i
4
=
1,2
,
…
,
Q
,
i
1
≠
N
,
-
(
β
+
i
4
γ
)
if
i
4
=
1,2
,
…
,
Q
,
i
1
=
N
,
or, simply, -
(
p
λ
δ
̅
i
1
N
+
β
+
i
4
γ
)
if
i
4
=
1,2
,
…
,
Q, for alli
1.Using arguments similar to the above, we can express the matrixA
i
,
i
=
1,2
,
…
,
N as follows:(11)where(i)
u
k
′
=
(
0,0
,
…
,
1
,
…
,
0,0
) a vector having 1 at kth place and 0 at other places,
(ii)
(12)
D
00
(
i
)
=
0
1
⋯
Q
-
1
Q
0
1
2
⋮
Q
-
1
Q
(
-
(
p
λ
+
β
)
0
⋯
0
0
γ
-
(
p
λ
+
γ
+
β
)
⋯
0
0
0
2
γ
⋯
0
0
⋮
⋮
⋱
⋮
⋮
0
0
⋯
-
(
p
λ
+
(
Q
-
1
)
γ
+
β
)
0
0
0
⋯
Q
γ
-
(
p
λ
+
Q
γ
+
β
)
)
,
(iii)
(13)
F
00
=
s
+
1
s
+
2
⋯
Q
Q
+
1
⋯
S
0
1
⋮
s
+
1
s
+
2
⋮
Q
(
0
0
⋯
0
0
⋯
0
0
0
⋯
0
0
⋯
0
⋮
⋮
⋱
⋮
⋮
⋱
⋮
β
0
⋯
0
0
⋯
0
0
β
⋯
0
0
⋯
0
⋮
⋮
⋱
⋮
⋮
⋱
⋮
0
0
⋯
β
0
⋯
0
)
,
(iv)
(14)
F
01
=
1
2
⋯
s
-
1
s
0
1
⋮
s
-
1
s
⋮
Q
(
0
0
⋯
0
0
β
0
⋯
0
0
⋮
⋮
⋮
⋮
⋮
0
0
⋯
β
0
0
0
⋯
0
β
⋮
⋮
⋱
⋮
⋮
0
0
⋯
0
0
)
,
(v)
(15)
H
00
(
i
)
=
s
+
1
s
+
2
⋯
S
-
1
S
s
+
1
s
+
2
⋮
⋮
S
(
-
(
λ
+
(
s
+
1
)
γ
+
α
i
)
0
⋯
0
0
λ
+
(
s
+
2
)
γ
-
(
λ
+
(
s
+
2
)
γ
+
α
i
)
⋯
0
0
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
0
0
⋯
λ
+
S
γ
-
(
λ
+
S
γ
+
α
i
)
)
,
(vi)
(16)
H
01
=
1
2
⋯
s
-
1
s
s
+
1
s
+
2
⋮
S
(
0
0
⋯
0
λ
+
(
s
+
1
)
γ
0
0
⋯
0
0
⋮
⋮
⋯
⋮
⋮
0
0
⋯
0
0
)
,
(vii)
(17)
H
10
=
s
+
1
⋯
Q
Q
+
1
⋯
S
1
⋮
s
(
0
⋯
0
μ
⋯
0
⋮
⋱
⋯
⋮
⋱
⋮
0
⋯
0
0
⋯
μ
)
,
(viii)
(18)
H
11
=
1
2
⋯
s
-
1
s
1
2
⋮
s
(
-
(
λ
+
γ
+
μ
)
0
⋯
0
0
λ
+
2
γ
-
(
λ
+
2
γ
+
μ
)
⋯
0
0
⋮
⋮
⋮
⋮
⋮
0
0
⋯
λ
+
s
γ
-
(
λ
+
s
γ
+
μ
)
)
.


## 4. Steady-State Analysis

It can be seen from the structure of the infinitesimal generatorA that the time-homogeneous Markov process {
(
X
(
t
)
,
Y
(
t
)
,
Z
(
t
)
,
L
(
t
)
)
;
t
≥
0
} on the finite state space E is irreducible. Hence, the limiting distribution(19)
ϕ
(
j
1
,
j
2
,
j
3
,
j
4
)
=
lim
⁡
t
→
∞
Pr
[
X
(
t
)
=
j
1
,
Y
(
t
)
=
j
2
,
Z
(
t
)
=
j
3
,
L
(
t
)
=
j
4
∣
X
(
0
)
,
Y
(
0
)
,
Z
(
0
)
,
L
(
0
)
]
exists. Let(20)
ϕ
(
i
1
)
=
(
ϕ
(
i
1
,
0,0
,
0
)
,
ϕ
(
i
1
,
0,0
,
1
)
,
…
,
ϕ
(
i
1
,
0,0
,
Q
)
,
ϕ
(
i
1
,
0,1
,
0
)
,
ϕ
(
i
1
,
1,0
,
s
+
1
)
,
ϕ
(
i
1
,
1,0
,
s
+
2
)
,
…
,
ϕ
(
i
1
,
1,0
,
S
)
,
ϕ
(
i
1
,
1,0
,
s
+
1
)
,
ϕ
(
i
1
,
1,0
,
s
+
2
)
,
…
,
ϕ
(
i
1
,
1,0
,
S
)
)
,
ϕ
(
i
1
,
1,1
,
1
)
,
ϕ
(
i
1
,
1,1
,
2
)
,
…
,
ϕ
(
i
1
,
1,1
,
s
)
,
Φ
=
(
ϕ
(
0
)
,
ϕ
(
1
)
,
…
,
ϕ
(
N
)
)
.
Then, the vector of limiting probabilities Φ satisfies(21)
Φ
A
=
0
,
Φ
e
=
1
.From the structure ofA, it is seen that the Markov process under study falls into the class of birth and death process in a Markovian environment as discussed by Gaver et al. [25]. Hence, using the same arguments, we can calculate the limiting probability vectors. For the sake of completeness, we provide the algorithm here.Algorithm.
(1) Determine recursively the matrices(22)
F
0
=
A
0
,
F
i
1
=
A
i
1
+
C
i
1
(
-
F
i
1
-
1
-
1
)
B
0
,
i
1
=
1,2
,
…
,
N
.
(2) Compute recursively the vectorsϕ
(
i
1
) using
(23)
ϕ
(
i
1
)
=
ϕ
(
i
1
+
1
)
C
i
1
+
1
(
-
F
i
1
-
1
)
,
i
1
=
N
-
1
,
N
-
2
,
…
,
0
.
(3) Solve the system of equations(24)
ϕ
(
N
)
F
N
=
0
,
(25)
∑
i
1
=
0
N
ϕ
(
i
1
)
e
=
1
.
From the system of (24), vector ϕ
(
N
) could be determined uniquely, up to a multiplicative constant. This constant is obtained by using (23) and (25).


## 5. System Performance Measures

In this section, we give some system performance measures, which are useful in the qualitative interpretation of the model under study.


### 5.1. Expected Inventory Level

Letη
I denote the mean inventory level in the steady-state. Since ϕ
(
i
,
k
,
p
,
m
) denotes the steady state probability when the number of demands in the pool is i, the server status is k, the status of reorder is p, and the inventory level is m, the mean inventory level is given by(26)
η
I
=
∑
i
=
0
N
(
∑
m
=
1
Q
m
ϕ
(
i
,
0,0
,
m
)
+
∑
m
=
s
+
1
S
m
ϕ
(
i
,
1,0
,
m
)
+
∑
m
=
1
s
m
ϕ
(
i
,
1,1
,
m
)
)
.


### 5.2. Expected Reorder Rate

Letη
R denote the expected reorder rate in the steady state. We note that a reorder is triggered(i)
when the inventory level drops froms
+
1 to s if the server is in service or
(ii)
when the server returns from the vacation, he/she finds the inventory level less than or equal tos and there is no order pending.This leads to(27)
η
R
=
∑
i
=
0
N
ϕ
(
i
,
1,0
,
s
+
1
)
(
λ
+
(
s
+
1
)
γ
+
δ
¯
i
0
α
i
)
+
∑
i
=
0
N
∑
m
=
0
s
β
ϕ
(
i
,
0,0
,
m
)
,
where δ
i
j is the Kronecker delta function.


### 5.3. Expected Perishable Rate

Letη
PR denote the expected reorder Perishable in the steady state. This is given by(28)
η
PR
=
∑
i
=
0
N
(
∑
m
=
1
Q
m
γ
ϕ
(
i
,
0,0
,
m
)
+
∑
m
=
s
+
1
S
m
γ
ϕ
(
i
,
1,0
,
m
)
+
∑
m
=
1
s
m
γ
ϕ
(
i
,
1,1
,
m
)
)
.


### 5.4. Mean Shortage Rate

Letη
SR denote the mean shortage rate in the steady state. It may be noted that a shortage occurs only during the server vacation. The customers who arrive during server vacation prefer to leave the system either by their own choice or the pool has reached its maximum capacity:(29)
η
SR
=
∑
i
=
0
N
-
1
q
λ
[
∑
m
=
0
Q
ϕ
(
i
,
0,0
,
m
)
+
ϕ
(
i
,
0,1
,
0
)
]
+
λ
[
∑
m
=
0
Q
ϕ
(
N
,
0,0
,
m
)
+
ϕ
(
N
,
0,1
,
0
)
]
.


### 5.5. Expected Number of Customers in the Pool

Letη
P denote the expected number of customers in the pool in the steady state. Since ϕ
(
i
) is the steady state probability vector for i customers in the pool with each component specifying the probability for the particular combination of the status of the server, status of the reorder, and the inventory level, the expected number of customers in the pool is given by(30)
η
P
=
∑
i
=
1
N
i
ϕ
(
i
)
e
.


### 5.6. Fraction of Time When the Server is on Vacation

Letη
F denote the fraction of time when the server is on vacation. This is given by(31)
η
F
=
∑
i
=
0
N
∑
m
=
0
Q
ϕ
(
i
,
0,0
,
m
)
+
∑
i
=
0
N
ϕ
(
i
,
0,1
,
0
)
.


## 5.1. Expected Inventory Level

Letη
I denote the mean inventory level in the steady-state. Since ϕ
(
i
,
k
,
p
,
m
) denotes the steady state probability when the number of demands in the pool is i, the server status is k, the status of reorder is p, and the inventory level is m, the mean inventory level is given by(26)
η
I
=
∑
i
=
0
N
(
∑
m
=
1
Q
m
ϕ
(
i
,
0,0
,
m
)
+
∑
m
=
s
+
1
S
m
ϕ
(
i
,
1,0
,
m
)
+
∑
m
=
1
s
m
ϕ
(
i
,
1,1
,
m
)
)
.


## 5.2. Expected Reorder Rate

Letη
R denote the expected reorder rate in the steady state. We note that a reorder is triggered(i)
when the inventory level drops froms
+
1 to s if the server is in service or
(ii)
when the server returns from the vacation, he/she finds the inventory level less than or equal tos and there is no order pending.This leads to(27)
η
R
=
∑
i
=
0
N
ϕ
(
i
,
1,0
,
s
+
1
)
(
λ
+
(
s
+
1
)
γ
+
δ
¯
i
0
α
i
)
+
∑
i
=
0
N
∑
m
=
0
s
β
ϕ
(
i
,
0,0
,
m
)
,
where δ
i
j is the Kronecker delta function.


## 5.3. Expected Perishable Rate

Letη
PR denote the expected reorder Perishable in the steady state. This is given by(28)
η
PR
=
∑
i
=
0
N
(
∑
m
=
1
Q
m
γ
ϕ
(
i
,
0,0
,
m
)
+
∑
m
=
s
+
1
S
m
γ
ϕ
(
i
,
1,0
,
m
)
+
∑
m
=
1
s
m
γ
ϕ
(
i
,
1,1
,
m
)
)
.


## 5.4. Mean Shortage Rate

Letη
SR denote the mean shortage rate in the steady state. It may be noted that a shortage occurs only during the server vacation. The customers who arrive during server vacation prefer to leave the system either by their own choice or the pool has reached its maximum capacity:(29)
η
SR
=
∑
i
=
0
N
-
1
q
λ
[
∑
m
=
0
Q
ϕ
(
i
,
0,0
,
m
)
+
ϕ
(
i
,
0,1
,
0
)
]
+
λ
[
∑
m
=
0
Q
ϕ
(
N
,
0,0
,
m
)
+
ϕ
(
N
,
0,1
,
0
)
]
.


## 5.5. Expected Number of Customers in the Pool

Letη
P denote the expected number of customers in the pool in the steady state. Since ϕ
(
i
) is the steady state probability vector for i customers in the pool with each component specifying the probability for the particular combination of the status of the server, status of the reorder, and the inventory level, the expected number of customers in the pool is given by(30)
η
P
=
∑
i
=
1
N
i
ϕ
(
i
)
e
.


## 5.6. Fraction of Time When the Server is on Vacation

Letη
F denote the fraction of time when the server is on vacation. This is given by(31)
η
F
=
∑
i
=
0
N
∑
m
=
0
Q
ϕ
(
i
,
0,0
,
m
)
+
∑
i
=
0
N
ϕ
(
i
,
0,1
,
0
)
.


## 6. Cost Analysis

The expected total cost per unit time (expected total cost rate) in the steady state for this model is defined to be(32)
TC
(
s
,
S
,
N
)
=
c
h
η
I
+
c
p
η
P
+
c
pr
η
PR
+
c
s
η
R
+
c
sh
η
SR
,
wherec
s: setup cost per order,
c
h: the inventory carrying cost per unit item per unit time,
c
p: cost of a waiting customer in the pool per unit time,
c
pr: perishable cost per unit item per unit time,
c
sh: cost per customer lost per unit time.Hence, we get(33)
TC
(
s
,
S
,
N
)
=
c
h
{
∑
i
=
0
N
(
∑
m
=
1
Q
m
ϕ
(
i
,
0,0
,
m
)
+
∑
m
=
s
+
1
S
m
ϕ
(
i
,
1,0
,
m
)
+
∑
m
=
1
s
m
ϕ
(
i
,
1,1
,
m
)
)
}
+
c
p
{
∑
i
=
1
N
i
ϕ
(
i
)
e
}
+
c
s
{
∑
i
=
0
N
ϕ
(
i
,
1,0
,
s
+
1
)
(
λ
+
(
s
+
1
)
γ
+
δ
¯
i
0
α
i
)
+
∑
i
=
0
N
∑
m
=
0
s
β
ϕ
(
i
,
0,0
,
m
)
}
+
c
pr
{
∑
i
=
0
N
(
∑
m
=
0
Q
m
γ
ϕ
(
i
,
0,0
,
m
)
+
∑
m
=
s
+
1
S
m
γ
ϕ
(
i
,
1,0
,
m
)
+
∑
m
=
1
s
m
γ
ϕ
(
i
,
1,1
,
m
)
)
}
+
c
sh
{
∑
i
=
0
N
-
1
q
λ
[
∑
m
=
0
Q
ϕ
(
i
,
0,0
,
m
)
+
ϕ
(
i
,
0,1
,
0
)
]
+
λ
[
∑
m
=
0
Q
ϕ
(
N
,
0,0
,
m
)
+
ϕ
(
N
,
0,1
,
0
)
]
}
.
Due to the complex form of the limiting distribution, it is difficult to discuss the properties of the cost function TC
(
s
,
S
,
N
) analytically. Hence, a detailed computational study of the expected cost function is carried out in the next section.


## 7. Numerical Illustrations

To study the behaviour of the model developed in this work, several examples were performed and a set of representative results is shown here. Although we have not shown the convexity ofTC
(
s
,
S
,
N
) analytically, our experience with considerable numerical examples indicates the function TC
(
s
,
S
,
N
) to be convex. In some cases, it turned out to be an increasing function of s, and we use simple numerical search procedure to get the optimal values of TC
,
s
,
S and N (say T
C
*
,
s
*
,
S
*, and N
*). A typical three-dimensional plot of the expected cost function is given in Figure 1. The optimal cost value T
C
*
=
37.352776 is obtained at (
s
*
,
S
*
,
N
*
)
=
(
11,63,11
).Figure 1
A three-dimensional plot of the cost function.λ
=
14,   μ
=
4,   γ
=
1.2,   β
=
1.3,   α
i
=
i
*
α,   i
=
1,2
,
…
,
N,   α
=
4,   p
=
0.85,   c
h
=
0.3,   c
s
=
15,   c
pr
=
0.3,   c
sh
=
5
,
c
p
=
3.We have studied the effect of varying the system parameters and costs on the optimal values, and the results agreed with what one would expect. Some of our results are presented in Tables1 through 20 where the upper entries in each cell give the s
*
,
S
*, and N
*, respectively, and the lower entry gives the corresponding T
C
*.Table 1
Optimal values corresponding to various demand rates and the replenishment rates(
γ
=
1.4
,
β
=
1.3
,
α
=
4
,
p
=
0.85
).
λ
12
13
14
15
16
μ
4.0
9
56
6
10
60
7
11
63
7
12
66
8
13
69
8
37.484085
39.299182
41.092056
42.865498
44.620871
4.5
9
55
8
9
58
8
10
61
9
11
64
10
12
66
10
36.545772
38.252507
39.930717
41.591928
43.235987
5.0
8
54
9
9
56
9
10
59
10
10
62
11
11
64
11
35.729504
37.353218
38.95615
40.53873
42.08412
5.5
8
52
9
8
55
10
9
57
11
10
60
12
11
62
12
35.077762
36.606215
38.119508
39.621105
41.11143
6.0
7
51
10
8
54
11
9
56
12
9
58
12
10
61
13
34.470629
35.957242
37.433969
38.854201
40.262687Table 2
Optimal values corresponding to various demand rates and perishable rate(
β
=
1.3
,
α
=
4
,
μ
=
4
,
p
=
0.85
).
λ
10
11
12
13
14
γ
1.325
7
50
6
8
54
7
9
57
8
10
60
8
11
60
7
32.654327
34.455224
36.227262
37.973244
39.725222
1.350
7
50
6
8
53
6
9
57
7
10
60
8
11
60
7
33.0285
34.852712
36.647164
38.416969
40.190166
1.375
7
50
5
8
53
6
9
57
7
10
60
7
11
60
6
33.401547
35.24819
37.066484
38.858742
40.654683
1.400
7
50
5
8
53
6
9
56
6
10
60
7
11
60
6
33.771515
35.643133
37.484085
39.299182
41.115929
1.425
7
50
5
8
53
5
9
56
6
10
59
6
11
60
6
34.140776
36.035305
37.900166
39.738532
41.576681Table 3
Optimal values corresponding to various demand rates and the vacation time rates(
γ
=
1.4
,
α
=
4
,
μ
=
4
,
p
=
0.85
).
λ
12
13
14
15
16
β
1.30
9
57
6
10
60
7
11
63
7
12
66
8
13
69
8
37.484085
39.299182
41.092056
42.865498
44.620871
1.35
9
57
8
10
60
9
11
63
9
12
66
10
13
69
10
37.167995
38.963248
40.734627
42.487454
44.221881
1.40
9
57
9
10
60
10
11
63
11
12
66
11
13
69
12
36.861364
38.635679
40.388064
42.120116
43.834419
1.45
9
56
10
9
59
11
10
62
12
11
66
13
12
69
14
36.566275
38.316645
40.041077
41.748416
43.440092
1.50
8
56
11
9
59
12
10
62
13
11
65
14
12
68
15
36.260058
37.985767
39.692538
41.381935
43.055515Table 4
Effect of primary and postponed demand rates on the optimal values(
γ
=
1.4
,
β
=
1.3
,
μ
=
4
,
p
=
0.85
).
λ
10
11
12
13
14
α
4.0
7
50
5
8
53
6
9
56
6
10
60
7
11
63
7
33.771515
35.643133
37.484085
39.299182
41.092056
4.5
7
50
6
8
54
7
9
57
8
10
60
8
11
63
9
33.690772
35.556042
37.392308
39.202171
40.990676
5.0
7
51
7
8
54
8
9
57
8
10
60
9
11
63
10
33.618558
35.478749
37.310908
39.116144
40.900342
5.5
7
51
8
8
54
8
9
57
9
10
60
10
11
63
10
33.555137
35.41211
37.2389
39.041211
40.821941
6.0
7
51
8
8
54
9
9
57
10
10
60
10
11
63
11
33.499545
35.352615
37.177226
38.976372
40.752532Table 5
Optimal values corresponding to various perishable rates and the replenishment rates(
λ
=
14
,
β
=
1.3
,
α
=
4
,
p
=
0.85
).
μ
4.0
4.5
5.0
5.5
6.0
γ
1.30
11
63
9
10
61
11
10
59
12
9
57
13
9
56
13
39.231174
38.117777
37.192369
36.39762
35.754118
1.35
11
63
8
10
61
10
10
59
11
9
57
12
9
56
12
40.164239
39.025748
38.075018
37.259064
36.594523
1.40
11
63
7
10
61
9
10
59
10
9
57
11
9
56
12
41.092056
39.930717
38.95615
38.119508
37.433969
1.45
11
63
6
10
61
8
10
59
9
9
57
10
9
56
11
42.013602
40.832153
39.835433
38.978705
38.272741
1.50
11
62
5
10
61
7
10
59
9
9
56
10
9
56
10
42.926105
41.7295
40.711703
39.83621
39.110758Table 6
Influence of the postponed demand rate and replenishment rate on the optimal values(
λ
=
14
,
γ
=
1.4
,
β
=
1.3
,
p
=
0.85
).
μ
4.0
4.5
5.0
5.5
6.0
α
3.50
11
62
4
10
60
7
10
59
9
9
57
9
9
56
10
41.194355
40.049416
39.065602
38.230701
37.531787
3.75
11
62
6
10
61
8
10
59
9
9
57
10
9
56
11
41.145131
39.988997
39.009046
38.17254
37.480279
4.00
11
63
7
10
61
9
10
59
10
9
57
11
9
56
12
41.092056
39.930717
38.95615
38.119508
37.433969
4.25
11
63
8
10
61
10
10
59
11
9
57
11
9
56
12
41.039824
39.876714
38.908033
38.071643
37.391719
4.50
11
63
9
10
61
10
10
59
11
9
57
12
8
56
13
40.990676
39.826149
38.863594
38.027026
37.349602Table 7
Influence of the vacation time rate and the replenishment rate on the optimal values(
λ
=
14
,
γ
=
1.4
,
α
=
4
,
p
=
0.85
).
μ
4.00
4.25
4.50
4.75
5.00
β
1.300
11
63
7
11
62
8
10
61
9
10
60
10
10
59
10
41.092056
40.484338
39.930717
39.41322
38.95615
1.325
11
63
8
11
62
9
10
61
10
10
60
10
10
59
11
40.912412
40.314336
39.754127
39.247492
38.800941
1.350
11
63
9
10
62
10
10
61
11
10
60
11
9
59
12
40.734627
40.145055
39.581237
39.084991
38.635694
1.375
11
63
10
10
62
11
10
60
11
10
60
12
9
59
12
40.559624
39.964654
39.412036
38.926868
38.469604
1.400
11
63
11
10
61
11
10
60
12
10
59
12
9
58
13
40.388064
39.787865
39.246274
38.773279
38.307351Table 8
Optimal values corresponding to various perishable rates and vacation time rates(
λ
=
14
,
α
=
4
,
μ
=
4
,
p
=
0.85
).
β
1.30
1.35
1.40
1.45
1.50
γ
1.30
11
63
9
11
63
11
11
63
12
10
62
14
10
62
15
39.231174
38.886899
38.556819
38.226251
37.897017
1.35
11
63
8
11
63
10
11
63
11
10
62
13
10
62
14
40.164239
39.812693
39.474237
39.134974
38.795693
1.40
11
63
7
11
63
9
11
63
11
10
62
12
10
62
13
41.092056
40.734627
40.388064
40.041077
39.692538
1.45
11
63
6
11
63
8
11
63
10
10
62
11
10
62
12
42.013602
41.652036
41.298322
40.944106
40.587172
1.50
11
62
5
11
63
7
11
63
9
10
63
11
10
62
12
42.926105
42.564207
42.204506
41.84354
41.478112Table 9
Influence of the postponed demand rate and vacation time rate on the optimal values(
λ
=
14
,
γ
=
1.4
,
μ
=
4
,
p
=
0.85
).
β
1.30
1.35
1.40
1.45
1.50
α
4.00
11
63
7
11
63
9
11
63
11
10
62
12
10
62
13
41.092056
40.734627
40.388064
40.041077
39.692538
4.25
11
63
8
11
63
10
11
63
11
10
62
13
10
62
14
41.039824
40.674551
40.321923
39.963846
39.611386
4.50
11
63
9
11
63
10
11
63
12
10
62
13
10
62
14
40.990676
40.618727
40.260928
39.893023
39.538037
4.75
11
63
9
11
63
11
10
63
12
10
63
14
10
62
15
40.943838
40.566448
40.204905
39.828332
39.470586
5.00
11
63
10
11
63
11
10
63
13
10
63
14
10
62
15
40.900342
40.518577
40.147674
39.769092
39.409429Example 1.
In the first example, we study the impact of the demand rateλ, the lead time rate μ, the perishable rate γ, the selection rate α
i, and the vacation time parameter β on the optimal values s
*
,
S
*
,
N
*, and T
C
*. Towards this end, we first fix α
i
=
i
α,i
=
1,2
,
…
,
N, and the cost values as c
h
=
0.3
,
c
s
=
15
,
c
pr
=
0.3
,
c
sh
=
5, and c
p
=
3. We observe the following from Table 1 to 10.(1)
The optimal valuess
*
,
S
*, and T
C
* increase monotonically with increase in λ. This is to be expected since, when the arrivals occurs closer and closer, the optimal reorder level and the maximum inventory level increase to prevent more waiting time of the demands in the pool. We also note that, as λ increases, N
* increases except for the case in which γ increases.
(2)
As is to be expected, the optimal cost rateT
C
* decreases monotonically, when μ
,
α, and β increase and monotonically increases when γ increases.
(3)
Whenμ increases, s
* and S
* decrease and N
* increases. This can be explained intuitively as follows. If the replenishments occur further and further apart, the inventory level becomes zero frequently. During the stock-out period, a part of arriving demands enter into the pool. To prevent the loss of demand, we have necessity to increase the pool size. For the same reason as β increases, s
* and S
* decrease and N
* increases.
(4)
We note that, asγ increases, s
* and N
* decrease monotonically. We also note that except for the combination of increasing γ, S
* decreases when α increases.
(5)
Asα increases, N
* increases and s
* decreases. We also note that, except for the combination of increasing μ, S
* increases when α increases.Table 10
Optimal values corresponding to various perishable rates and postponed demand rates(
λ
=
14
,
β
=
1.3
,
μ
=
4
,
p
=
0.85
).
α
4.00
4.25
4.50
4.75
5.00
γ
1.0
11
62
15
11
62
16
11
62
16
11
62
17
11
62
17
33.566483
33.512828
33.46372
33.419215
33.378523
1.1
11
62
13
11
62
13
11
63
14
11
63
15
11
63
15
35.463386
35.407926
35.357122
35.310631
35.268167
1.2
11
63
11
11
63
12
11
63
12
11
63
13
11
63
13
37.352776
37.296972
37.245085
37.197543
37.153723
1.3
11
63
9
11
63
10
11
63
10
11
63
11
11
63
11
39.231174
39.176278
39.125134
39.077093
39.033214
1.4
11
63
7
11
63
8
11
63
9
11
63
9
11
63
10
41.092056
41.039824
40.990676
40.943838
40.900342Example 2.
In this example, we study the impact of the setup costc
s, holding cost c
h, perishable cost c
pr, shortage cost c
sh, and the waiting cost c
p on the optimal values s
*
,
S
*
,
N
*, and T
C
*. Towards this end, we first fix the parameter values as λ
=
14
,
γ
=
1.4
,
β
=
1.3
,
α
i
=
i
α
,
i
=
1,2
,
…
,
N
,
α
=
4
,
μ
=
4, and p
=
0.85. We observe the following from Table 11 to 20:(1)
The optimal cost increases, whenc
h
,
c
s
,
c
pr
,
c
sh, and c
p increase. The optimal cost is more sensitive to c
h than to c
s
,
c
pr
,
c
sh, and c
p.
(2)
Asc
h increases, as is to be expected, the optimal values s
*
,
S
*, and N
* decrease monotonically. This is to be expected since the holding cost increases, we resort to maintain low stock in the inventory.
(3)
Whenc
sh increases, as is to be expected, the optimal values s
*
,
S
*, and N
* increase monotonically. This is to be expected since for the high shortage cost to reduce the number of customer to be lost, we have to increase the pool size, maintain high inventory, and place order in the higher level.
(4)
If the waiting costc
p increases, the optimal values of s
* increase and N
* decreases monotonically. We also note that, except for the combination of increasing the holding cost c
h, S
* decreases when c
p increases.
(5)
If the setup costc
s increases, it is a common decision that we have to maintain more stock to avoid frequent ordering. This fact is also observed in our model.
(6)
We note that, when the perishable costc
pr increases s
*
,
S
*, and N
* decrease monotonically.Table 11
Variation in optimal values for different values ofc
h and c
p
(
c
s
=
15
,
c
sh
=
5
,
c
pr
=
0.3
).
c
h
0.20
0.25
0.30
0.35
0.40
c
p
2.2
11
69
24
10
66
22
10
63
21
10
61
21
10
59
20
37.185505
38.452295
39.665423
40.837851
41.974311
2.4
11
69
19
11
66
18
10
64
18
10
61
17
10
59
16
37.599174
38.884988
40.125458
41.301575
42.440661
2.6
11
69
16
11
66
15
11
64
14
11
61
13
10
59
13
37.972831
39.260922
40.503633
41.705424
42.851175
2.8
11
69
12
11
66
12
11
63
11
11
61
10
11
59
9
38.303582
39.590386
40.831047
42.028107
43.188119
3.0
12
69
9
11
66
8
11
63
7
11
60
6
11
58
5
38.563014
39.86128
41.092056
42.278538
43.424151Table 12
Optimal values corresponding to various holding and perishable costs(
c
s
=
15
,
c
sh
=
5
,
c
p
=
3
).
c
h
0.20
0.25
0.30
0.35
0.40
c
pr
0.250
12
74
10
12
70
9
12
67
8
11
64
8
11
61
6
36.642829
38.025913
39.353537
40.605434
41.809696
0.275
12
71
10
12
68
9
11
65
8
11
62
7
11
60
6
37.617513
38.960668
40.235119
41.452256
42.627321
0.300
12
69
9
11
66
8
11
63
7
11
60
6
11
58
5
38.563014
39.86128
41.092056
42.278538
43.424151
0.325
11
66
8
11
63
7
11
61
6
11
58
5
11
56
4
39.482691
40.72827
41.927648
43.085434
44.202381
0.350
11
65
8
11
62
7
11
59
6
11
57
5
10
54
3
40.359663
41.571843
42.741939
43.871344
44.952053Table 13
Optimal values corresponding to various setup and holding costs(
c
sh
=
5
,
c
p
=
3
,
c
pr
=
0.3
).
c
h
0.20
0.25
0.30
0.35
0.40
c
s
13
12
66
8
12
63
7
12
61
7
11
58
6
11
55
4
36.775469
38.037617
39.253444
40.42214
41.527663
14
12
68
9
12
65
8
11
62
7
11
59
6
11
57
5
37.674734
38.959441
40.189174
41.355166
42.481256
15
12
69
9
11
66
8
11
63
7
11
60
6
11
58
5
38.563014
39.86128
41.092056
42.278538
43.424151
16
11
70
9
11
67
8
11
64
7
11
61
6
10
58
5
39.433976
40.734783
41.986075
43.19263
44.356396
17
11
72
9
11
68
9
11
65
8
10
62
7
10
60
6
40.282373
41.600143
42.871236
44.094213
45.249004Table 14
Variation in optimal values for different values ofc
h and c
sh
(
c
s
=
15
,
c
p
=
3
,
c
pr
=
0.3
).
c
h
0.20
0.25
0.30
0.35
0.40
c
sh
5.00
12
69
9
11
66
8
11
63
7
11
60
6
11
58
5
38.563014
39.86128
41.092056
42.278538
43.424151
5.25
12
70
12
12
67
11
11
64
10
11
62
10
11
60
9
38.811315
40.138832
41.406634
42.618679
43.790972
5.50
12
71
14
12
68
13
12
66
13
11
63
12
11
61
11
39.026709
40.371016
41.668858
42.905412
44.097789
5.75
12
72
16
12
69
16
12
66
15
12
64
14
11
62
14
39.219457
40.577343
41.889389
43.15853
44.366545
6.00
12
73
19
12
70
18
12
67
17
12
65
16
11
63
16
39.394802
40.764441
42.08824
43.370604
44.60868Table 15
Effect ofc
s and c
p on the optimal cost rate (
c
h
=
0.3
,
c
sh
=
5
,
c
pr
=
0.3
).
c
s
14.0
14.5
15.0
15.5
16.0
c
p
2.2
10
63
21
10
63
21
10
64
21
10
64
22
10
65
22
38.786969
39.226893
39.665423
40.100898
40.535658
2.4
11
63
17
11
63
17
10
64
18
10
64
18
10
65
18
39.22218
39.674866
40.125458
40.559503
40.991876
2.6
11
63
14
11
63
14
11
64
14
11
64
14
11
65
14
39.601437
40.053153
40.503633
40.951362
41.397236
2.8
11
62
11
11
63
11
11
63
11
11
64
11
11
65
11
39.929298
40.380592
40.831047
41.277607
41.723459
3.0
11
62
7
11
62
7
11
63
7
11
63
7
11
64
7
40.189174
40.641335
41.092056
41.540189
41.986075Table 16
Effect ofc
s and c
pr on the optimal cost rate(
c
h
=
0.3
,
c
sh
=
5
,
c
p
=
3
).
c
s
13.0
13.5
14.0
14.5
15.0
c
pr
0.30
12
61
7
12
61
7
11
62
7
11
62
7
11
63
7
39.253444
39.726359
40.189174
40.641335
41.092056
0.32
12
59
6
11
59
6
11
60
6
11
61
7
11
61
6
39.914108
40.386834
40.847306
41.306302
41.762515
0.34
11
57
5
11
58
6
11
59
6
11
59
6
11
60
6
40.557466
41.026401
41.492639
41.956904
42.418051
0.36
11
56
5
11
57
5
11
57
5
11
58
5
11
59
6
41.178068
41.652787
42.125454
42.594898
43.062709
0.38
11
55
4
11
55
4
11
56
4
11
57
5
11
57
5
41.786343
42.267918
42.745509
43.220775
43.693443Table 17
Sensitivity of optimal values to variation inc
s and c
sh
(
c
h
=
0.3
,
c
p
=
3
,
c
pr
=
0.3
).
c
s
11
12
13
14
15
c
sh
5.00
12
58
6
12
59
6
12
61
7
11
62
7
11
63
7
37.330038
38.297398
39.253444
40.189174
41.092056
5.25
13
60
9
12
61
9
12
62
10
12
63
10
11
64
10
37.656022
38.617201
39.561709
40.495801
41.406634
5.50
13
61
11
12
62
12
12
63
12
12
64
12
12
66
13
37.918543
38.88736
39.824379
40.751743
41.668858
5.75
13
62
14
13
63
14
12
64
14
12
65
15
12
66
15
38.149135
39.121004
40.056997
40.978042
41.889389
6.00
13
63
16
13
64
16
12
65
16
12
66
17
12
67
17
38.357356
39.322828
40.267002
41.182026
42.08824Table 18
Variation in optimal values for different values ofc
p and c
sh
(
c
s
=
15
,
c
h
=
0.3
,
c
pr
=
0.3
).
c
sh
4.8
5.0
5.2
5.4
5.6
c
p
2.5
10
63
14
11
64
16
11
64
18
11
65
20
11
66
22
40.130057
40.320642
40.495101
40.655041
40.804415
2.6
11
63
12
11
64
14
11
64
16
11
65
18
11
66
20
40.299295
40.503633
40.689262
40.859223
41.017575
2.7
11
63
10
11
64
13
11
64
14
11
65
16
11
66
18
40.453574
40.674434
40.873015
41.053664
41.221074
2.8
11
62
8
11
63
11
11
64
13
11
65
15
11
66
17
40.588856
40.831047
41.044544
41.237658
41.415324
2.9
11
62
6
11
63
9
11
64
11
11
65
13
12
66
15
40.701943
40.97071
41.203531
41.410894
41.598786Table 19
Effect ofc
sh and c
pr on the optimal cost rate (
c
s
=
15
,
c
h
=
0.3
,
c
p
=
3
).
c
sh
5.0
5.2
5.4
5.6
5.8
c
pr
0.20
12
72
10
12
73
12
12
73
14
12
74
16
12
75
18
37.479871
37.668234
37.836697
37.989434
38.130738
0.22
12
70
9
12
71
12
12
72
14
12
72
15
12
73
17
38.242077
38.439857
38.615717
38.774912
38.92153
0.24
12
68
9
12
69
11
12
70
13
12
71
15
12
71
17
38.987028
39.193609
39.377128
39.543759
39.696724
0.26
11
66
8
12
67
10
12
68
13
12
69
14
12
70
16
39.709844
39.932901
40.12401
40.296785
40.455714
0.28
11
64
8
11
65
10
12
67
12
12
67
14
12
68
16
40.40882
40.653872
40.855615
41.03516
41.199655Table 20
Variation in Optimal values for different values ofc
p and c
pr
(
c
s
=
15
,
c
h
=
0.3
,
c
sh
=
5
).
c
p
2.6
2.7
2.8
2.9
3.0
c
pr
0.32
11
62
14
11
62
12
11
62
10
11
62
9
11
61
6
41.181731
41.350929
41.506426
41.644627
41.762515
0.34
11
61
13
11
61
12
11
61
10
11
60
8
11
60
6
41.846597
42.015719
42.169423
42.30555
42.418051
0.36
10
60
13
11
60
11
11
60
10
11
59
8
11
59
6
42.4908
42.66911
42.821582
42.953905
43.062709
0.38
10
59
13
10
58
11
11
58
9
11
58
7
11
57
5
43.120144
43.301145
43.460842
43.590387
43.693443
0.40
10
57
12
10
57
10
10
57
9
11
57
7
11
56
4
43.737882
43.918076
44.077826
44.215885
44.3119Example 3.
In this example, we look to the impact of the demand rateλ, the reorder rate μ, the selection rate α, and the mean vacation time 1
/
β on the fraction, η
F, of time the server is on vacation for the combination of various values of postponed demand rate α
(
n
)
(
n
=
1,2
,
3
), α
(
n
) is the increasing, constant, and decreasing functions, where α
(
1
)
=
5
+
i
*
0.1, α
(
2
)
=
5, and α
(
3
)
=
5
+
i
*
(
-
0.1
), i
=
1
,
…
,
N. From Figures 2 to 5, we observe the following.(i)
As is to be expected asλ increases, η
F increases. This is because, if the demands rate increases, the stockout occurs frequently, and hence the server takes vacation frequently. Hence, η
F increases. Similarly the same thing is observed in the case γ.
(ii)
Asμ increases, η
F decreases. This happens because, if μ increases, the ordered items are received at higher rate. Hence, fewer stock-out periods occur. Hence the server returns to the system more often. Hence, the intervacation time increases. Therefore η
F decreases. Similarly, the same thing is observed in the case β.
(iii)
In these cases, we observeα
(
1
)
>
α
(
2
)
>
α
(
3
).Figure 2
Fraction of time server is on vacation—λ versus α
(
n
)
(
S
=
90,   s
=
15,   N
=
25,   β
=
1.4,   γ
=
4,   μ
=
0.8,   p
=
0.75
.
)Figure 3
Fraction of time server is on vacation—β versus α
(
n
)
(
S
=
90
,
s
=
15
,
N
=
25
,
λ
=
8
,
γ
=
4
,
μ
=
0.8
,
p
=
0.75
.
)Figure 4
Fraction of time server is on vacation—μ versus α
(
n
)
(
S
=
90
,
s
=
15
,
N
=
25
,
β
=
1.4
,
γ
=
4
,
λ
=
8
,
p
=
0.75
.
)Figure 5
Fraction of time server is on vacation—γ versus α
(
n
)
(
S
=
90
,
s
=
15
,
N
=
25
,
β
=
1.4
,
λ
=
8
,
μ
=
0.8
,
p
=
0.75
.
)Example 4.
In this example, we now study the impact ofp on the performance measures, expected Inventory level η
I, expected reorder rate η
R, expected perishable rate η
PR, mean shortage rate η
SR, expected number of customers in the pool η
P, fraction of time the server is on vacation η
F and the total expected cost TC
(
s
,
S
,
N
), for the combination of various values of postponed demand rate α
(
n
)
(
n
=
1,2
,
3
), α
(
n
) is the increasing, constant, and decreasing functions, where α
(
1
)
=
5
+
i
*
0.1, α
(
2
)
=
5 and α
(
3
)
=
5
+
i
*
(
-
0.1
), i
=
1
,
…
,
N. From Figure 6 to 12, we observe the following.(i)
AsTC
(
s
,
S
,
N
), η
SR and η
P are monotonically increasing, when p increases. In these cases, we observe α
(
1
)
<
α
(
2
)
<
α
(
3
).
(ii)
Asη
R and η
F are monotonically increasing, when p increases. In these cases, we observe α
(
1
)
>
α
(
2
)
>
α
(
3
).
(iii)
Asη
I and η
PR are monotonically decreasing, when p increases. In these cases, we observe α
(
1
)
<
α
(
2
)
<
α
(
3
).Figure 6
The total expected cost rate—p versus α
(
n
)
(
S
=
90
,
s
=
15
,
N
=
25
,
λ
=
8
,
β
=
1.4
,
γ
=
0.7
,
λ
=
8
.
)Figure 7
Expected Inventory Level—p versus α
(
n
)
(
S
=
90
,
s
=
15
,
N
=
25
,
λ
=
8
,
β
=
1.4
,
γ
=
0.7
,
μ
=
0.8
.
)Figure 8
Expected reorder rate—p versus α
(
n
)
(
S
=
90
,
s
=
15
,
N
=
25
,
λ
=
8
,
β
=
1.4
,
γ
=
0.7
,
μ
=
0.8
.
)Figure 9
Expected number of customers in the pool—p versus α
(
n
)
(
S
=
90
,
s
=
15
,
N
=
25
,
λ
=
8
,
β
=
1.4
,
γ
=
0.7
,
μ
=
0.8
.
)Figure 10
Expected perishable rate—p versus α
(
n
)
(
S
=
90
,
s
=
15
,
N
=
25
,
λ
=
8
,
β
=
1.4
,
γ
=
0.7
,
μ
=
0.8
.
)Figure 11
Mean loss of demand rate—p versus α
(
n
)
(
S
=
90
,
s
=
15
,
N
=
25
,
λ
=
8
,
β
=
1.4
,
γ
=
0.7
,
μ
=
0.8
.
)Figure 12
Fraction of time server is on vacation—p versus α
(
n
)
(
S
=
90
,
s
=
15
,
N
=
25
,
λ
=
8
,
β
=
1.4
,
γ
=
0.7
,
μ
=
0.8
.
)



---

*Source: 620960-2012-04-22.xml*