Finite perishable inventory system with repeated demands

International Journal of Mathematics and Statistics Invention (IJMSI)
E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759
www.ijmsi.org Volume 1 Issue 2ǁ December. 2013ǁ PP 01-15

Perishable Inventory System with a Finite Population and
Repeated Attempts
K. Jeganathan And N. Anbazhagan
Department of Mathematics, Alagappa University, Karaikudi-630 004, Tamil Nadu, India.

ABSTRACT : In this article, we consider a two commodity continuous review perishable inventory system
with a finite number of homogeneous sources of demands. The maximum storage capacity of S i units for the
i

th

commodity (i = 1,2) . The life time of items of each commodity is assumed to be exponentially distributed

with parameter  i (i = 1,2) . The time points of primary demand occurrences form independent quasi random
distributions each with parameter  i ( i = 1,2). A joint reordering policy is adopted with a random lead time
for orders with exponential distribution. When the inventory position of both commodities are zero, any arriving
primary demand enters into an orbit. The demands in the orbit send out signal to compete for their demand
which is distributed as exponential. We assume that the two commodities are both way substitutable. The joint
probability distribution for both commodities and number of demands in the orbit is obtained for the steady
state case. Various system performance measures are derived and the results are illustrated with numerical
examples.

KEYWORDS: Retrial Demand, Positive Lead-Time, Finite Population, Perishable Inventory, Substitutable,
Markov Process, Continuous Review.

I.

INTRODUCTION

The analysis of perishable inventory systems has been the theme of many articles due to its potential
applications in sectors like food industries, drug industries, chemical industries, photographic materials,
pharmaceuticals, blood bank management and even electronic items such as memory chips. The often quoted
review articles ([21], [23]) and the recent review articles ([24], [16]) provide excellent summaries of many of
these modelling efforts. Most of these models deal with either the periodic review systems with fixed life times
or continuous review systems with instantaneous supply of reorders. One of the factors that contribute the
complexity of the present day inventory system is the multitude of items stocked and this necessitated the multicommodity systems. In dealing with such systems, in the earlier days models were proposed with independently
established reorder points. But in situations were several product compete for limited storage space or share the
same transport facility or are produced on (procured from) the same equipment (supplier) the above strategy
overlooks the potential savings associated with joint ordering and, hence, will not be optimal. Thus, the
coordinated, or what is known as joint replenishment, reduces the ordering and setup costs and allows the user
to take advantage of quantity discounts [17].
Inventory system with multiple items have been subject matter for many investigators in the past. Such
studies vary from simple extensions of EOQ analysis to sophisticated stochastic models. References may be
found in ([7], [17], [20], [22], [25], [28]) and the references therein. Multi commodity inventory system has
received more attention on the researchers on the last five decades. In continuous review inventory systems,
Ballintfy [11] and Silver [25] have considered a coordinated reordering policy which is represented by the triplet
( S , c , s ) , where the three parameters S i , c i and s i are specified for each item i with s i  c i  S i , under the
unit sized Poisson demand and constant lead time. In this policy, if the level of i th commodity at any time is
below s i , an order is placed for S i  s i items and at the same time, any other item j (  i ) with available
inventory at or below its can-order level c j , an order is placed so as to bring its level back to its maximum
capacity S j . Subsequently many articles have appeared with models involving the above policy and another
article of interest is due to Federgruen et al. [14], which deals with the general case of compound Poisson
demands and non-zero lead times. A review of inventory models under joint replenishment is provided by Goyal
and Satir [15].

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Perishable Inventory System With A Finite..
Kalpakam and Arivarignan [17] have introduced ( s , S ) policy with a single reorder level s defined
in terms of the total number of items in the stock. This policy avoids separate ordering for each commodity and
hence a single processing of orders for both commodities has some advantages in situation where in
procurement is made from the same supplies, items are produced on the same machine, or items have to be
supplied by the same transport facility. Krishnamoorthy and Varghese [18] have considered a two commodity
inventory problem without lead time and with Markov shift in demand for the type of commodity namely
’‘commodity-1’’, ‘‘commodity-2’’ or ‘‘both commodity’’, using the direct Markov renewal theoretical results.
Anbazhagan and Arivarignan ([3], [4], [5], [6]) have analyzed two commodity inventory system under various
ordering policies. Yadavalli et al. [29] have analyzed a model with joint ordering policy and varying order
quantities. Yadavalli et al. [30] have considered a two-commodity substitutable inventory system with Poisson
demands and arbitrarily distributed lead time. All the models considered in the two-commodity inventory
system assumed lost sales of demands during stock out periods.
Traditionally the inventory models incorporate the stream of customers (either at fixed time intervals
or random intervals of time) whose demands are satisfied by the items from the stock and those demands which
cannot be satisfied are either backlogged or lost. But in recent times due to the changes in business
environments in terms of technology such as Internet, the customer may retry for his requirements at random
time points. The concept of retrial demands in inventory was introduced in [9] and only few papers ([2], [26],
[27], [31] ) have appeared in this area. Moreover product such as bath soaps, body spray, etc., may have
different flavours and the customer would be willing to settle for one only when the other is not available. These
aspects provided the motivation for writing this paper. We will focus on the case in which the population under
study is finite so each individual customer generates his own flow of primary demands. For the analysis of finite
source retrial queue in continuous time, the interested reader is referred to Falin and Templeton [12], Artalejo
and Lopez-Herrero [10], Falin and Artalejo [13], Almasi et al. [1] and Artalejo [8] and references therein.
The rest of the paper is organized as follows. In the next section, we describe the mathematical model. 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 in the steady state. Several
numerical results that illustrate the influence of the system parameters on the system performance are discussed
in section 6. The last section is meant for conclusion.

II.

MATHEMATICAL MODEL

We consider a continuous review perishable inventory system with a maximum stock of S i units for
the i th commodity (i = 1,2) and the demands originated from a finite population of sources N . Each source
th

is either free or in the orbit at any time. The primary demand for i commodity is of unit size and the time
points of primary demand occurrences form independent Quasi-random distributions each with parameter  i
(i = 1,2) . The items are perishable in nature and the life time of items of each commodity is assumed to be

exponentially distributed with parameter  i (i = 1,2) . The reorder level for the i

th

commodity is fixed as s i

(1  s i  S i ) and an order is placed for both commodities when both the inventory levels are less than or equal

to

their

respective

th

reorder

levels. The ordering quantity for the i
commodity is
Q i (= S i  s i > s i  1, i = 1,2) items. The requirement S i  s i > s i  1 , ensures that after a replenishment
the inventory level will always be above the respective reorder levels; otherwise it may not be possible to place
reorder (according to this policy) which leads to perpetual shortage. The lead time is assumed to be
exponentially distributed with parameter  > 0 . Both the commodities are assumed to be both way
substitutable in the sense that at the time of zero stock of any one commodity, the other one is used to meet the
demand. If the inventory position of both the commodities are zero thereafter any arriving primary demand
enters into the orbit. These orbiting demands send out signal to compete for their demand which is distributed as
exponential with parameter  (> 0) . In this article, the classical retrial policy is followed, that is, the probability
of a repeated attempt is depend on the number of demands in the orbit. The retrial demand may accept an item
of commodity- i with probability p i (i = 1,2) , where p 1  p 2 = 1 . We also assume that the inter demand
times between the primary demands, lead times, life time of each items and retrial demand times are mutually
independent random variables.

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2.1 Notations:
e : a column vec

tor of appropriat

e dimension

containing

all ones

0 : Zero matrix

 A ij

: entry

at ( i , j )

th

position

of a matrix

A

k  V i : k = i , i  1,  j
j

 ij : 1   ij
1

if j = i

0

otherwise

 ij : 

1,
H (x) : 
 0,
E 1 : {0,1,2,

if

x  0,

otherwise

.

 , S 1}

E 2 : {0,1,2,  , S 2 }

E 3 : {0,1,2,  , N }
E : E1  E 2  E 3

III.

Analysis

Let L 1 ( t ) , L 2 ( t ) and X ( t ) denote the inventory position of commodity-I, the inventory position of
commodity-II and the number of demands in the orbit at time t , respectively. From the assumptions made on
the input and output processes it can be shown that the triplet {( L 1 ( t ), L 2 ( t ), X ( t )), t  0} is a continuous
time Markov chain with state space given by

E.

To determine the infinitesimal generator

 = a (( i1 , i 2 , i 3 ), ( j1 , j 2 , j 3 )) , ( i1 , i 2 , i 3 ), ( j1 , j 2 , j 3 )  E , of this process.

Theorem 1:
The infinitesimal generator of this Markov process is given by,

a ( ( i1 , i 2 , i 3 ), ( j1 , j 2 , j 3 , ) ) =

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 ( N  i 3 )(  1   2 ),




 i 3 ,








 i 3 p 1 ,




 i 3 p 2 ,









 ( N  i 3 )(  1   2 )  i 1  1 ,



 ( N  i )(    )  i  ,
3
1
2
2
2




  (( N  i 3 )(  1   2 )  i 2 

  i 0 i 3   H ( s 2  i 2 )  ),



2

j1 = 0 ,

j 3 = i 3  1,

j2 = 0,

i 1 = 0,

i3  V 0

i 2 = 0,

j 2 = i 2  1,

j 1 = i1 ,

i2  V1

i 1 = 0,

S

2

N 1

,

j 3 = i 3  1,
i3  V 1

,

N

,

or
j 1 = i 1  1,
i1  V 1

S

1

S

1

i2  V1

S

1

,

i2  V1

s

1

S

1

i1  V 0

2



1

2

s

2

S

2

i2  V1

,

S

2

2

,

i2  V 0

i 1 = 0,

j 1 = i1 ,
i1  V 1

S

1

2

i3  V 0

,

i3  V 0

,

N

,

,

N

,

j3 = i3 ,

i3  V 0

,

N

,

j3 = i3 ,
N

,

j3 = i3 ,
i3  V 0

,

j2 = i2 ,
,

N

j3 = i3 ,

,

j2 = i2 ,
S

,

j 3 = i 3  1,
i3  V 1

,

j 2 = i 2  1,

j 1 = i1 ,

  (( N  i 3 )(  1   2 )  i1  1  i 2  2 

i 3    H ( s 1  i1 ) H ( s 2  i 2 )),




0




S

j2 = i2 ,
i2  V 0

,

j 1 = i1 ,
S

i3  V 1

,

j2 = i2  Q
i2  V 0

,

j 1 = i 1  1,
i1  V 1

2

j 2 = i 2  1,

j 1 = i1  Q 1 ,
i1  V 0

S

N

j 3 = i 3  1,

j2 = i2 ,

,

j 1 = i1 ,
i1  V 1

i3  V 1

i 2 = 0,

j 1 = i 1  1,
i1  V 1

j 3 = i 3  1,

j2 = i2 ,

,

i2  V 0

S

2

N

,

j3 = i3 ,
,

i3  V 0 ,
N

otherwise

Proof:
The infinitesimal generator a (( i1 , i 2 , i 3 ), ( j1 , j 2 , j 3 )) of this process can be obtained using the
following arguments:
1: Let i1 > 0, i 2 > 0, i 3  0 .
A primary demand from any one of the ( N  i 3 ) sources or due to perishability takes the inventory level
( i1 , i 2 , i 3 ) to ( i1  1, i 2 , i 3 ) with intensity ( N  i 3 )  1  i1 1 for I-commodity or ( i1 , i 2 , i 3 ) to ( i1 , i 2  1, i 3 )

with intensity ( N  i 3 )  2  i 2  2 for II-commodity.
The level (0, i 2 , i 3 ) , and ( i1 ,0, i 3 ) , respectively, is taken to (0, i 2  1, i 3 ) , with intensity
( N  i 3 )(  1   2 )  i 2  2 , and ( i1  1,0, i 3 ) with intensity ( N  i 3 )(  1   2 )  i1 1 .

2: If the inventory position of both the commodities are zero then any arriving primary demand enters
into the orbit. Hence a transition takes place from (0,0, i 3 ) to (0,0, i 3  1) with intensity ( N  i 3 )(  1   2 ) ,

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0  i3  N  1 .

3: Let i1 > 0, i 2 > 0, i 3  1 .
A demand from orbit takes the inventory level ( i1 , i 2 , i 3 ) to ( i1  1, i 2 , i 3  1) with intensity i 3 p 1 for Icommodity or ( i1 , i 2 , i 3 ) to ( i1 , i 2  1, i 3  1) with intensity i 3 p 2 for II-commodity.
The level (0, i 2 , i 3 ) , and ( i1 ,0, i 3 ) , respectively, is taken to (0, i 2  1, i 3  1) and ( i1  1,0, i 3  1) with
intensity i 3 .
4: From a state ( i1 , i 2 , i 3 ) with ( i1 , i 2 )  ( s 1 , s 2 ) , i 3  0 a replenishment by the delivery of orders
for both commodities takes the inventory level to ( i1  Q 1 , i 2  Q 2 , i 3 ) , Q 1 = S 1  s 1 , Q 2 = S 2  s 2 , with
intensity of this transition  .
We observe that no transition other than the above is possible.
Finally the value of a ( ( i1 , i 2 , i 3 ), ( j1 , j 2 , j 3 ) ) is obtained by
a (( i1 , i 2 , i 3 ), ( i1 , i 2 , i 3 )) = 


j
j
j
1
2
3
( i ,i ,i )  ( j , j , j )
1 2 3
1 2 3

a (( i1 , i 2 , i 3 ), ( j1 , j 2 , j 3 ))

□

Hence we get the infinitesimal generator a ( ( i1 , i 2 , i 3 ), ( j1 , j 2 , j 3 ) ).

In order to write down the infinitesimal generator  in a matrix form, we arrange the states in
lexicographic order and group ( S 2  1)( N  1) states as:
< q >= (( q ,0,0), ( q ,0,1),  , ( q ,0, N ), ( q ,1,0), ( q ,1,1),  , ( q ,1, N ),  , ( q , S 2 ,0),
( q , S 2 ,1),  , ( q , S 2 , N ))

for

q = 0,1,  , S 1 .

Then the rate matrix  has the block partitioned form with the following sub matrix [  ] i

j
1 1

at the i1 -

the row and j1 -th column position.

[  ]i

j
1 1

A ,
i
 1
B ,
=  i1
C ,

0,

j 1 = i1 ,

i1  V 0

S

j1 = i1  1,

i1  V 1

S

j 1 = i1  Q 1 ,

i1  V 0 1

otherwise

1

1

s

.

where
[C ]i

j
2 2

[ H ]i

j
3 3


W
= 
 0,


j2 = i2  Q 2 ,
otherwise

i2  V 0

s

2

.

 ( N  i 3 )(  1   2 ),

=   (( N  i 3 )(  1   2 )   ),
 0,


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N 1

j 3 = i 3  1,

i3  V 0

j 3 = i3 ,

i3  V 0 ,

otherwise

,

N

.

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[ A0 ]i

H

 Fi
=  2
 H i2
 0,


j

2 2

For

[ B i ]i

i 2 = 0,

j 2 = i 2  1,

i2  V1

S

i2  V1

S

j2 = i2 ,
otherwise

2

,

2

,

.

i1 = 1,  , S 1

V i
1

= M i
1
 0,


j 2 = 0,

1

j
2 2

[ Ai ]i
1

G
i
 2
Ji
=  1
 Li i
1 2

0,


j
2 2

j2 = i2 ,

i2  V1

S

j 2 = i 2  1,

i2  V1

S

j2 = i2 ,

i 2 = 0,

j2 = i2 ,

i2  V1

otherwise

2

,

2

,

2

,

.

otherwise

S

.

i1 = 1,2,  , S 1

For

[V i ] i
1

j
3 3

 i 3 ,

=  ( N  i 3 )(  1   2 )  i1  1 ,
 0,


j 3 = i 3  1,

i3  V 1 ,

j 3 = i3 ,

i3  V 0 ,

N

N

otherwise

.

i1 = 1,2,  , S 1

For

i
1

]i

[ Fi ]i
2

For

 0,


j 3 = i 3  1,

i3  V 1 ,

j3 = i3 ,

i3  V 0 ,

otherwise

N

N

.

j
3 3

 i 3 ,

=  ( N  i 3 )(  1   2 )  i 2  2 ,
 0,


j 3 = i 3  1,

i3  V 1 ,

j 3 = i3 ,

i3  V 0 ,

otherwise

N

N

.

i 2 = 1,2,  , S 2

For

i

j
3 3

 p 1 ,

=  ( N  i 3 )  1  i1  1 ,

i 2 = 1,2,  , S 2

For

[H

i 2 = 0,

i1 = 1,2,  , S 1

For

[M

j2 = i2 ,

2

]i

j
3 3

  (( N  i 3 )(  1   2 )  i 2 

=  i 3  H ( s 2  i 2 )  ),

2

 0,



i3  V 0 ,
N

j 3 = i3 ,
otherwise

.

i 2 = 1,2,  , S 2

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[G i ]i
2

[ J i ]i

For

[ Li

i
1 2

j
3 3

j 3 = i 3  1,

i3  V 1 ,

j 3 = i3 ,

i3  V 0 ,

N

N

otherwise

.

i1 = 1,2,  , S 1

For

1

 i 3 p 2 ,

=  ( N  i3 )  2  i 2  2 ,
 0,


  (( N  i 3 )(  1   2 )  i1  1  i 3 

=   H ( s 1  i1 ))
 0,


j
3 3

i3  V 0 ,
N

j 3 = i3 ,
otherwise

.

i1 = 1,2,  , S 1 ; i 2 = 1,2,  , S 2 ,

]i

  (( N  i 3 )(  1   2 )  i1  1  i 2 

=  i 3   H ( s 1  i1 ) H ( s 2  i 2 )),

j
3 3

2



 0,


i3  V 0 ,
N

j 3 = i3 ,
otherwise

.

W =  I N 1

It may be noted that the matrices A i , B i , i1 = 1,2,  , S 1 , A 0 and C are square matrices of order
1

( S 2  1)( N  1) . The sub matrices V i , M
1

H

i

2

1

i
1

, J i , L i i , i1 = 1,2,  , S 1 , i 2 = 1,2,  , S 2 , W , H , F i ,
1

1 2

2

, G i , i 2 = 1,2,  , S 2 , are square matrices of order ( N  1) .
2

It

can

be

seen

from

the

structure

of



that

the homogeneous

Markov process

{( L 1 ( t ), L 2 ( t ), X ( t )) : t  0} on the finite space E is irreducible, aperiodic and persistent non-null.

Hence the limiting distribution


( i ,i ,i )
1 2 3

= lim Pr [ L 1 ( t ) = i1 , L 2 ( t ) = i 2 , X ( t ) = i 3 | L 1 (0), L 2 (0), X (0)],

exists.

t 

Let Π = ( 

(0)

,

(1)

, , 

partitioning the vector, 


(i )
1

= (

( i ,0)
1

(S )
1

(i )
1

,

),

into as follows:
( i ,1)
1

,

( i ,2)
1

, , 

(i ,S )
1 2

), i1 = 0,1,2,  , S 1

which is partitioned as follows:


( i ,i )
1 2

= (

( i , i ,0)
1 2

, , 

( i ,i , N )
1 2

), i1 = 0,1,2,  , S 1 ; i 2 = 0,1,2,  , S 2 .

Then the vector of limiting probabilities Π satisfies
Π  = 0 and

Π e = 1.

(1)

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Theorem 2:
The limiting distribution Π is given by,


(i )
1

(Q )
1

= 

i ,

i1 = 0,1,  , S 1 ,

1

where
Q i



i
1

1
 (  1) 1 1 B A  1 B
 B i  1 A i , i1 = 0,1,  , Q 1  1,
Q
Q 1
Q 1
1
1
1
1
1


I,
i1 = Q 1 ,


= 
S i
1 1
2 Q  i 1
1
1
1

(  1) 1 1  B Q A Q  1 B Q  1  B s  1  j A s  j CA S  j
1
1
1
1
1
1 1
1 1

j =0
1

1
1
 B S  j A S  j 1 B S  j 1  B i  1 A i ,

1 1
1 1
1
1
1
1

i1 = Q 1  1,  , S 1 ,






(2)





(Q )
1

The value of 



(Q )
1

can be obtained from the relation


 (  1)



Q

s 1
1

1

 B

1

Q

1

AQ

1

1

BQ

1

1

 Bs

1

1

1 j
1

As

1

 j
1

CA

j =0
1



 BS

1

 j
1

1

AS

1

 j 1
1

BS

1

 j 1
1

 BQ

1

1

AQ

2

1

1

B

(3)
 (  1)

Q

1

1

B Q AQ
1

1

1

BQ

1

1

1

1
S  j
1
1

Q 1
1

 AQ

1



 B 1 A 0 C = 0,

and


(Q )
1


   (  1)

i = Q 1
1
1 
S

1

 Q1  1
  (  1)
 i1 = 0




2 Q  i 1
1 1

S i
1 1



Q i
1 1

1

B Q AQ
1

B

j =0
1

1

Q

1

AQ



 BS

1

1

1

1

BQ

1

BQ

1

1

 j
1

AS

1

 Bi

 j 1
1

BS

1

 I

1

1

 Bs

1

1

1

1

Ai

1

1

1

1 j
1

 j 1
1

As

1

 j
1

 Bi

1

CA
1

1

Ai

1

1

(4)

S  j
1
1

e = 1.

Proof:
The first equation of (1) yields the following set of equations :


( i  1)
1

Bi

1



Bi

1



Bi

1



1



( i  1)
1

1



( i  1)
1

1



(i )
1

A i = 0, i1 = 0,1,  , Q 1  1,

(5)

1

(i )
1

Ai  

(i Q )
1
1

1

Ai  
1

(i )
1

(i )
1

Ai  
1

(i Q )
1
1

(i Q )
1
1

C = 0, i1 = Q 1 ,

(6)

C = 0, i1 = Q 1  1,  , S 1  1,

(7)
(8)

C = 0, i1 = S 1 .

Solving the above set of equations we get the required solution.

www.ijmsi.org

□

8|Pa ge


IV.

SYSTEM PERFORMANCE MEASURES

In this section some performance measures of the system under consideration in the steady state are
derived.
4.1 Expected inventory level
Let  i and  i denote the average inventory level for the first commodity and the second commodity
1

2

respectively in the steady state. Then
S

S

1

N

2

  i

i =

( i ,i ,i )
1 2 3

(9)

1

1

i =1 i = 0 i = 0
1
2
3

and
S

S

1

N

2

  i 

i =

( i ,i ,i )
1 2 3

(10)

2

2

i = 0 i =1i = 0
1
2
3

4.2 Expected reorder rate
Let  r denote the mean reorder rate in the steady state. Then
s

2

 (N

r =

1

 i

2

0

N  2  ( s 1  1)  1 ) 

( s  1, i ,0)
1
2

i =0
2
s



1

 (N

  i 0 N  1  ( s 2  1)  2 ) 

2

( i , s  1,0)
1 2

1

i =0
1

s



N

2

  (( N

 i 3 )  1  ( s 1  1)  1   i

2

0

(( N  i 3 )  2  i 3 p 2 )  i 3 p 1 ) 

( s  1, i , i )
1
2 3

i = 0 i =1
2
3

(11)
s



N

1

  (( N

 i 3 )  2  ( s 2  1) 

2

  i 0 (( N  i 3 )  1  i 3 p 1 )  i 3 p 2 ) 

( i , s  1, i )
1 2
3

1

i = 0 i =1
1
3

4.3 Expected perishable rate
Let  p and  p denote the expected perishable rates for the first commodity and the second
1

2

commodity respectively in the steady state. Then
S



p

S

1

N

2

  i

=

1

1

1



( i ,i ,i )
1 2 3

(12)

i =1 i = 0 i = 0
1
2
3

and
S



p

=
2

S

1

N

2

  i

2

 2

( i ,i ,i )
1 2 3

(13)

i = 0 i =1i = 0
1
2
3

4.4 Expected number of demands in the orbit
Let  o denote the expected number of demands in the orbit. Then
S

o =

1

S

2

N

  i 

( i ,i ,i )
1 2 3

3

(14)

i = 0 i = 0 i =1
1
2
3

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9|Pa ge

4.5 Expected an arriving demand enters into the orbit
The expected an arriving primary demand enters into the orbit is given by
N 1

a =

 (N

 i 3 )(  1   2 ) 

(0,0, i )
3

(15)

i =0
3

4.6 The overall rate of retrials
The overall rate of retrials for the orbit customers in the steady state. Then
S

 or =

1

S

2

N

   i 

( i ,i ,i )
1 2 3

(16)

3

i = 0 i = 0 i =1
1
2
3

4.7 The successful rate of retrials
The successful rate of retrials for the orbit customers in the steady state. Then
S

 sr =

2



S

N

 i 3

(0, i , i )
2 3



1



i =1i =1
2
3

S

N

 i 3

( i ,0, i )
1
3



i =1 i =1
1
3

S

1

N

2

   i 

( i ,i ,i )
1 2 3

(17)

3

i =1 i =1i =1
1
2
3

4.8 Fraction of successful rate of retrials
Let  fr denote the fraction of successful rate of retrials is given by


fr

=

 sr

(18)

 or

V.

COST ANALYSIS

To compute the total expected cost per unit time (total expected cost rate),
the following costs, are considered.
c h 1 : The inventory holding cost per unit item per unit time for I-commodity.
c h 2 : The inventory holding cost per unit item per unit time for II-commodity.
c s : The setup cost per order.
c p 1 : Perishable cost of the I - commodity per unit item per unit time.
c p 2 : Perishable cost of the II- commodity per unit item per unit time.

c w : Waiting cost of an orbiting demand per unit time.

The long run total expected cost rate is given by
TC ( S 1 , S 2 , s 1 , s 2 , N ) = c h  i  c h  i  c s r  c p 
1

1

2

2

1

p

1

 cp 
2

p

2

 c w o .

(19)
Substituting the values of  ’s we get TC ( S 1 , S 2 , s 1 , s 2 , N )
 S1 S 2 N
= c h     i1
1
 i1 = 1 i 2 = 0 i3 = 0

 S1 S 2 N
 c w     i3
 i1 = 0 i 2 = 0 i3 = 1


( i ,i ,i )
1 2 3

( i ,i ,i )
1 2 3

 S1 S 2 N
 c p 2     i2 2

 i1 = 0 i 2 = 1 i3 = 0


  ch
2



 S1 S 2 N
    i2
 i1 = 0 i 2 = 1 i3 = 0


( i ,i ,i )
1 2 3


 S1 S 2 N
  c p 1     i1  1

 i1 = 1 i 2 = 0 i3 = 0



( i ,i ,i )
1 2 3











( i ,i ,i )
1 2 3






 s2
c s    ( N  1   i 0 N  2  ( s 1  1)  1 ) 
2
  i2 = 0


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( s  1, i ,0)
1
2






10 | P a g e

 s1
   ( N  2   i 0 N  1  ( s 2  1)  2 
1
 i =0
1

( i , s  1,0)
1 2






(20)

 s2 N
    (( N  i 3 )  1  i 3 p 1  ( s 1  1)  1   i 0 (( N  i 3 )  2  i 3 p 2 ))
2
 i = 0 i =1
 2 3
 s1 N
    (( N  i 3 )  2  i 3 p 2  ( s 2  1) 
 i = 0 i =1
1 3

2






  i 0 (( N  i 3 )  1  i 3 p 1 )) 

( i , s  1, i )
1 2
3

1






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 in the next
section.

VI.

NUMERICAL ILLUSTRATIONS

In this section we discuss some interesting numerical examples that qualitatively describe the
performance of this inventory model under study. Our experience with considerable numerical examples
indicates that the function TC ( S 1 , S 2 ), to be convex. Appropriate numerical search procedures are used to
*

*

*

obtain the optimal values of TC , S 1 and S 2 (say TC , S 1 and S 2 ). The effect of varying the system
parameters and costs on the optimal values have been studied and the results agreed with what one would
expect. A typical three dimensional plot of the total expected cost function is given in Figure 1 .In Table 1 gives
*

*

the total expected cost rate as a function of S 1 and S 2 by fixing the parameters and the cost values:
s 1 = 2, s 2 = 3, N = 10,  1 = 0.01,

 2 = 0.02,

 = 0.01,

c p 1 = 0.4,

c p 2 = 0.5,

c w = 6,

c h 1 = 0.01,

c h 2 = 0.04,

 1 = 0.2,



2

= 0.1,

 = 0.02,

c s = 12,

p 1 = 0.4 and p 2 = 0.6 .
*

*

From the Table 1 the total expected cost rate is more sensitive to the changes in S 2 than that of in S 1 .
Some of the results are presented in Tables 2 through 6 where the lower entry in each cell gives the total
*

*

expected cost rate and the upper entries the corresponding S 1 and S 2 .

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11 | P a g e

s 1 = 2, s 2 = 3, N = 10,  1 = 0.01,

 2 = 0.02,  = 0.01,  1 = 0.2, 

2

= 0.1,  = 0.02,

c s = 12,
c h 1 = 0.01,

c h 2 = 0.04,

c p 1 = 0.4, c p 2 = 0.5, c w = 6, p 1 = 0.4,

p 2 = 0.6 .

Figure 1: A three dimensional plot of the cost function TC ( S 1 , S 2 )
Table 1: Total expected cost rate as a function of S 1 and S 2
S2

29

30

31

32

33

S1

88
89
90
91
92

52.866093
52.866088
52.866167
52.866586
52.867052

52.863418
52.863289
52.863315
52.863494
52.8638820

52.862517
52.862247
52.862134
52.862227
52.862361

52.863258
52.862850
52.862599
52.862502
52.862555

52.865527
52.864984
52.864598
52.864368
52.864288

6.1 Example 1
In the first example, we look at the impact of  1 ,  2 ,  1 , and 

*

2

*

on the optimal values ( S 1 , S 2 )

*

and the corresponding total expected cost rate TC . For this, first by fixing the parameters and cost values as
s 1 = 2, s 2 = 3, N = 10,  = 0.02 ,  = 0.01 , p 1 = 0.4 , p 2 = 0.6 , c h 1 = 0.01 , c h 2 = 0.04 , c s = 12 ,
c w = 6 , c p 1 = 0.4 and c p 2 = 0.5 . Observe the following from Tables 2 and 3 :

1. From the Table 2 , it is observed that the TC , S 1 and S 2 increase when  1 and  2 increase. The result
*

*

*

is obvious as  1 and  2 increase it has impact on higher re-ordering and the cost of carrying to orbit
customers. Hence arrival rates are vital to this system. Also the TC

*

is more sensitive to changes in  1 than

that of in  2 .
2. From the Table 3 , it is observed that if  1 and 

*

2

*

increase then S 1 and S 2 decrease, and the TC

increases, in a significant amount. This results is obvious as  1 and 

2

*

increase, more items will be perished

that finally incurred a substantial amount of costs to the system. From the observation it seems that the TC
very sensitive to changes in  2 than that of in  1 .

*

is

6.2 Example 2
In this example, we study the impact of c s , c h 1 , c h 2 , c p 1 , c p 2 and c w on the optimal values
*

*

(S1 , S 2 )

*

and the corresponding TC . Towards this end, first by fixing the parameter values as

s 1 = 2, s 2 = 3, N = 10,  1 = 0.01 ,  2 = 0.02 ,  = 0.02 ,  = 0.01 ,  1 = 0.2 , 

2

= 0.1 , p 1 = 0.4

and p 2 = 0.6 .
Observe the following from Tables 4  6 :
1. The total expected cost rate increases when c h 1 , c h 2 , c s , c w , c p 1 and c p 2 increase monotonically.
*

*

2. As c h 1 and c h 2 increase, the optimal values S 1 and S 2 decrease monotonically. This is to be expected
since c h 1 and c h 2 increase, we resort to maintain low stock in the inventory.
*

*

3. Similarly, when c w increases, the values of S 1 and S 2 increase monotonically. This is because if c w
increases then we have to maintain high inventory to reduce the number of waiting customers in the orbit.

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12 | P a g e

*

*

4. As S 1 and S 2 increase monotonically, c s increases. This is a common decision that we have to maintain
more stock to avoid frequent ordering.
*

*

5. If c p 1 and c p 2 increase monotonically then S 1 and S 2 decrease and TC

*

increases. We also note that the

total expected cost rate is more sensitive to changes in c p 1 than that of in c p 2 .
Table 2: Sensitivity of  1 and  2 on the optimal values
2

0.010

0.015

0.020

0.025

104
30
52.404997
104
30
52.677218
105
30
52.916048
106
30
53.128612
107
30
53.319961

104
30
52.616266
104
31
52.862134
105
31
53.080348
106
31
53.276383
107
32
53.454181

105
52.805787
105
53.030387
105
53.231281
106
53.413216
107
53.579277

0.030

1
0.005
0.010
0.015
0.020
0.025

104
52.166935
104
52.471973
105
52.735666
106
52.967782
107
53.174930

28
28
28
28
28

33
33
33
34
34

Table 3: Variation in optimal values for different values of  1 and 
0.08

0.10

105
31
50.734492
99
31
51.777332
67
27
52.403399
49
26
52.802492
37
25

105
31
51.983923
95
31
53.098322
65
27
53.768192
47
24
54.198838
35
22

104
52.862134
90
54.023159
62
54.715274
44
55.153269
32

53.067357



0.06

54.498210

55.446462

105
52.977922
105
53.184492
106
53.370828
106
53.540578
107
53.696383

34
34
34
34
35

2

0.12

0.14

102
30
53.512248
87
30
54.718720
60
26
55.438513
42
24
55.880546
31
21

100
26
53.989839
86
26
55.247406
60
23
56.014391
41
23
56.468607
29
20

56.169518

56.753754

2

1

0.20
0.25
0.30
0.35
0.40

31
31
27
24
21

Table 4: Effect of varying c h 1 and c h 2 on the optimal values
c h1

0.005

0.010

0.015

0.020

93
36
52.655217
91
33
52.761980
90
31
52.862134
89
29
52.956645
89
28
53.045806

87
35
52.730014
86
33
52.836322
85
31
52.936195
84
29
53.030349
84
28
53.119568

83
35
52.800966
82
33
52.907085
81
31
53.006699
80
29
53.100529
79
27
53.189435

0.025

ch2

0.02
0.03
0.04
0.05
0.06

98
36
52.576308
98
34
52.683493
97
32
52.784094
95
30
52.878746
94
28
52.968191

www.ijmsi.org

79
35
52.868637
77
32
52.974514
76
30
53.073884
76
29
53.1673640
75
27
53.256085

13 | P a g e

Table 5: Influence of c w and c s on the optimal values
2

4

6

23
13
18.666668
23
17
18.732696
22
19
18.792123
20
20
18.845248
19
21
18.892830

56
23
35.866303
56
24
35.926358
56
25
35.984528
55
26
36.040683
55
27
36.094711

91
30
52.753058
90
30
52.808218
90
31
52.862134
90
32
52.915075
90
33
52.966959

cw

8

10

cs

8
10
12
14
16

127
36
69.441440
127
36
69.493279
127
37
69.544235
127
38
69.594647
126
38
69.644419

164
41
85.985889
164
42
86.035232
164
42
86.084255
164
43
86.132520
163
44
86.180477

Table 6: Variation in optimal values for different values of c p 1 and c p 2
c p2

0.2

0.5

0.8

1.1

1.4

c p1

2
4
6
8
1.0

175
43
51.737733
94
39
52.545497
63
36
53.058156
47
35
53.431616
37
34
53.723445

167 35
52.071328
90
31
52.862134
61
29
53.368609
45
28
53.739166
35
27
54.029803

VII.

162 29
52.346578
87
26
53.126890
59
25
53.628758
45
27
53.829995
35
23
54.287583

159 24
52.577109
86
23
53.353545
58
22
53.853509
44
21
54.221504
34
20
54.511543

159 20
52.769276
85
20
53.549672
57
19
54.049595
43
18
54.418675
34
18
54.700904

CONCLUSIONS

In this paper we consider a finite source two commodity perishable inventory system with
substitutable and retrial demands. This model is most suitable to two different items which are substitutable. The
joint probability distribution for both commodities and number of demands in the orbit is obtained in the steady
state case. Finally, we give numerical examples to illustrate the effect of the parameters on several performance
characteristics.

ACKNOWLEDGMENT
N. Anabzhagan’s research was supported by the National Board for Higher Mathematics (DAE), Government
of India through research project 2/48(11)/2011/R&D II/1141. K. Jeganathan’s research was supported by
University Grants Commission of India under Rajiv Gandhi National Fellowship F.16-1574/2010(SA-III).

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