International Journal of Mathematics and Statistics Invention (IJMSI)

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

www.ijmsi.org 1 | P a g e

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 i

S units for the

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

with parameter i

1,2)=( i . The time points of primary demand occurrences form independent quasi random

distributions each with parameter i

1,2).=( i 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 multi-

commodity 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

),,( scS , where the three parameters i

S , i

c and i

s are specified for each item i with iii

Scs , 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 i

s , an order is placed for ii

sS items and at the same time, any other item j )( i with available

inventory at or below its can-order level j

c , an order is placed so as to bring its level back to its maximum

capacity j

S . 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].

Perishable Inventory System With A Finite..


Kalpakam and Arivarignan [17] have introduced ),( Ss 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 i

S units for

the i th commodity 1,2)=( i and the demands originated from a finite population of sources N . Each source

is either free or in the orbit at any time. The primary demand for th

i commodity is of unit size and the time

points of primary demand occurrences form independent Quasi-random distributions each with parameter i

1,2)=( i . The items are perishable in nature and the life time of items of each commodity is assumed to be

exponentially distributed with parameter i

1,2)=( i . The reorder level for the th

i commodity is fixed as i

s

)(1ii

Ss and an order is placed for both commodities when both the inventory levels are less than or equal

to their respective reorder levels. The ordering quantity for the th

i commodity is

1,2)= 1,>(= issSQiiii items. The requirement 1>

iiissS , 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 i

p 1,2)=( i , where 1=21

pp . 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.



2.1 Notations:

ones all containingdimension eappropriat oftor column vec a:e

matrix Zero:0

Amatrix a ofposition ),( atentry :th

ijjiA

jiikVkj

i1,,=:

ijij

1:

otherwise0

= if1:

ij

ij

.0,

0,1,:)(

otherwise

xifxH

},{0,1,2,:11

SE

},{0,1,2,:22

SE

},{0,1,2,:3

NE

321

: EEEE

III. Analysis

Let )(1

tL , )(2

tL and )( tX 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 0})),(),(),({(21

ttXtLtL is a continuous

time Markov chain with state space given by .E To determine the infinitesimal generator

)),,(),,,((=321321

jjjiiia , Ejjjiii ),,(),,,(321321

, of this process.

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

=) ),,,(),,,( (321321

jjjiiia



,

,,0,=),)(

,=,=,=))(((

,,,

,=1,=,=,))((

,,,

,=,=1,=,))((

,,,

,=,=,=

,,,

1,=1,=,=,

,,,

1,=,=1,=,

,0,=,

1,=,=1,=

or

,,0,=

1,=1,=,=,

,0,=0,=

1,=,0=,0=),)((

03

2

02122302

33221122213

03

2

12

1

01

33221122213

03

2

02

1

11

33221111213

03

2

02

1

01

33222111

13

2

12

1

11

33221123

13

2

12

1

11

33221113

132

1

11

332211

13

2

121

3322113

1

0321

3321213

NS

i

NSS

NSS

Nss

NSS

NSS

NS

NS

N

ViViiisHi

ijijijiiN

ViViVi

ijijijiiN

ViViVi

ijijijiiN

ViViVi

ijQijQij

ViViVi

ijijijpi

ViViVi

ijijijpi

ViiVi

ijijij

ViVii

ijijiji

Viii

ijjjiN

otherwise

ViViViisHisHi

ijijijiiiN

NSS

0

,,,)),()(

,=,=,=))(((

03

2

02

1

1122113

3322112211213

Proof:

The infinitesimal generator )),,(),,,((321321

jjjiiia of this process can be obtained using the

following arguments:

1: Let 00,>0,>321iii .

A primary demand from any one of the )(3

iN sources or due to perishability takes the inventory level

),,(321

iii to ),1,(321

iii with intensity 1113

)( iiN for I-commodity or ),,(321

iii to )1,,(321

iii

with intensity 2223

)( iiN for II-commodity.

The level ),(0,32

ii , and ),0,(31

ii , respectively, is taken to )1,(0,32

ii , with intensity

22213))(( iiN , and )1,0,(

31ii with intensity

11213))(( iiN .

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 )(0,0,3

i to 1)(0,0,3i with intensity ))((

213 iN ,



103

Ni .

3: Let 10,>0,>321iii .

A demand from orbit takes the inventory level ),,(321

iii to 1),1,(321 iii with intensity

13pi for I-

commodity or ),,(321

iii to 1)1,,(321 iii with intensity

23pi for II-commodity.

The level ),(0,32

ii , and ),0,(31

ii , respectively, is taken to 1)1,(0,32 ii and 1)1,0,(

31 ii with

intensity 3

i .

4: From a state ),,(321

iii with ),(),(2121

ssii , 03i a replenishment by the delivery of orders

for both commodities takes the inventory level to ),,(32211

iQiQi , 111

= sSQ , 222

= sSQ , with

intensity of this transition .

We observe that no transition other than the above is possible.

Finally the value of ) ),,(),,,( (321321

jjjiiia is obtained by

)),,(),,,((=)),,(),,,((321321

321

)3

,2

,1

()3

,2

,1

(

321321jjjiiiaiiiiiia

jjj

jjjiii

Hence we get the infinitesimal generator ). ),,(),,,( (321321

jjjiiia □

In order to write down the infinitesimal generator in a matrix form, we arrange the states in

lexicographic order and group 1)1)((2

NS states as:

,0),,(,),,1,(,,1,1),(,1,0),(),,0,(,,0,1),(,0,0),((>=<2

SqNqqqNqqqq

.,0,1,= )),,(,,1),,(122

SqforNSqSq

Then the rate matrix has the block partitioned form with the following sub matrix [11

]ji

at the 1

i -

the row and 1

j -th column position.

.,

,=,

1,=,

,=,

=][1

01111

1

11111

1

01111

11

otherwise

ViQijC

ViijB

ViijA

s

S

i

S

i

ji

0

where

.0,

,==][

2

02222

22otherwise

ViQijWC

s

ji

.0,

,,=),))(((

,1,=),)((

=][0333213

1

0333213

33

otherwise

ViijiN

ViijiN

HN

N

ji



.0,

,,=

,1,=

0,=0,=

=][2

12222

2

12222

22

220

otherwise

ViijH

ViijF

ijH

AS

i

S

i

ji

11

,1,= SiFor

.0,

,,=

0,=,=

=][ 2

12221

2221

221

otherwise

ViijM

iijV

BS

i

i

jii

11

,1,2,= SiFor

.0,

,,=

0,=,=

,1,=

=][2

122221

2221

2

12222

221

otherwise

ViijL

iijJ

ViijG

AS

ii

i

S

i

jii

11

,1,2,= SiFor

.0,

,,=,))((

,1,=,

=][033311213

13333

331

otherwise

ViijiiN

Viiji

VN

N

jii

11

,1,2,= SiFor

.0,

,,=,)(

,1,=,

=][03331113

13331

331

otherwise

ViijiiN

Viijp

MN

N

jii

22

,1,2,= SiFor

.0,

,,=,))((

,1,=,

=][033322213

13333

332

otherwise

ViijiiN

Viiji

FN

N

jii

22

,1,2,= SiFor

.0,

,,=),)(

))(((

=][0333223

22213

332

otherwise

ViijisHi

iiN

HN

jii

22

,1,2,= SiFor



.0,

,,=,)(

,1,=,

=][03332223

133323

332

otherwise

ViijiiN

Viijpi

GN

N

jii

11

,1,2,= SiFor

.0,

,,=))(

))(((

=][033311

311213

331

otherwise

ViijisH

iiiN

JN

jii

,,1,2,= ;,1,2,=2211

SiSiFor

.0,

,,=)),()(

))(((

=][033322113

2211213

3321

otherwise

ViijisHisHi

iiiN

LN

jiii

1

=N

IW

It may be noted that the matrices 1i

A , ,1i

B 11

,1,2,= Si , 0

A and C are square matrices of order

1)1)((2

NS . The sub matrices 1i

V , 1i

M , 1i

J , ,21

iiL

11,1,2,= Si , ,,1,2,=

22Si W , H ,

2i

F ,

2i

H , 2

iG ,

22,1,2,= Si , are square matrices of order 1)( N .

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

0}:))(),(),({(21

ttXtLtL on the finite space E is irreducible, aperiodic and persistent non-null.

Hence the limiting distribution

(0)],(0),(0),|=)(,=)(,=)([lim=2132211

)3

,2

,1

(

XLLitXitLitLPrt

iii

exists.

Let ),,,,(=)

1((1)(0) S

Π

partitioning the vector, )

1( i

into as follows:

11

)2

,1

(,2)1

(,1)1

(,0)1

()1

(

,0,1,2,=),,,,,(= SiSiiiii

which is partitioned as follows:

.,0,1,2,=;,0,1,2,=),,,(=2211

),2

,1

(,0)2

,1

()2

,1

(

SiSiNiiiiii

Then the vector of limiting probabilities Π satisfies

1.== eand Π0Π (1)



Theorem 2:

The limiting distribution Π is given by,

,,0,1,=,=11

1

)1

()1

(

Sii

Qi

where

,,1,=

,

1)(

,=,

1,,0,1,= ,1)(

=

111

1

11

11

11

1

11111

1

11

1

1111

11

1

1

111

11

0=1

111

2

11

11

1

11

11

1

1

111

11

1

SQi

ABBAB

CAABBAB

QiI

QiABBAB

iijSjSjS

jSjsjsQQQ

iS

j

iQ

iiQQQ

iQ

i

(2)

The value of )

1( Q

can be obtained from the relation

1

11

1

1111

11

1

1

111

11

0=1

1)

1(

1)(jSjsjsQQQ

s

j

QQ

CAABBAB

1

11

1

11

21

111

1

11111

QQQQjSjSjSABABBAB

(3)

0,=1)(1

0111

1

111

1 CABBABQQQ

Q

and

IABBABiiQQQ

iQ

Q

i

Q 1

11

11

1

1

111

11

11

0=1

)1

(

1)(

1

11

1

1111

11

1

1

111

11

0=1

111

21

11

=1

1)(jSjsjsQQQ

iS

j

iQ

S

Qi

CAABBAB (4)

1.=1

11

11

11

1

11111

eABBABiijSjSjS

Proof:

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

1,,0,1,=0,=11

1

)1

(

11

1)1

(

QiABi

i

i

i

(5)

,=0,=11

)11

(

1

)1

(

11

1)1

(

QiCABQi

i

i

i

i

(6)

1,,1,=0,=111

)11

(

1

)1

(

11

1)1

(

SQiCABQi

i

i

i

i

(7)

.=0,=11

)11

(

1

)1

(

SiCAQi

i

i

(8)

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



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 1i

and 2

i denote the average inventory level for the first commodity and the second commodity

respectively in the steady state. Then

)

3,

2,

1(

1

0=3

2

0=2

1

1=1

1

=iii

N

i

S

i

S

i

ii (9)

and

)

3,

2,

1(

2

0=3

2

1=2

1

0=1

2

=iii

N

i

S

i

S

i

ii (10)

4.2 Expected reorder rate

Let r

denote the mean reorder rate in the steady state. Then

,0)

21,

1(

11202

1

2

0=2

)1)((=is

i

s

i

rsNN

1,0)

2,

1(

22101

2

1

0=1

)1)((

si

i

s

i

sNN

)

3,

21,

1(

13232302

1113

1=3

2

0=2

)))((1)()((iis

i

N

i

s

i

pipiiNsiN

(11)

)

31,

2,

1(

23131301

2223

1=3

1

0=1

)))((1)()((isi

i

N

i

s

i

pipiiNsiN

4.3 Expected perishable rate

Let 1

p and

2p

denote the expected perishable rates for the first commodity and the second

commodity respectively in the steady state. Then

)

3,

2,

1(

11

0=3

2

0=2

1

1=1

1

=iii

N

i

S

i

S

i

pi (12)

and

)

3,

2,

1(

22

0=3

2

1=2

1

0=1

2

=iii

N

i

S

i

S

i

pi (13)

4.4 Expected number of demands in the orbit

Let o

denote the expected number of demands in the orbit. Then

)

3,

2,

1(

3

1=3

2

0=2

1

0=1

=iii

N

i

S

i

S

i

oi (14)



4.5 Expected an arriving demand enters into the orbit

The expected an arriving primary demand enters into the orbit is given by

)

3(0,0,

213

1

0=3

))((=i

N

i

aiN

(15)

4.6 The overall rate of retrials

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

)

3,

2,

1(

3

1=3

2

0=2

1

0=1

=iii

N

i

S

i

S

i

ori (16)

4.7 The successful rate of retrials

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

)

3,

2,

1(

3

1=3

2

1=2

1

1=1

)3

,0,1

(

3

1=3

1

1=1

)3

,2

(0,

3

1=3

2

1=2

=iii

N

i

S

i

S

i

iiN

i

S

i

iiN

i

S

i

sriii (17)

4.8 Fraction of successful rate of retrials

Let fr

denote the fraction of successful rate of retrials is given by

or

sr

fr

= (18)

V. COST ANALYSIS To compute the total expected cost per unit time (total expected cost rate),

the following costs, are considered.

1hc : The inventory holding cost per unit item per unit time for I-commodity.

2hc : The inventory holding cost per unit item per unit time for II-commodity.

sc : The setup cost per order.

1pc : Perishable cost of the I - commodity per unit item per unit time.

2pc : Perishable cost of the II- commodity per unit item per unit time.

wc : Waiting cost of an orbiting demand per unit time.

The long run total expected cost rate is given by

.=),,,,(22112211

2121 owpppprsihihccccccNssSSTC

(19)

Substituting the values of ’s we get ),,,,(2121

NssSSTC

)

3,

2,

1(

2

0=3

2

1=2

1

0=1

2

)3

,2

,1

(

1

0=3

2

0=2

1

1=1

1

=iii

N

i

S

i

S

i

h

iiiN

i

S

i

S

i

hicic

)

3,

2,

1(

11

0=3

2

0=2

1

1=1

1

)3

,2

,1

(

3

1=3

2

0=2

1

0=1

iiiN

i

S

i

S

i

p

iiiN

i

S

i

S

i

wicic

)

3,

2,

1(

22

0=3

2

1=2

1

0=1

2

iiiN

i

S

i

S

i

pic

,0)

21,

1(

11202

1

2

0=2

)1)((is

i

s

i

ssNNc



1,0)

2,

1(

22101

2

1

0=1

1)((si

i

s

i

sNN (20)

)))((1)()((23230

2111313

1=3

2

0=2

piiNspiiNi

N

i

s

i

)

31,

2,

1(

131301

222323

1=3

1

0=1

)))((1)()((isi

i

N

i

s

i

piiNspiiN

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 ),,(21

SSTC to be convex. Appropriate numerical search procedures are used to

obtain the optimal values of ,TC 1

S and 2

S (say ,*

TC *

1S and

*

2S ). 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 *

1S and

*

2S by fixing the parameters and the cost values:

2,=1

s 3,=2

s 10,=N 0.01,=1

0.02,=2

0.01,= 0.2,=1

0.1,=2

0.02,= 12,=s

c

0.01,=1h

c 0.04,=2h

c 0.4,=1p

c 0.5,=2p

c 6,=w

c 0.4=1

p and 0.6=2

p .

From the Table 1 the total expected cost rate is more sensitive to the changes in *

2S than that of in

*

1S .

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 *

1S and

*

2S .



2,=1

s 3,=2

s 10,=N 0.01,=1

0.02,=2

0.01,= 0.2,=1

0.1,=2

0.02,=

12,=s

c

0.01,=1h

c 0.04,=2h

c 0.4,=1p

c 0.5,=2p

c 6,=w

c 0.4,=1

p 0.6=2

p .

Figure 1: A three dimensional plot of the cost function ),(21

SSTC

Table 1: Total expected cost rate as a function of 1

S and 2

S

6.1 Example 1

In the first example, we look at the impact of 1

, 2

, 1

, and 2

on the optimal values ),(*

2

*

1SS

and the corresponding total expected cost rate*

TC . For this, first by fixing the parameters and cost values as

2,=1

s 3,=2

s 10,=N 0.02= , 0.01= , 0.4=1

p , 0.6=2

p , 0.01=1h

c , 0.04=2h

c , 12=s

c ,

6=w

c , 0.4=1p

c and 0.5=2p

c . Observe the following from Tables 2 and 3 :

1. From the Table 2 , it is observed that the *

TC , *

1S and

*

2S 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 *

1S and

*

2S 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 is

very sensitive to changes in 2

than that of in 1

.

6.2 Example 2

In this example, we study the impact of s

c , 1h

c , 2h

c , 1p

c , 2p

c and w

c on the optimal values

),(*

2

*

1SS and the corresponding

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

2,=1

s 3,=2

s 10,=N 0.01=1

, 0.02=2

, 0.02= , 0.01= , 0.2=1

, 0.1=2

, 0.4=1

p

and 0.6=2

p .

Observe the following from Tables 64 :

1. The total expected cost rate increases when 1h

c , 2h

c , s

c , w

c , 1p

c and 2p

c increase monotonically.

2. As 1h

c and 2h

c increase, the optimal values *

1S and

*

2S decrease monotonically. This is to be expected

since 1h

c and 2h

c increase, we resort to maintain low stock in the inventory.

3. Similarly, when w

c increases, the values of *

1S and

*

2S increase monotonically. This is because if

wc

increases then we have to maintain high inventory to reduce the number of waiting customers in the orbit.

2S

1S

29 30 31 32 33

88 52.866093 52.863418 52.862517 52.863258 52.865527

89 52.866088 52.863289 52.862247 52.862850 52.864984

90 52.866167 52.863315 52.862134 52.862599 52.864598

91 52.866586 52.863494 52.862227 52.862502 52.864368

92 52.867052 52.8638820 52.862361 52.862555 52.864288



4. As *

1S and

*

2S increase monotonically,

sc increases. This is a common decision that we have to maintain

more stock to avoid frequent ordering.

5. If 1p

c and 2p

c increase monotonically then *

1S and

*

2S decrease and

*TC increases. We also note that the

total expected cost rate is more sensitive to changes in 1p

c than that of in 2p

c .

Table 2: Sensitivity of 1

and 2

on the optimal values

Table 3: Variation in optimal values for different values of 1

and 2

Table 4: Effect of varying 1h

c and 2h


2

1

0.010 0.015 0.020 0.025 0.030

0.005 104 28 104 30 104 30 105 33 105 34

52.166935 52.404997 52.616266 52.805787 52.977922

0.010 104 28 104 30 104 31 105 33 105 34

52.471973 52.677218 52.862134 53.030387 53.184492

0.015 105 28 105 30 105 31 105 33 106 34

52.735666 52.916048 53.080348 53.231281 53.370828

0.020 106 28 106 30 106 31 106 34 106 34

52.967782 53.128612 53.276383 53.413216 53.540578

0.025 107 28 107 30 107 32 107 34 107 35

53.174930 53.319961 53.454181 53.579277 53.696383

2

1

0.06 0.08 0.10 0.12 0.14

0.20

105 31 105 31 104 31 102 30 100 26

50.734492 51.983923 52.862134 53.512248 53.989839

0.25

99 31 95 31 90 31 87 30 86 26

51.777332 53.098322 54.023159 54.718720 55.247406

0.30

67 27 65 27 62 27 60 26 60 23

52.403399 53.768192 54.715274 55.438513 56.014391

0.35

49 26 47 24 44 24 42 24 41 23

52.802492 54.198838 55.153269 55.880546 56.468607

0.40 37 25 35 22 32 21 31 21 29 20

53.067357 54.498210 55.446462 56.169518 56.753754

1hc

2hc

0.005 0.010 0.015 0.020 0.025

0.02

98 36 93 36 87 35 83 35 79 35

52.576308 52.655217 52.730014 52.800966 52.868637

0.03

98 34 91 33 86 33 82 33 77 32

52.683493 52.761980 52.836322 52.907085 52.974514

0.04

97 32 90 31 85 31 81 31 76 30

52.784094 52.862134 52.936195 53.006699 53.073884

0.05

95 30 89 29 84 29 80 29 76 29

52.878746 52.956645 53.030349 53.100529 53.1673640

0.06 94 28 89 28 84 28 79 27 75 27

52.968191 53.045806 53.119568 53.189435 53.256085



Table 5: Influence of w

c and s


Table 6: Variation in optimal values for different values of 1p

c and 2p

c

VII. 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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23 17 56 24 90 30 127 36 164 42

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22 19 56 25 90 31 127 37 164 42

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20 20 55 26 90 32 127 38 164 43

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51.737733 52.071328 52.346578 52.577109 52.769276

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94 39 90 31 87 26 86 23 85 20

52.545497 52.862134 53.126890 53.353545 53.549672

6

63 36 61 29 59 25 58 22 57 19

53.058156 53.368609 53.628758 53.853509 54.049595

8

47 35 45 28 45 27 44 21 43 18

53.431616 53.739166 53.829995 54.221504 54.418675

1.0 37 34 35 27 35 23 34 20 34 18

53.723445 54.029803 54.287583 54.511543 54.700904



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International Journal of Mathematics and Statistics Invention (IJMSI)

Technology

s i c i s i

s i s i items

parameters s i

s policy

review of inventory

commodity inventory

item j i

triplet s