TWO-COMMODITY PERISHABLE INVENTORY
SYSTEM WITH PARTIAL BACKLOG DEMANDS
N. Anbazhagan*
Department of Mathematics, Alagappa University, Karaikudi, (India).
E-mail: anbazhagann@alagappauniversity.ac.in
ORCID: 0000-0002-9146-2483
* Corresponding author
B. Sivakumar
Department of Applied Mathematics and Statistics, Madurai Kamaraj University, Madurai, (India).
E-mail: sivabkumar@yahoo.com
ORCID: 0000-0002-7062-4252
S. Amutha
Ramanujan Center for Higher Mathematics, Alagappa University, Karaikudi, (India).
E-mail: amuthas@alagappauniversity.ac.in
ORCID: 0000-0003-4985-179X
R. Suganya
Department of Mathematics, Alagappa University, Karaikudi, (India).
E-mail: saisugan92@gmail.com
Reception: 06/08/2022 Acceptance: 21/08/2022 Publication: 29/12/2022
Suggested citation:
N. Anbazhagan, B. Sivakumar, S. Amuthaand and R. Suganya (2022). Two-commodity perishable inventory system with
partial backlog demands. 3C Empresa. Investigación y pensamiento crítico,11 (2), 33-48.
https://doi.org/10.17993/3cemp.2022.110250.33-48
https://doi.org/10.17993/3cemp.2022.110250.33-48
33
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
ABSTRACT
This article examines a two-commodity continuous review perishable inventory system. The demands
are arrived at for each product by independent Markovian arrival processes (MAP). Lifetimes follow an
exponential distribution. The commodities are assumed to be substitutable. If both commodities have
reached zero, demand is backlogged up to predetermined levels. This article’s novelty has been a local
purchase, which is made to clear the backlog instantaneously when demand reaches a predetermined level.
In the steady-state, the joint probability distribution of inventory levels of both commodities is obtained.
Several metrics of system performance in steady-state are derived and also provided as numerical
examples to explain the optimum values of the system’s parameters.
KEYWORDS
Two-commodity Inventory system, Substitutable items, Joint ordering Policy, Markov arrival demands,
Partial backlog.
https://doi.org/10.17993/3cemp.2022.110250.33-48
1 INTRODUCTION
Over the last few decades, researchers have been fascinated with the study of a two-commodity inventory
system. It has more importance because these systems are more sophisticated than single commodity
inventory systems due to the large number of items held and their coordinated behaviors. Also, many
organisations have increasingly used multi-commodity inventory systems. However, the correlation of
the reorder points for each item is the major challenge in multi-product inventory systems. Unlike
systems that deal with a single commodity, the reordering methods in these systems are more com-
plicated. So, replenishment orders for groups of products must be well coordinated. Initially, research
focused on inventory models with independently defined reorder points. The individual ordering policy
includes calculating the best order quantity and reordering duration for each item. This ordering policy
implementation provides the system with significant flexibility in picking the appropriate inventory
models for each item and separately modifying the policy. However, joint ordering policies are preferred
over individual ordering policies when the products share the same storage space and transportation
facilities. Joint replenishment has several advantages because the joint ordering policy allows for the
simultaneous replenishment of several commodities, quantity discounts, and significant savings in
ordering and purchasing expenses. The joint replenishment was proposed by Balintfy and developed by
Silver. More details about joint replenishment can be seen in Anbazhagan et al. (2012, 2015), Senthil
Kumar, and Sivakumar. Various models with two-commodity readers can read in Anbazhagan and
Arivarignan, Benny et al., Krishnamoorthy et al., and Ozkar et al.
In the earlier literature on inventory systems, it has generally been recognized that inventory models
built under the presumption of a product’s lifetime being indefinite until its storage, i.e., an item once
placed in a storeroom stays unmodified and entirely functional for supplying future demand. However,
this is not the case. When constructing inventory models, one aspect for consideration is an item’s
perishability, as commodities do not necessarily retain their properties when held for future use. In
general, perishability is the outcome of stock depletion, which consists of obsolescence, breakage, decay,
losing usefulness, and many other factors. Some examples of perishable objects are meals, evaporative
fluids, chemicals, drugs, and radioactive substances. For more details about perishable product readers
can refer Karthikeyan and Sudesh, Nahmias, Sivakumar et al., Smrutirekha Debataa et al., Umay and
Bahar, Yadavalli et al. (2010, 2015), Zhang et al.
Several research articles examine inventory systems in which required products are directly provided
from stock if the item is available. Demand that appears during stock-out times results in either lost
sales or a backlog (demand satisfied immediately after the arrival of ordered items). Initially, it is
believed that there is a total backlog of unfilled demand. In actuality, many customers are willing to
wait until the end of the shortage period to pick up their orders, while others are not. As a result, it is
presumed that any predefined quantity of demand (partial backlog) that appeared during the stock-out
time is satisfied. For more details about backlog concept readers an refer Adak Sudip and Mahapatra,
C
´a
rdenas-Barr
´o
n Leopoldo et al., Khan et al., Kurt et al., San Jos
´e
et al., Stanley et al. and Tai et al.
Generally, customer satisfaction generates a lot of profit for the system. So the shopkeeper does the
maximum amount of work to satisfy the customers. In a practical situation, the local purchase is made
by the shopkeeper when the shop runs out of stock and that item’s replenishment has been delayed.
We can see this act in clothing stores, supermarkets, and all the retailers’ shops.
In this article we assume that demands during the stock-out periods are backlogged. We further
assume that when the number of backlogged demands reaches a prefixed level a local purchase is
made to clear the backlog instantaneously so that the inventory level of the corresponding commodity
becomes zero. In the following sections, We have obtained the joint probability distribution for the
inventory levels of both commodities in the steady state case in section 3. Various system performance
measures in the steady state are derived in section 4 and the cost analysis and the results are illustrated
numerically in section 5 and 6.
https://doi.org/10.17993/3cemp.2022.110250.33-48
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
34
ABSTRACT
This article examines a two-commodity continuous review perishable inventory system. The demands
are arrived at for each product by independent Markovian arrival processes (MAP). Lifetimes follow an
exponential distribution. The commodities are assumed to be substitutable. If both commodities have
reached zero, demand is backlogged up to predetermined levels. This article’s novelty has been a local
purchase, which is made to clear the backlog instantaneously when demand reaches a predetermined level.
In the steady-state, the joint probability distribution of inventory levels of both commodities is obtained.
Several metrics of system performance in steady-state are derived and also provided as numerical
examples to explain the optimum values of the system’s parameters.
KEYWORDS
Two-commodity Inventory system, Substitutable items, Joint ordering Policy, Markov arrival demands,
Partial backlog.
https://doi.org/10.17993/3cemp.2022.110250.33-48
1 INTRODUCTION
Over the last few decades, researchers have been fascinated with the study of a two-commodity inventory
system. It has more importance because these systems are more sophisticated than single commodity
inventory systems due to the large number of items held and their coordinated behaviors. Also, many
organisations have increasingly used multi-commodity inventory systems. However, the correlation of
the reorder points for each item is the major challenge in multi-product inventory systems. Unlike
systems that deal with a single commodity, the reordering methods in these systems are more com-
plicated. So, replenishment orders for groups of products must be well coordinated. Initially, research
focused on inventory models with independently defined reorder points. The individual ordering policy
includes calculating the best order quantity and reordering duration for each item. This ordering policy
implementation provides the system with significant flexibility in picking the appropriate inventory
models for each item and separately modifying the policy. However, joint ordering policies are preferred
over individual ordering policies when the products share the same storage space and transportation
facilities. Joint replenishment has several advantages because the joint ordering policy allows for the
simultaneous replenishment of several commodities, quantity discounts, and significant savings in
ordering and purchasing expenses. The joint replenishment was proposed by Balintfy and developed by
Silver. More details about joint replenishment can be seen in Anbazhagan et al. (2012, 2015), Senthil
Kumar, and Sivakumar. Various models with two-commodity readers can read in Anbazhagan and
Arivarignan, Benny et al., Krishnamoorthy et al., and Ozkar et al.
In the earlier literature on inventory systems, it has generally been recognized that inventory models
built under the presumption of a product’s lifetime being indefinite until its storage, i.e., an item once
placed in a storeroom stays unmodified and entirely functional for supplying future demand. However,
this is not the case. When constructing inventory models, one aspect for consideration is an item’s
perishability, as commodities do not necessarily retain their properties when held for future use. In
general, perishability is the outcome of stock depletion, which consists of obsolescence, breakage, decay,
losing usefulness, and many other factors. Some examples of perishable objects are meals, evaporative
fluids, chemicals, drugs, and radioactive substances. For more details about perishable product readers
can refer Karthikeyan and Sudesh, Nahmias, Sivakumar et al., Smrutirekha Debataa et al., Umay and
Bahar, Yadavalli et al. (2010, 2015), Zhang et al.
Several research articles examine inventory systems in which required products are directly provided
from stock if the item is available. Demand that appears during stock-out times results in either lost
sales or a backlog (demand satisfied immediately after the arrival of ordered items). Initially, it is
believed that there is a total backlog of unfilled demand. In actuality, many customers are willing to
wait until the end of the shortage period to pick up their orders, while others are not. As a result, it is
presumed that any predefined quantity of demand (partial backlog) that appeared during the stock-out
time is satisfied. For more details about backlog concept readers an refer Adak Sudip and Mahapatra,
C
´a
rdenas-Barr
´o
n Leopoldo et al., Khan et al., Kurt et al., San Jos
´e
et al., Stanley et al. and Tai et al.
Generally, customer satisfaction generates a lot of profit for the system. So the shopkeeper does the
maximum amount of work to satisfy the customers. In a practical situation, the local purchase is made
by the shopkeeper when the shop runs out of stock and that item’s replenishment has been delayed.
We can see this act in clothing stores, supermarkets, and all the retailers’ shops.
In this article we assume that demands during the stock-out periods are backlogged. We further
assume that when the number of backlogged demands reaches a prefixed level a local purchase is
made to clear the backlog instantaneously so that the inventory level of the corresponding commodity
becomes zero. In the following sections, We have obtained the joint probability distribution for the
inventory levels of both commodities in the steady state case in section 3. Various system performance
measures in the steady state are derived in section 4 and the cost analysis and the results are illustrated
numerically in section 5 and 6.
https://doi.org/10.17993/3cemp.2022.110250.33-48
35
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
2 THE MODEL
We consider a two-commodity inventory system with the maximum capacity
Si
units for
i−
th commodity
(
i
=1
,
2)
.
The demands for
i−
th commodity is of unit size. The demands for commodity-1 arrive
according to a Markovian arrival process (MAP) with representation (
D0,D
1
)where
D
’s are of order
m1×m1.
The underlying Markov chain
J1
(
t
)of the
MAP
has the generator
D
(=
D0
+
D1
)and a
stationary row vector
λ1
of length
m1.
Independently of this process, demands for commodity-2 arrive
according to a MAP with representation (
F0,F
1
)where
F
’s are of order
m2×m2.
The underlying
Markov chain
J2
(
t
)of this
MAP
has the generator
F
(=
F0
+
F1
)and a stationary row vector
λ2
of
length
m2.
The items are perishable in nature. The life time of each commodity is assumed to be
distributed as exponential with parameter
γi,
(
i
=1
,
2)
.
The two-commodities serve as substitute for
each other, that is, a demand for a commodity that is sold out, is satisfied with the other commodity
when still in stock. If both the commodities are out of stock, any arriving demands are backlogged. The
backlog is allowed up to the level
Ni
(
<∞
)for the
i−
th commodity (
i
=1
,
2). Whenever the backlog
level reaches
Ni,
(
i
=1
,
2) an order for
Ni
items is placed which is replenished instantaneously. The
reorder level for the
i
-th commodity is fixed at
si
(1
≤si≤Si
) with an ordering quantity for the
i
-th
commodity is
Qi
(=
Si−si>s
i
+
Ni
+ 1) items when both inventory levels are less than or equal to their
respective reorder levels. The requirement
Si−si>s
i
+
Ni
+1ensures that after the replenishment
the inventory levels of both commodities will be always above the respective reorder levels; otherwise
it may not be possible to place reorder (according to this policy) which leads to perpetual shortage.
More explicitly if
Li
(
t
)represents inventory level of
i
-th commodity at time t, then a reorder for both
commodities is made when
L1
(
t
)
≤s1
and
L2
(
t
)
≤s2.
The lead time is assumed to be distributed as
negative exponential with parameter β(>0).
Notations
[A]ij : The element/submatrix at (i, j)-th position of A.
0: Zero matrix.
I: An identity matrix.
Ik: An identity matrix of order k.
A⊗B: Kronecker product of matrices Aand B.
A⊕B: Kronecker sum of matrices Aand B.
e: A column vector of 1′s of appropriate dimension.
3 ANALYSIS
From the assumptions made on the input and output processes it can be shown that the quadruple
(L1,L
2,J
1,J
2)={(L1(t),L
2(t),J
1(t),J
2(t)),t≥0}is a Markov process with state space given by
E={(i,k,j
1,j
2)|i=1,2,...,S
1,k =0,1,...,S
2,j
1=1,2,...,m
1,j
2=1,2,...,m
2}
∪{(i,k,j
1,j
2)|i=0,k =−(N2−1),−(N2−2),...,S
2,j
1=1,2,...,m
1,j
2=1,2,...,m
2}
∪{(i,k,j
1,j
2)|i=−(N1−1),−(N1−2),...,−1,k =−(N2−1),−(N2−2),...,0,
j1=1,2,...,m
1,j
2=1,2,...,m
2}.
Define the following ordered sets :
i= ((i, 0),(i, 1),...,(i, S2)),
<i> = ((i, −N2+ 1),(i, −N2+ 2),...,(i, S2)),
[i]=((i, −N2+ 1),(i, −N2+ 2),...,(i, 0) ) ,
(i, j)=((i, j, 1),(i, j, 2),...,(i,j,m
1)),
(i,j,k)=((i,j,k,1),(i,j,k,2),...,(i,j,k,m
2)),
https://doi.org/10.17993/3cemp.2022.110250.33-48
Then the state space is ordered as ([−N1+ 1],[−N1+ 2],...,[−1],<0>, 1,
2,...,S1).
The infinitesimal generator of
P
of the Markov process (
L1,L
2,J
1,J
2
)has the following
block partitioned form :
[P]ij =
Bi,j=i−1,i=0,1,...,S
1,
B, j =i−1,i=−(N1−2),−(N1−3),...,−1,
B, j =i+(N1−1),i=−(N1−1),
C, j =i+Q1,i=1,2,...,s
1,
C, j =i+Q1,i=0,
C, j =i+Q1,i=−(N1−1),−(N1−2),...,−1,
Ai,j=i, i =0,1,...,S,
A, j =i, i =−(N1−1),−(N1−2),...,−1,
0,otherwise,
where
[C]kl =βIm1⊗Im2,l=k+Q2,k=0,1,...,s
2,
0,otherwise.
[
C]kl =βIm1⊗Im2l=k+Q2,k=−(N2−1),−(N2−2),...,s
2,
0,otherwise.
[
C]kl =βIm1⊗Im2,l=k+Q2,k=−(N2−1),−(N2−2),...,0,
0,otherwise.
For i=2,3,...,S
1,
[Bi]kl =
D1⊗Im2+iγ1Im1⊗Im2,l=k, k =1,2,...,S
2,
D1⊕F1+iγ1Im1⊗Im2,l=k, k =0,
0,otherwise.
For i=1,
[Bi]kl =
D1⊗Im2+iγ1Im1⊗Im2,l=k, k =1,2,...,S
2,
D1⊕F1+iγ1Im1⊗Im2,l=k, k =0,
0,otherwise.
For i=0,
[Bi]kl =D1⊗Im2,l=k, k =−(N2−1),(N2−2),...,0,
0,otherwise.
ˆ
Bkl =D1⊗Im2,l=k, k =−(N2−1),(N2−2),...,0,
0,otherwise.
˜
Bkl =D1⊗Im2,l=k, k =−(N2−1),(N2−2),...,0,
0,otherwise.
For i=1,2,...s
1,
[Ai]kl =
Im1⊗F1+kγ2Im1⊗Im2,l=k−1,k=1,2,...,S
2,
D0⊕F0−(iγ1+β)Im1⊗Im2,l=k, k =0,
D0⊕F0−(iγ1+β+kγ2)Im1⊗Im2,l=k, k =1,2,...,s
2
D0⊕F0−(iγ1+kγ2)Im1⊗Im2,l=k, k =s2+1,s
2+2,...,S
2
0,otherwise.
https://doi.org/10.17993/3cemp.2022.110250.33-48
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
36
2 THE MODEL
We consider a two-commodity inventory system with the maximum capacity
Si
units for
i−
th commodity
(
i
=1
,
2)
.
The demands for
i−
th commodity is of unit size. The demands for commodity-1 arrive
according to a Markovian arrival process (MAP) with representation (
D0,D
1
)where
D
’s are of order
m1×m1.
The underlying Markov chain
J1
(
t
)of the
MAP
has the generator
D
(=
D0
+
D1
)and a
stationary row vector
λ1
of length
m1.
Independently of this process, demands for commodity-2 arrive
according to a MAP with representation (
F0,F
1
)where
F
’s are of order
m2×m2.
The underlying
Markov chain
J2
(
t
)of this
MAP
has the generator
F
(=
F0
+
F1
)and a stationary row vector
λ2
of
length
m2.
The items are perishable in nature. The life time of each commodity is assumed to be
distributed as exponential with parameter
γi,
(
i
=1
,
2)
.
The two-commodities serve as substitute for
each other, that is, a demand for a commodity that is sold out, is satisfied with the other commodity
when still in stock. If both the commodities are out of stock, any arriving demands are backlogged. The
backlog is allowed up to the level
Ni
(
<∞
)for the
i−
th commodity (
i
=1
,
2). Whenever the backlog
level reaches
Ni,
(
i
=1
,
2) an order for
Ni
items is placed which is replenished instantaneously. The
reorder level for the
i
-th commodity is fixed at
si
(1
≤si≤Si
) with an ordering quantity for the
i
-th
commodity is
Qi
(=
Si−si>s
i
+
Ni
+ 1) items when both inventory levels are less than or equal to their
respective reorder levels. The requirement
Si−si>s
i
+
Ni
+1ensures that after the replenishment
the inventory levels of both commodities will be always above the respective reorder levels; otherwise
it may not be possible to place reorder (according to this policy) which leads to perpetual shortage.
More explicitly if
Li
(
t
)represents inventory level of
i
-th commodity at time t, then a reorder for both
commodities is made when
L1
(
t
)
≤s1
and
L2
(
t
)
≤s2.
The lead time is assumed to be distributed as
negative exponential with parameter β(>0).
Notations
[A]ij : The element/submatrix at (i, j)-th position of A.
0: Zero matrix.
I: An identity matrix.
Ik: An identity matrix of order k.
A⊗B: Kronecker product of matrices Aand B.
A⊕B: Kronecker sum of matrices Aand B.
e: A column vector of 1′s of appropriate dimension.
3 ANALYSIS
From the assumptions made on the input and output processes it can be shown that the quadruple
(L1,L
2,J
1,J
2)={(L1(t),L
2(t),J
1(t),J
2(t)),t≥0}is a Markov process with state space given by
E={(i,k,j
1,j
2)|i=1,2,...,S
1,k =0,1,...,S
2,j
1=1,2,...,m
1,j
2=1,2,...,m
2}
∪{(i,k,j
1,j
2)|i=0,k =−(N2−1),−(N2−2),...,S
2,j
1=1,2,...,m
1,j
2=1,2,...,m
2}
∪{(i,k,j
1,j
2)|i=−(N1−1),−(N1−2),...,−1,k =−(N2−1),−(N2−2),...,0,
j1=1,2,...,m
1,j
2=1,2,...,m
2}.
Define the following ordered sets :
i= ((i, 0),(i, 1),...,(i, S2)),
<i> = ((i, −N2+ 1),(i, −N2+ 2),...,(i, S2)),
[i]=((i, −N2+ 1),(i, −N2+ 2),...,(i, 0) ) ,
(i, j)=((i, j, 1),(i, j, 2),...,(i,j,m
1)),
(i,j,k)=((i,j,k,1),(i,j,k,2),...,(i,j,k,m
2)),
https://doi.org/10.17993/3cemp.2022.110250.33-48
Then the state space is ordered as ([−N1+ 1],[−N1+ 2],...,[−1],<0>, 1,
2,...,S1).
The infinitesimal generator of
P
of the Markov process (
L1,L
2,J
1,J
2
)has the following
block partitioned form :
[P]ij =
Bi,j=i−1,i=0,1,...,S
1,
B, j =i−1,i=−(N1−2),−(N1−3),...,−1,
B, j =i+(N1−1),i=−(N1−1),
C, j =i+Q1,i=1,2,...,s
1,
C, j =i+Q1,i=0,
C, j =i+Q1,i=−(N1−1),−(N1−2),...,−1,
Ai,j=i, i =0,1,...,S,
A, j =i, i =−(N1−1),−(N1−2),...,−1,
0,otherwise,
where
[C]kl =βIm1⊗Im2,l=k+Q2,k=0,1,...,s
2,
0,otherwise.
[
C]kl =βIm1⊗Im2l=k+Q2,k=−(N2−1),−(N2−2),...,s
2,
0,otherwise.
[
C]kl =βIm1⊗Im2,l=k+Q2,k=−(N2−1),−(N2−2),...,0,
0,otherwise.
For i=2,3,...,S
1,
[Bi]kl =
D1⊗Im2+iγ1Im1⊗Im2,l=k, k =1,2,...,S
2,
D1⊕F1+iγ1Im1⊗Im2,l=k, k =0,
0,otherwise.
For i=1,
[Bi]kl =
D1⊗Im2+iγ1Im1⊗Im2,l=k, k =1,2,...,S
2,
D1⊕F1+iγ1Im1⊗Im2,l=k, k =0,
0,otherwise.
For i=0,
[Bi]kl =D1⊗Im2,l=k, k =−(N2−1),(N2−2),...,0,
0,otherwise.
ˆ
Bkl =D1⊗Im2,l=k, k =−(N2−1),(N2−2),...,0,
0,otherwise.
˜
Bkl =D1⊗Im2,l=k, k =−(N2−1),(N2−2),...,0,
0,otherwise.
For i=1,2,...s
1,
[Ai]kl =
Im1⊗F1+kγ2Im1⊗Im2,l=k−1,k=1,2,...,S
2,
D0⊕F0−(iγ1+β)Im1⊗Im2,l=k, k =0,
D0⊕F0−(iγ1+β+kγ2)Im1⊗Im2,l=k, k =1,2,...,s
2
D0⊕F0−(iγ1+kγ2)Im1⊗Im2,l=k, k =s2+1,s
2+2,...,S
2
0,otherwise.
https://doi.org/10.17993/3cemp.2022.110250.33-48
37
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
For i=s1+1,s
1+2,...S
1,
[Ai]kl =
Im1⊗F1+kγ2Im1⊗Im2,l=k−1,k=1,2,...,S
2,
D0⊕F0−iγ1Im1⊗Im2,l=k, k =0,
D0⊕F0−(iγ1+kγ2)Im1⊗Im2,l=k, k =1,2,...,S
2
0,otherwise.
For i=0,
[Ai]kl =
D1⊕F1+kγ2Im1⊗Im2,l=k−1,k=1,2,...,S
2,
Im1⊗F1,l=k−1,k=−(N2−2),−(N2−3),...,−1,0,
(or)
l=k+N2−1,k=−(N2−1),
D0⊕F0−βIm1⊗Im2,l=k, k =−(N2−1),−(N2−2),...,0,
D0⊕F0−(β+kγ2)Im1⊗Im2,l=k, k =1,2,...,s
2,
D0⊕F0−kγ2Im1⊗Im2,l=k, k =s2+1,s
2+2,...,S
2,
0,otherwise.
Akl =
Im1⊗F1,l=k−1,k=−(N2−2),−(N2−3),...,−1,0,
(or)
l=k+N2−1,k=−(N2−1),
D0⊕F0−βIm1⊗Im2,l=k, k =−(N2−1),−(N2−2),...,0,
0,otherwise.
It may be noted that the matrices
Ai,i
=1
,
2
,...,S
1,B
i,i
=2
,
3
,...,S
1
and
C
are of size (
S2
+
1)
m1m2×
(
S2
+ 1)
m1m2,B
1
is of size (
S2
+ 1)
m1m2×
(
S1
+
N2
)
m1m2,B
0
is of size (
S2
+
N2
)
m1m2×
N2m1m2,
B
is of size
N2m1m2×N2m1m2,
B
is of size
N2m1m2×
(
S2
+
N2
)
m1m2,
C
is of size
(
S2
+
N2
)
m1m2×
(
S2
+ 1)
m1m2,
C
is of size
N2m1m2×
(
S2
+ 1)
m1m2,A
0
is of size (
S2
+
N2
)
m1m2×
(S2+N2)m1m2and
Ais of size N2m1m2×N2m1m2.
3.1 STEADY STATE ANALYSIS
It can be seen from the structure of
P
that the homogeneous Markov process
{
(
L1
(
t
)
,L
2
(
t
)
,J
1
(
t
)
,J
2
(
t
))
,t≥
0
}
on the finite state space
E
is irreducible. Hence the limiting distribu-
tion ϕ(i,k,j1,j2)=
lim
t→∞ Pr[L1(t)=i, L2(t)=k, J1(t)=j1,J
2(t)=j2|L1(0),L
2(0),J
1(0),J
2(0)]
exists. Let
ϕ(i,k,j1)=(ϕ(i,k,j1,1),ϕ
(i,k,j1,2),...,ϕ
(i,k,j1,m2)),j
1=1,2,...,m
1,
ϕ(i,k)=�ϕ(i,k,1),ϕ
(i,k,2),...,ϕ
(i,k,m1),k =−N2+1,−N2+2,...,S
2,
ϕ(i)=
(ϕ(i,0),ϕ
(i,1),...,ϕ
(i,S2)),if i=1,2,...,S
1,
(ϕ(i,−N2+1),ϕ
(i,−N2+2),...,ϕ
(i,S2)),if i=0,
(ϕ(i,−N2+1),ϕ
(i,−N2+2),...,ϕ
(i,0)),if i=−N1+1,−N1+2,...,−1.
and
Φ=(ϕ(−N1+1),ϕ(−N1+2),...,ϕ(S1−1),ϕ(S1)).
Then the vector of limiting probabilities Φsatisfies
ΦP=0and Φe=1.(1)
The first equation of the above yields the following set of equations:
https://doi.org/10.17993/3cemp.2022.110250.33-48
ϕ(i+1)
B+ϕ(i)
A=0,i=−N1+1,−N1+2,...,−2,
ϕ(i+1)Bi+1 +ϕ(i)
A=0,i=−1,
ϕ(i+1)Bi+1 +ϕ(i)Ai+ϕ(i−N1+1)
B=0,i=0,
ϕ(i+1)Bi+1 +ϕ(i)Ai=0,i=1,2,...,Q
1−N1,
ϕ(i+1)Bi+1 +ϕ(i)Ai+ϕ(i−Q1)
C=0,i=Q1−N1+1,Q
1−N1+2,...,Q
1−1,
ϕ(i+1)Bi+1 +ϕ(i)Ai+ϕ(i−Q1)
C=0,i=Q1,(2)
ϕ(i+1)Bi+1 +ϕ(i)Ai+ϕ(i−Q1)C=0,i=Q1+1,Q
1+2...,S
1−1,
ϕ(i)Ai+ϕ(i−Q1)C=0,i=S1.
The equations (except (2)) can be recursively solved to get
ϕ(i)=ϕ(Q1)θi,i=−N1+1,−N1+2,...,S
1,
where
θi=
−θi+1
B
A−1,i=−(N1−1),−(N1−2),...,−2,
−θi+1B0
A−1,i=−1,
−θi+1Bi+1 +θi−N1+1
BA−1
i,i=0,
−θi+1Bi+1A−1
i,i=1,2,...,Q
1−N1,
−θi+1Bi+1 +θi−Q1
CA−1
i,i=Q1−N1+1,Q
1−N1+2,...,Q
1−1,
I, i =Q1,
−(θi+1Bi+1 +θi−Q1C)A−1
i,i=Q1+1,Q
1+2,...,S
1−1,
−θi−Q1CA−1
i,i=S1.
Substituting the values of
θi
in equation (2) and in the normalizing condition we get the value of
ϕ(Q1).
4 SYSTEM PERFORMANCE MEASURES
In this section we derive some stationary performance measures of the system. Using these measures,
we can construct the total expected cost per unit time.
4.1 MEAN INVENTORY LEVEL
Let
ηIi
denote the mean inventory level of
i−
th commodity in the steady state (
i
=1
,
2). Since
ϕ(i,j)
is
the steady state probability vector for inventory level of first commodity is
i
and the second commodity
is j, we have
ηI1=
S1
i=1
S2
k=0
iϕ(i,k)e.
and
ηI2=
S1
i=0
S2
k=1
kϕ(i,k)e.
4.2 MEAN REORDER RATE
A reorder for both commodities is made when the joint inventory level, drops to either (
s1,s
2
)or
(
s1,j
)
,j <s
2
or (
i, s2
)
,i<s
1.
Let
ζR
denote the mean joint reorder rate for both commodities in the
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3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
38
For i=s1+1,s
1+2,...S
1,
[Ai]kl =
Im1⊗F1+kγ2Im1⊗Im2,l=k−1,k=1,2,...,S
2,
D0⊕F0−iγ1Im1⊗Im2,l=k, k =0,
D0⊕F0−(iγ1+kγ2)Im1⊗Im2,l=k, k =1,2,...,S
2
0,otherwise.
For i=0,
[Ai]kl =
D1⊕F1+kγ2Im1⊗Im2,l=k−1,k=1,2,...,S
2,
Im1⊗F1,l=k−1,k=−(N2−2),−(N2−3),...,−1,0,
(or)
l=k+N2−1,k=−(N2−1),
D0⊕F0−βIm1⊗Im2,l=k, k =−(N2−1),−(N2−2),...,0,
D0⊕F0−(β+kγ2)Im1⊗Im2,l=k, k =1,2,...,s
2,
D0⊕F0−kγ2Im1⊗Im2,l=k, k =s2+1,s
2+2,...,S
2,
0,otherwise.
Akl =
Im1⊗F1,l=k−1,k=−(N2−2),−(N2−3),...,−1,0,
(or)
l=k+N2−1,k=−(N2−1),
D0⊕F0−βIm1⊗Im2,l=k, k =−(N2−1),−(N2−2),...,0,
0,otherwise.
It may be noted that the matrices
Ai,i
=1
,
2
,...,S
1,B
i,i
=2
,
3
,...,S
1
and
C
are of size (
S2
+
1)
m1m2×
(
S2
+ 1)
m1m2,B
1
is of size (
S2
+ 1)
m1m2×
(
S1
+
N2
)
m1m2,B
0
is of size (
S2
+
N2
)
m1m2×
N2m1m2,
B
is of size
N2m1m2×N2m1m2,
B
is of size
N2m1m2×
(
S2
+
N2
)
m1m2,
C
is of size
(
S2
+
N2
)
m1m2×
(
S2
+ 1)
m1m2,
C
is of size
N2m1m2×
(
S2
+ 1)
m1m2,A
0
is of size (
S2
+
N2
)
m1m2×
(S2+N2)m1m2and
Ais of size N2m1m2×N2m1m2.
3.1 STEADY STATE ANALYSIS
It can be seen from the structure of
P
that the homogeneous Markov process
{
(
L1
(
t
)
,L
2
(
t
)
,J
1
(
t
)
,J
2
(
t
))
,t≥
0
}
on the finite state space
E
is irreducible. Hence the limiting distribu-
tion ϕ(i,k,j1,j2)=
lim
t→∞ Pr[L1(t)=i, L2(t)=k, J1(t)=j1,J
2(t)=j2|L1(0),L
2(0),J
1(0),J
2(0)]
exists. Let
ϕ(i,k,j1)=(ϕ(i,k,j1,1),ϕ
(i,k,j1,2),...,ϕ
(i,k,j1,m2)),j
1=1,2,...,m
1,
ϕ(i,k)=�ϕ(i,k,1),ϕ
(i,k,2),...,ϕ
(i,k,m1),k =−N2+1,−N2+2,...,S
2,
ϕ(i)=
(ϕ(i,0),ϕ
(i,1),...,ϕ
(i,S2)),if i=1,2,...,S
1,
(ϕ(i,−N2+1),ϕ
(i,−N2+2),...,ϕ
(i,S2)),if i=0,
(ϕ(i,−N2+1),ϕ
(i,−N2+2),...,ϕ
(i,0)),if i=−N1+1,−N1+2,...,−1.
and
Φ=(ϕ(−N1+1),ϕ(−N1+2),...,ϕ(S1−1),ϕ(S1)).
Then the vector of limiting probabilities Φsatisfies
ΦP=0and Φe=1.(1)
The first equation of the above yields the following set of equations:
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ϕ(i+1)
B+ϕ(i)
A=0,i=−N1+1,−N1+2,...,−2,
ϕ(i+1)Bi+1 +ϕ(i)
A=0,i=−1,
ϕ(i+1)Bi+1 +ϕ(i)Ai+ϕ(i−N1+1)
B=0,i=0,
ϕ(i+1)Bi+1 +ϕ(i)Ai=0,i=1,2,...,Q
1−N1,
ϕ(i+1)Bi+1 +ϕ(i)Ai+ϕ(i−Q1)
C=0,i=Q1−N1+1,Q
1−N1+2,...,Q
1−1,
ϕ(i+1)Bi+1 +ϕ(i)Ai+ϕ(i−Q1)
C=0,i=Q1,(2)
ϕ(i+1)Bi+1 +ϕ(i)Ai+ϕ(i−Q1)C=0,i=Q1+1,Q
1+2...,S
1−1,
ϕ(i)Ai+ϕ(i−Q1)C=0,i=S1.
The equations (except (2)) can be recursively solved to get
ϕ(i)=ϕ(Q1)θi,i=−N1+1,−N1+2,...,S
1,
where
θi=
−θi+1
B
A−1,i=−(N1−1),−(N1−2),...,−2,
−θi+1B0
A−1,i=−1,
−θi+1Bi+1 +θi−N1+1
BA−1
i,i=0,
−θi+1Bi+1A−1
i,i=1,2,...,Q
1−N1,
−θi+1Bi+1 +θi−Q1
CA−1
i,i=Q1−N1+1,Q
1−N1+2,...,Q
1−1,
I, i =Q1,
−(θi+1Bi+1 +θi−Q1C)A−1
i,i=Q1+1,Q
1+2,...,S
1−1,
−θi−Q1CA−1
i,i=S1.
Substituting the values of
θi
in equation (2) and in the normalizing condition we get the value of
ϕ(Q1).
4 SYSTEM PERFORMANCE MEASURES
In this section we derive some stationary performance measures of the system. Using these measures,
we can construct the total expected cost per unit time.
4.1 MEAN INVENTORY LEVEL
Let
ηIi
denote the mean inventory level of
i−
th commodity in the steady state (
i
=1
,
2). Since
ϕ(i,j)
is
the steady state probability vector for inventory level of first commodity is
i
and the second commodity
is j, we have
ηI1=
S1
i=1
S2
k=0
iϕ(i,k)e.
and
ηI2=
S1
i=0
S2
k=1
kϕ(i,k)e.
4.2 MEAN REORDER RATE
A reorder for both commodities is made when the joint inventory level, drops to either (
s1,s
2
)or
(
s1,j
)
,j <s
2
or (
i, s2
)
,i<s
1.
Let
ζR
denote the mean joint reorder rate for both commodities in the
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39
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
steady state and it is given by
ηR=1
λ1
s2
k=0
ϕ(s1+1,k)(D1⊗Im2)e+1
λ2
s1
i=0
ϕ(i,s2+1) (Im1⊗F1)e
+1
λ1
ϕ(0,s2+1) (Im1⊗F1)e+1
λ2
ϕ(s1+1,0) (D1⊗Im2)e
+(s1+ 1)γ1
s2
k=0
ϕ(s1+1,k)e+(s2+ 1)γ2
s1
i=0
ϕ(i,s2+1)e.
Let
ηRi
denote the mean individual reorder rate for commodity-
i
in the steady state (
i
=1
,
2). When
the inventory level of commodity-1 is
−
(
N1−
1)
,
a demand for commodity-1 will trigger the individual
reorder for commodity-1. Hence we get
ηR1=1
λ1
0
k=−N2+1
ϕ(−N1+1,k)(D1⊗Im2)e.
Similar arguments lead to
ηR2=1
λ2
0
i=−N1+1
ϕ(i,−N2+1) (Im1⊗F1)e.
4.3 AVERAGE BACKLOG
Let ηBidenote the mean backlog of commodity-iin the steady state (i=1,2). Then we have
ηB1=
−1
i=−N1+1
0
k=−N2+1 |i|ϕ(i,k)e.
and
ηB2=
0
i=−N1+1
−1
k=−N2+1 |k|ϕ(i,k)e.
4.4 MEAN PERISHABLE RATE
Let the mean perishable rate of commodity-
i
in the steady state de denoted by
ζFi,
(
i
=1
,
2)
.
Then we
have
ηF1=
S1
i=1
S2
k=0
iγ1ϕ(i,k)e.
and
ηF2=
S1
i=0
S2
k=1
kγ2ϕ(i,k)e.
5 COST ANALYSIS
The expected total cost per unit time (expected total cost rate) in the steady state for this model is
defined to be
TC(S1,S
2,s
1,s
2,N
1,N
2)=ch1ηI1+ch2ηI2+crηR+cr1ηR1+cr2ηR2
+cb1ηB1+cb2ηB2+cp1ηF1+cp2ηF2,
where
https://doi.org/10.17993/3cemp.2022.110250.33-48
cr: Setup cost per order.
cri: Setup cost for the i-th commodity under local purchase (i=1,2).
chi: Holding cost for the i-th commodity per unit time, i=1,2.
cpi: Perishable cost per unit item per unit time of i-th commodity (i=1,2).
cbi: Cost per unit backlog for the i-th commodity per unit time, i=1,2.
By substituting the values for η’s we can compute the value of TC(S1,S
2,s
1,s
2,N
1,N
2).
Since the evaluation of the
ϕ
’s involve recursive computations, it is quite difficult to show the
convexity of the total expected cost rate. However we present the following example to demonstrate the
computability of the results derived in our work, and to illustrate the existence of local optima when
the total cost function is treated as a function of only two variables.
6 NUMERICAL ILLUSTRATION
We consider the following numerical example : The demand for first commodity is given by (
D0,D
1
)
where
D0=−50 0
0−5,D
1=39 11
3.91.1.
The demand for second commodity is given by (F0,F
1)where
F0=−20 0
0−2,F
1=19 1
1.90.1.
In the following tables, the optimal cost for each row is shown in underlined and the optimal cost for
each column is shown in bold.
Let
γ1
=1
,γ
2
=1
,β
= 25
,s
1
=2
,s
2
=2
,N
1
=3
,N
2
=3
,c
h1
=0
.
01
,c
h2
=0
.
01
,c
r
= 75
,c
r1
=2
,c
r2
=
2,c
b1=1,c
b2=1,c
p1=2,c
p2=1.
Let TC(S1,S
2)=TC(S1,S
2,2,2,3,3).
From table 1, the numerical values shows that
TC
(
S1,S
2
)is a convex function in (
S1,S
2
)and the
Table 1 – Total Expected Cost Rate of S1and S2
S210 11 12 13 14
S1
13 9.872429 9.630808 9.596855 9.743616 9.775500
14 9.709404 9.520833 9.501561 9.633814 9.684700
15 9.594168 9.451205 9.446881 9.569767 9.634135
16 9.517610 9.413976 9.424364 9.541746 9.616449
17 9.472896 9.403231 9.427757 9.542487 9.625701
18 9.454763 9.414471 9.452342 9.566473 9.657093
19 9.459070 9.444205 9.494506 9.609474 9.706772
(possibly local) optimum occurs at (S1,S
2)=(17,11).
Let
γ1
=0
.
01
,γ
2
=0
.
8
,β
= 18
,S
2
= 20
,s
2
=3
,N
1
=3
,N
2
=3
,c
h1
=0
.
01
,c
h2
=0
.
01
,c
r
=0
.
55
,c
r1
=
0.45,c
r2=0.5,c
b1=0.1,c
b2=0.1,c
p1=0.1,c
p2=0.4.
Let TC(S1,s
1)=TC(S1,20,s
1,3,3,3).
From table 2, the numerical values shows that
TC
(
S1,s
1
)is a convex function in (
S1,s
1
)and the
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3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
40
steady state and it is given by
ηR=1
λ1
s2
k=0
ϕ(s1+1,k)(D1⊗Im2)e+1
λ2
s1
i=0
ϕ(i,s2+1) (Im1⊗F1)e
+1
λ1
ϕ(0,s2+1) (Im1⊗F1)e+1
λ2
ϕ(s1+1,0) (D1⊗Im2)e
+(s1+ 1)γ1
s2
k=0
ϕ(s1+1,k)e+(s2+ 1)γ2
s1
i=0
ϕ(i,s2+1)e.
Let
ηRi
denote the mean individual reorder rate for commodity-
i
in the steady state (
i
=1
,
2). When
the inventory level of commodity-1 is
−
(
N1−
1)
,
a demand for commodity-1 will trigger the individual
reorder for commodity-1. Hence we get
ηR1=1
λ1
0
k=−N2+1
ϕ(−N1+1,k)(D1⊗Im2)e.
Similar arguments lead to
ηR2=1
λ2
0
i=−N1+1
ϕ(i,−N2+1) (Im1⊗F1)e.
4.3 AVERAGE BACKLOG
Let ηBidenote the mean backlog of commodity-iin the steady state (i=1,2). Then we have
ηB1=
−1
i=−N1+1
0
k=−N2+1 |i|ϕ(i,k)e.
and
ηB2=
0
i=−N1+1
−1
k=−N2+1 |k|ϕ(i,k)e.
4.4 MEAN PERISHABLE RATE
Let the mean perishable rate of commodity-
i
in the steady state de denoted by
ζFi,
(
i
=1
,
2)
.
Then we
have
ηF1=
S1
i=1
S2
k=0
iγ1ϕ(i,k)e.
and
ηF2=
S1
i=0
S2
k=1
kγ2ϕ(i,k)e.
5 COST ANALYSIS
The expected total cost per unit time (expected total cost rate) in the steady state for this model is
defined to be
TC(S1,S
2,s
1,s
2,N
1,N
2)=ch1ηI1+ch2ηI2+crηR+cr1ηR1+cr2ηR2
+cb1ηB1+cb2ηB2+cp1ηF1+cp2ηF2,
where
https://doi.org/10.17993/3cemp.2022.110250.33-48
cr: Setup cost per order.
cri: Setup cost for the i-th commodity under local purchase (i=1,2).
chi: Holding cost for the i-th commodity per unit time, i=1,2.
cpi: Perishable cost per unit item per unit time of i-th commodity (i=1,2).
cbi: Cost per unit backlog for the i-th commodity per unit time, i=1,2.
By substituting the values for η’s we can compute the value of TC(S1,S
2,s
1,s
2,N
1,N
2).
Since the evaluation of the
ϕ
’s involve recursive computations, it is quite difficult to show the
convexity of the total expected cost rate. However we present the following example to demonstrate the
computability of the results derived in our work, and to illustrate the existence of local optima when
the total cost function is treated as a function of only two variables.
6 NUMERICAL ILLUSTRATION
We consider the following numerical example : The demand for first commodity is given by (
D0,D
1
)
where
D0=−50 0
0−5,D
1=39 11
3.91.1.
The demand for second commodity is given by (F0,F
1)where
F0=−20 0
0−2,F
1=19 1
1.90.1.
In the following tables, the optimal cost for each row is shown in underlined and the optimal cost for
each column is shown in bold.
Let
γ1
=1
,γ
2
=1
,β
= 25
,s
1
=2
,s
2
=2
,N
1
=3
,N
2
=3
,c
h1
=0
.
01
,c
h2
=0
.
01
,c
r
= 75
,c
r1
=2
,c
r2
=
2,c
b1=1,c
b2=1,c
p1=2,c
p2=1.
Let TC(S1,S
2)=TC(S1,S
2,2,2,3,3).
From table 1, the numerical values shows that
TC
(
S1,S
2
)is a convex function in (
S1,S
2
)and the
Table 1 – Total Expected Cost Rate of S1and S2
S210 11 12 13 14
S1
13 9.872429 9.630808 9.596855 9.743616 9.775500
14 9.709404 9.520833 9.501561 9.633814 9.684700
15 9.594168 9.451205 9.446881 9.569767 9.634135
16 9.517610 9.413976 9.424364 9.541746 9.616449
17 9.472896 9.403231 9.427757 9.542487 9.625701
18 9.454763 9.414471 9.452342 9.566473 9.657093
19 9.459070 9.444205 9.494506 9.609474 9.706772
(possibly local) optimum occurs at (S1,S
2)=(17,11).
Let
γ1
=0
.
01
,γ
2
=0
.
8
,β
= 18
,S
2
= 20
,s
2
=3
,N
1
=3
,N
2
=3
,c
h1
=0
.
01
,c
h2
=0
.
01
,c
r
=0
.
55
,c
r1
=
0.45,c
r2=0.5,c
b1=0.1,c
b2=0.1,c
p1=0.1,c
p2=0.4.
Let TC(S1,s
1)=TC(S1,20,s
1,3,3,3).
From table 2, the numerical values shows that
TC
(
S1,s
1
)is a convex function in (
S1,s
1
)and the
https://doi.org/10.17993/3cemp.2022.110250.33-48
41
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
Table 2 – Total Expected Cost Rate of S1and s1
s145678
S1
49 5.076500 5.075467 5.080127 5.088899 5.100611
50 5.076459 5.075080 5.079444 5.087956 5.099435
51 5.076589 5.074872 5.078945 5.087203 5.098452
52 5.076888 5.074837 5.078626 5.086635 5.097661
53 5.077351 5.074974 5.078484 5.086248 5.097055
54 5.077976 5.075278 5.078514 5.086039 5.096631
55 5.078760 5.075747 5.078714 5.086004 5.096385
56 5.079700 5.076377 5.079080 5.086140 5.096314
57 5.080793 5.077165 5.079609 5.086443 5.096414
(possibly local) optimum occurs at (S1,s
1) = (52,5).
let
γ1
=0
.
01
,γ
2
=0
.
9
,β
= 10
,S
2
= 20
,s
2
=2
,S
1
= 20
,s
1
=2
,c
h1
=0
.
01
,c
h2
=0
.
01
,c
r
= 21
,c
r1
=
15,c
r2= 18,c
b1=5,c
b2=5,c
p1=0.8,c
p2=0.75.
Let TC(N1,N
2)=TC(20,20,2,2,N
1,N
2).
From table 3, the numerical values shows that
TC
(
N1,N
2
)is a convex function in (
N1,N
2
)and the
Table 3 – Total Expected Cost Rate of N1and N2
N234567
N1
4 10.565306 10.543407 10.533310 10.531192 10.535652
5 10.509154 10.491453 10.486547 10.490709 10.502146
6 10.476386 10.465856 10.468392 10.481047 10.501967
7 10.456879 10.454703 10.465625 10.487919 10.520371
810.453162 10.449303 10.468540 10.500486 10.545501
9 10.461411 10.459465 10.492293 10.541267 10.609235
(possibly local) optimum occurs at (N1,N
2)=(8,4).
Let
γ1
=0
.
1
,γ
2
=0
.
8
,β
= 18
,S
1
= 20
,s
1
=3
,s
2
=2
,N
1
= 3;
ch1
=0
.
1
,c
h2
=0
.
1
,c
r
=0
.
11
,c
r1
=
0.1,c
r2=0.1,c
b1=0.1,c
b2=0.1,c
p1=0.1,c
p2=0.1.
Let TC(S2,N
2)=TC(20,S
2,3,2,3,N
2).
From table 4, the numerical values shows that
TC
(
S2,N
2
)is a convex function in (
S2,N
2
)and the
Table 4 – Total Expected Cost Rate of S2and N2
N256 789
S2
39 8.654678 8.653849 8.654060 8.654295 8.654545
40 8.517849 8.517004 8.517197 8.517414 8.517645
41 8.689863 8.680004 8.680181 8.680380 8.680593
42 8.843738 8.842865 8.843027 8.843210 8.843407
43 9.005788 9.005604 9.005752 9.005920 9.006102
https://doi.org/10.17993/3cemp.2022.110250.33-48
(possibly local) optimum occurs at (S2,N
2)=(40,6).
Let
γ1
=0
.
1
,γ
2
=0
.
8
,β
= 18
,S
1
= 20
,s
1
=3
,N
2
=3
,N
1
= 3;
ch1
=0
.
1
,c
h2
=0
.
1
,c
r
=0
.
11
,c
r1
=
0.1,c
r2=0.1,c
b1=0.1,c
b2=0.1,c
p1=0.1,c
p2=0.1.
Let TC(S2,s
2)=TC(20,S
2,3,s
2,3,3).
From table 5, the numerical values shows that
TC
(
S2,s
2
)is a convex function in (
S2,s
2
)and the
Table 5 – Total Expected Cost Rate of S2and s2
s223 456
S2
39 8.656617 8.655266 8.656616 8.657680 8.658492
40 8.519803 8.518349 8.519626 8.520642 8.521425
41 8.699830 8.681279 8.682486 8.683454 8.684207
42 8.846717 8.844073 8.845214 8.846135 8.846858
43 9.007478 9.006747 9.007824 9.008700 9.009393
(possibly local) optimum occurs at (S2,s
2) = (40,3).
Let
γ1
=0
.
01
,γ
2
=0
.
8
,β
= 18
,S
2
= 20
,s
1
=2
,s
2
=3
,N
2
= 3;
ch1
=0
.
01
,c
h2
=0
.
01
,c
r
=0
.
55
,c
r1
=
0.45,c
r2=0.5,c
b1=0.1,c
b2=0.1,c
p1=0.1,c
p2=0.4.
Let TC(S1,N
1)=TC(S1,20,2,3,N
1,3).
From table 6, the numerical values shows that
TC
(
S1,N
1
)is a convex function in (
S1,N
1
)and the
Table 6 – Total Expected Cost Rate of S1and N1
N156789
S1
49 5.034028 5.023524 5.021157 5.023724 5.029325
50 5.033895 5.023100 5.020510 5.022892 5.028327
51 5.033935 5.022856 5.020048 5.022249 5.027524
52 5.034146 5.022789 5.019767 5.021791 5.026910
53 5.034523 5.022894 5.019662 5.021515 5.026482
54 5.035065 5.023168 5.019732 5.021415 5.026235
55 5.035768 5.023608 5.019971 5.021490 5.026165
56 5.036629 5.024210 5.020377 5.021734 5.026268
(possibly local) optimum occurs at (S1,N
1)=(53,7).
The Figure 1 grants the impact of the perishable rate
γ1
, on the total expected cost rate TC via four
curves which relate to
β
= 18.5,18.6,18.7,18.8. Since figure 1, we perceive that the total cost value
decreases when the perishable rate γ1and the replenishment rate βincreases.
The Figure 2 grants the impact of the perishable rate
γ2
, on the total expected cost rate TC via
three curves which relate to
β
= 19,20 and 21. Since figure 2, we perceive that the total cost value
decreases when the perishable rate γ2and the replenishment rate βincreases.
In tables 7 and 8, we show that the impact of the cost values on the optimal values (
S∗
1,s
∗
1
)and the
corresponding total expected cost rate. Towards this end, we first fix the parameters and cost value as
S2= 20,s
2=3,N
1=3,N
2=3,β = 18,γ
1=0.01,γ
2=0.8,c
r1=0.45,c
r2=0.5.
https://doi.org/10.17993/3cemp.2022.110250.33-48
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
42
Table 2 – Total Expected Cost Rate of S1and s1
s145678
S1
49 5.076500 5.075467 5.080127 5.088899 5.100611
50 5.076459 5.075080 5.079444 5.087956 5.099435
51 5.076589 5.074872 5.078945 5.087203 5.098452
52 5.076888 5.074837 5.078626 5.086635 5.097661
53 5.077351 5.074974 5.078484 5.086248 5.097055
54 5.077976 5.075278 5.078514 5.086039 5.096631
55 5.078760 5.075747 5.078714 5.086004 5.096385
56 5.079700 5.076377 5.079080 5.086140 5.096314
57 5.080793 5.077165 5.079609 5.086443 5.096414
(possibly local) optimum occurs at (S1,s
1) = (52,5).
let
γ1
=0
.
01
,γ
2
=0
.
9
,β
= 10
,S
2
= 20
,s
2
=2
,S
1
= 20
,s
1
=2
,c
h1
=0
.
01
,c
h2
=0
.
01
,c
r
= 21
,c
r1
=
15,c
r2= 18,c
b1=5,c
b2=5,c
p1=0.8,c
p2=0.75.
Let TC(N1,N
2)=TC(20,20,2,2,N
1,N
2).
From table 3, the numerical values shows that
TC
(
N1,N
2
)is a convex function in (
N1,N
2
)and the
Table 3 – Total Expected Cost Rate of N1and N2
N234567
N1
4 10.565306 10.543407 10.533310 10.531192 10.535652
5 10.509154 10.491453 10.486547 10.490709 10.502146
6 10.476386 10.465856 10.468392 10.481047 10.501967
7 10.456879 10.454703 10.465625 10.487919 10.520371
810.453162 10.449303 10.468540 10.500486 10.545501
9 10.461411 10.459465 10.492293 10.541267 10.609235
(possibly local) optimum occurs at (N1,N
2)=(8,4).
Let
γ1
=0
.
1
,γ
2
=0
.
8
,β
= 18
,S
1
= 20
,s
1
=3
,s
2
=2
,N
1
= 3;
ch1
=0
.
1
,c
h2
=0
.
1
,c
r
=0
.
11
,c
r1
=
0.1,c
r2=0.1,c
b1=0.1,c
b2=0.1,c
p1=0.1,c
p2=0.1.
Let TC(S2,N
2)=TC(20,S
2,3,2,3,N
2).
From table 4, the numerical values shows that
TC
(
S2,N
2
)is a convex function in (
S2,N
2
)and the
Table 4 – Total Expected Cost Rate of S2and N2
N256 789
S2
39 8.654678 8.653849 8.654060 8.654295 8.654545
40 8.517849 8.517004 8.517197 8.517414 8.517645
41 8.689863 8.680004 8.680181 8.680380 8.680593
42 8.843738 8.842865 8.843027 8.843210 8.843407
43 9.005788 9.005604 9.005752 9.005920 9.006102
https://doi.org/10.17993/3cemp.2022.110250.33-48
(possibly local) optimum occurs at (S2,N
2)=(40,6).
Let
γ1
=0
.
1
,γ
2
=0
.
8
,β
= 18
,S
1
= 20
,s
1
=3
,N
2
=3
,N
1
= 3;
ch1
=0
.
1
,c
h2
=0
.
1
,c
r
=0
.
11
,c
r1
=
0.1,c
r2=0.1,c
b1=0.1,c
b2=0.1,c
p1=0.1,c
p2=0.1.
Let TC(S2,s
2)=TC(20,S
2,3,s
2,3,3).
From table 5, the numerical values shows that
TC
(
S2,s
2
)is a convex function in (
S2,s
2
)and the
Table 5 – Total Expected Cost Rate of S2and s2
s22 3 4 5 6
S2
39 8.656617 8.655266 8.656616 8.657680 8.658492
40 8.519803 8.518349 8.519626 8.520642 8.521425
41 8.699830 8.681279 8.682486 8.683454 8.684207
42 8.846717 8.844073 8.845214 8.846135 8.846858
43 9.007478 9.006747 9.007824 9.008700 9.009393
(possibly local) optimum occurs at (S2,s
2) = (40,3).
Let
γ1
=0
.
01
,γ
2
=0
.
8
,β
= 18
,S
2
= 20
,s
1
=2
,s
2
=3
,N
2
= 3;
ch1
=0
.
01
,c
h2
=0
.
01
,c
r
=0
.
55
,c
r1
=
0.45,c
r2=0.5,c
b1=0.1,c
b2=0.1,c
p1=0.1,c
p2=0.4.
Let TC(S1,N
1)=TC(S1,20,2,3,N
1,3).
From table 6, the numerical values shows that
TC
(
S1,N
1
)is a convex function in (
S1,N
1
)and the
Table 6 – Total Expected Cost Rate of S1and N1
N156789
S1
49 5.034028 5.023524 5.021157 5.023724 5.029325
50 5.033895 5.023100 5.020510 5.022892 5.028327
51 5.033935 5.022856 5.020048 5.022249 5.027524
52 5.034146 5.022789 5.019767 5.021791 5.026910
53 5.034523 5.022894 5.019662 5.021515 5.026482
54 5.035065 5.023168 5.019732 5.021415 5.026235
55 5.035768 5.023608 5.019971 5.021490 5.026165
56 5.036629 5.024210 5.020377 5.021734 5.026268
(possibly local) optimum occurs at (S1,N
1)=(53,7).
The Figure 1 grants the impact of the perishable rate
γ1
, on the total expected cost rate TC via four
curves which relate to
β
= 18.5,18.6,18.7,18.8. Since figure 1, we perceive that the total cost value
decreases when the perishable rate γ1and the replenishment rate βincreases.
The Figure 2 grants the impact of the perishable rate
γ2
, on the total expected cost rate TC via
three curves which relate to
β
= 19,20 and 21. Since figure 2, we perceive that the total cost value
decreases when the perishable rate γ2and the replenishment rate βincreases.
In tables 7 and 8, we show that the impact of the cost values on the optimal values (
S∗
1,s
∗
1
)and the
corresponding total expected cost rate. Towards this end, we first fix the parameters and cost value as
S2= 20,s
2=3,N
1=3,N
2=3,β = 18,γ
1=0.01,γ
2=0.8,c
r1=0.45,c
r2=0.5.
https://doi.org/10.17993/3cemp.2022.110250.33-48
43
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
Table 7 – Impact of Cost Values
Ch20.01
Cp20.4 0.5
Cb20.09 0.1 0.11 0.09 0.1 0.11
CrCh1Cp1Cb1
0.4 0.01 0.10 0.09 55 6 55 6 55 6 55 6 55 6 55 6
5.0730 5.0733 5.0736 6.1763 6.1765 6.1768
0.10 55 6 55 6 55 6 55 6 55 6 55 6
5.0734 5.0737 5.0740 6.1767 6.1770 6.1773
0.11 55 6 55 6 55 6 55 6 55 6 55 6
5.0735 5.0738 5.0741 6.1768 6.1771 6.1774
0.20 0.09 55 6 55 6 55 6 55 6 55 6 55 6
5.1261 5.1264 5.1267 6.2202 6.2205 6.2208
0.10 55 6 54 6 54 6 54 6 54 6 54 6
5.1266 5.1269 5.1272 6.2206 6.2209 6.2212
0.11 54 6 54 6 54 6 54 6 54 6 54 6
5.1271 5.1273 5.1276 6.2211 6.2214 6.2217
0.02 0.10 0.09 54 6 54 6 54 6 53 6 53 6 53 6
5.5517 5.5520 5.5523 6.6793 6.6795 6.6798
0.10 54 6 54 6 54 6 53 6 53 6 53 6
5.5522 5.5524 5.5527 6.6797 6.6800 6.6873
0.11 54 6 54 5 54 5 53 5 53 5 53 5
5.5526 5.5529 5.5532 6.6817 6.6819 6.6893
0.20 0.09 54 6 53 5 53 5 53 5 53 5 53 5
5.6281 5.6284 5.6287 6.6822 6.6835 6.6900
0.10 53 6 53 5 53 5 53 5 53 5 53 5
5.6296 5.6299 5.6309 6.6826 6.6837 6.6912
0.11 53 6 53 5 53 5 53 5 53 5 52 5
5.6371 5.6373 5.6376 6.6829 6.6838 6.6915
0.5 0.01 0.10 0.09 55 6 55 6 55 6 55 6 55 6 55 6
5.6462 5.6468 5.6471 6.6983 6.6985 6.6988
0.10 55 6 55 6 55 6 55 6 55 6 55 6
5.6771 5.6777 5.6780 6.7067 6.7079 6.7083
0.11 55 6 55 6 55 6 55 6 55 6 55 6
5.6784 5.6787 5.6790 6.7177 6.7178 6.7179
0.20 0.09 55 6 55 6 55 6 55 6 55 6 55 6
5.6861 5.6884 5.6887 6.7202 6.7265 6.7268
0.10 55 6 54 6 54 6 54 6 54 6 54 6
5.6886 5.6889 5.6890 6.7566 6.7569 6.7570
0.11 54 6 54 6 54 6 54 6 54 6 54 6
5.6887 5.6890 5.6891 6.7571 6.7574 6.7577
0.02 0.10 0.09 54 6 54 6 54 6 53 6 53 6 53 6
5.6892 5.6894 5.6896 6.7692 6.7698 6.7791
0.10 54 6 54 6 54 6 53 6 53 6 53 6
5.6893 5.6902 5.6907 6.7857 6.7860 6.7893
0.11 54 6 54 5 54 5 53 5 53 5 53 5
5.6896 5.6909 5.6912 6.7919 6.7942 6.7958
0.20 0.09 54 5 53 5 53 5 53 5 53 5 53 5
5.7281 5.7284 5.7287 6.8012 6.8015 6.8018
0.10 53 6 53 5 53 5 53 5 53 5 53 5
5.7296 5.7299 5.7309 6.8026 6.8029 6.8032
0.11 53 6 53 5 53 5 53 5 53 5 52 5
5.8371 5.8373 5.8376 6.8036 6.8044 6.8047
https://doi.org/10.17993/3cemp.2022.110250.33-48
Table 8 – Impact of Cost Value
Ch20.02
Cp20.4 0.5
Cb20.09 0.1 0.11 0.09 0.1 0.11
CrCh1Cp1Cb1
0.4 0.01 0.10 0.09 54 6 54 6 54 6 54 6 54 6 54 6
5.2189 5.2191 5.2194 6.2197 6.2213 6.2232
0.10 54 6 54 6 54 6 54 6 54 6 54 6
5.2196 5.2199 5.3101 6.2367 6.2370 6.2373
0.11 54 6 54 6 54 6 54 6 54 6 54 6
5.2234 5.2237 5.3240 6.2767 6.2770 6.2773
0.20 0.09 54 6 54 6 54 6 54 6 54 6 54 6
5.2281 5.2284 5.3287 6.3902 6.3913 6.3956
0.10 53 6 53 6 53 6 53 6 53 6 53 6
5.3296 5.3299 5.3302 6.4106 6.4109 6.4112
0.11 53 6 53 6 53 6 53 6 53 6 53 6
5.3311 5.3323 5.3346 6.4152 6.4163 6.4177
0.02 0.10 0.09 53 6 53 6 53 6 52 6 52 6 52 6
5.6517 5.6520 5.6523 6.7793 6.7795 6.7798
0.10 53 6 53 6 53 6 52 6 52 6 52 6
5.6522 5.6524 5.6527 6.7797 6.7800 6.7873
0.11 53 5 53 5 53 5 52 5 52 5 52 5
5.6526 5.6529 5.6532 6.7817 6.7819 6.7883
0.20 0.09 53 5 53 5 53 5 52 5 52 5 52 5
5.7281 5.7284 5.7287 6.7912 6.7915 6.7918
0.10 52 5 52 5 52 5 52 5 52 5 52 5
5.7296 5.7299 5.7309 6.8026 6.8029 6.8112
0.11 52 5 52 5 52 5 52 5 52 5 52 5
5.7371 5.7373 5.7376 6.8116 6.8134 6.8147
0.5 0.01 0.10 0.09 54 6 54 6 54 6 54 6 54 6 54 6
5.8762 5.8768 5.8771 6.8783 6.8785 6.8788
0.10 54 6 54 6 54 6 54 6 54 6 54 6
5.8771 5.8777 5.8780 6.9767 6.9790 6.9793
0.11 54 6 54 6 54 6 54 6 54 6 54 6
5.8784 5.8787 5.8790 6.9777 6.9870 6.9873
0.20 0.09 54 6 54 6 54 6 54 6 54 6 54 6
6.1761 6.1784 6.1787 7.2202 7.2565 7.2568
0.10 53 6 53 6 53 6 53 6 53 6 53 6
6.1786 6.1789 6.1792 7.2566 7.2569 7.2572
0.11 53 6 53 6 53 6 53 6 53 6 53 6
6.1791 6.1793 6.1796 7.2671 7.2674 7.2677
0.02 0.10 0.09 53 6 53 6 53 6 52 6 52 6 52 6
6.5627 6.5670 6.5823 7.6797 7.6894 7.6898
0.10 53 6 53 6 53 6 52 6 52 6 52 6
6.5792 6.5802 6.5907 7.6857 7.6920 7.6923
0.11 53 5 53 5 53 5 52 5 52 5 52 5
6.5906 6.5929 6.5942 7.6919 7.6942 7.6958
0.20 0.09 53 5 53 5 53 5 52 5 52 5 52 5
6.7281 6.7284 6.7287 7.7312 7.7315 7.7318
0.10 52 5 52 5 52 5 52 5 52 5 52 5
6.7296 6.7299 6.7309 7.7426 7.7429 7.7512
0.11 52 5 52 5 52 5 52 5 52 5 52 5
6.8371 6.8373 6.8376 7.7516 7.7534 7.7547
https://doi.org/10.17993/3cemp.2022.110250.33-48
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
44
Table 7 – Impact of Cost Values
Ch20.01
Cp20.4 0.5
Cb20.09 0.1 0.11 0.09 0.1 0.11
CrCh1Cp1Cb1
0.4 0.01 0.10 0.09 55 6 55 6 55 6 55 6 55 6 55 6
5.0730 5.0733 5.0736 6.1763 6.1765 6.1768
0.10 55 6 55 6 55 6 55 6 55 6 55 6
5.0734 5.0737 5.0740 6.1767 6.1770 6.1773
0.11 55 6 55 6 55 6 55 6 55 6 55 6
5.0735 5.0738 5.0741 6.1768 6.1771 6.1774
0.20 0.09 55 6 55 6 55 6 55 6 55 6 55 6
5.1261 5.1264 5.1267 6.2202 6.2205 6.2208
0.10 55 6 54 6 54 6 54 6 54 6 54 6
5.1266 5.1269 5.1272 6.2206 6.2209 6.2212
0.11 54 6 54 6 54 6 54 6 54 6 54 6
5.1271 5.1273 5.1276 6.2211 6.2214 6.2217
0.02 0.10 0.09 54 6 54 6 54 6 53 6 53 6 53 6
5.5517 5.5520 5.5523 6.6793 6.6795 6.6798
0.10 54 6 54 6 54 6 53 6 53 6 53 6
5.5522 5.5524 5.5527 6.6797 6.6800 6.6873
0.11 54 6 54 5 54 5 53 5 53 5 53 5
5.5526 5.5529 5.5532 6.6817 6.6819 6.6893
0.20 0.09 54 6 53 5 53 5 53 5 53 5 53 5
5.6281 5.6284 5.6287 6.6822 6.6835 6.6900
0.10 53 6 53 5 53 5 53 5 53 5 53 5
5.6296 5.6299 5.6309 6.6826 6.6837 6.6912
0.11 53 6 53 5 53 5 53 5 53 5 52 5
5.6371 5.6373 5.6376 6.6829 6.6838 6.6915
0.5 0.01 0.10 0.09 55 6 55 6 55 6 55 6 55 6 55 6
5.6462 5.6468 5.6471 6.6983 6.6985 6.6988
0.10 55 6 55 6 55 6 55 6 55 6 55 6
5.6771 5.6777 5.6780 6.7067 6.7079 6.7083
0.11 55 6 55 6 55 6 55 6 55 6 55 6
5.6784 5.6787 5.6790 6.7177 6.7178 6.7179
0.20 0.09 55 6 55 6 55 6 55 6 55 6 55 6
5.6861 5.6884 5.6887 6.7202 6.7265 6.7268
0.10 55 6 54 6 54 6 54 6 54 6 54 6
5.6886 5.6889 5.6890 6.7566 6.7569 6.7570
0.11 54 6 54 6 54 6 54 6 54 6 54 6
5.6887 5.6890 5.6891 6.7571 6.7574 6.7577
0.02 0.10 0.09 54 6 54 6 54 6 53 6 53 6 53 6
5.6892 5.6894 5.6896 6.7692 6.7698 6.7791
0.10 54 6 54 6 54 6 53 6 53 6 53 6
5.6893 5.6902 5.6907 6.7857 6.7860 6.7893
0.11 54 6 54 5 54 5 53 5 53 5 53 5
5.6896 5.6909 5.6912 6.7919 6.7942 6.7958
0.20 0.09 54 5 53 5 53 5 53 5 53 5 53 5
5.7281 5.7284 5.7287 6.8012 6.8015 6.8018
0.10 53 6 53 5 53 5 53 5 53 5 53 5
5.7296 5.7299 5.7309 6.8026 6.8029 6.8032
0.11 53 6 53 5 53 5 53 5 53 5 52 5
5.8371 5.8373 5.8376 6.8036 6.8044 6.8047
https://doi.org/10.17993/3cemp.2022.110250.33-48
Table 8 – Impact of Cost Value
Ch20.02
Cp20.4 0.5
Cb20.09 0.1 0.11 0.09 0.1 0.11
CrCh1Cp1Cb1
0.4 0.01 0.10 0.09 54 6 54 6 54 6 54 6 54 6 54 6
5.2189 5.2191 5.2194 6.2197 6.2213 6.2232
0.10 54 6 54 6 54 6 54 6 54 6 54 6
5.2196 5.2199 5.3101 6.2367 6.2370 6.2373
0.11 54 6 54 6 54 6 54 6 54 6 54 6
5.2234 5.2237 5.3240 6.2767 6.2770 6.2773
0.20 0.09 54 6 54 6 54 6 54 6 54 6 54 6
5.2281 5.2284 5.3287 6.3902 6.3913 6.3956
0.10 53 6 53 6 53 6 53 6 53 6 53 6
5.3296 5.3299 5.3302 6.4106 6.4109 6.4112
0.11 53 6 53 6 53 6 53 6 53 6 53 6
5.3311 5.3323 5.3346 6.4152 6.4163 6.4177
0.02 0.10 0.09 53 6 53 6 53 6 52 6 52 6 52 6
5.6517 5.6520 5.6523 6.7793 6.7795 6.7798
0.10 53 6 53 6 53 6 52 6 52 6 52 6
5.6522 5.6524 5.6527 6.7797 6.7800 6.7873
0.11 53 5 53 5 53 5 52 5 52 5 52 5
5.6526 5.6529 5.6532 6.7817 6.7819 6.7883
0.20 0.09 53 5 53 5 53 5 52 5 52 5 52 5
5.7281 5.7284 5.7287 6.7912 6.7915 6.7918
0.10 52 5 52 5 52 5 52 5 52 5 52 5
5.7296 5.7299 5.7309 6.8026 6.8029 6.8112
0.11 52 5 52 5 52 5 52 5 52 5 52 5
5.7371 5.7373 5.7376 6.8116 6.8134 6.8147
0.5 0.01 0.10 0.09 54 6 54 6 54 6 54 6 54 6 54 6
5.8762 5.8768 5.8771 6.8783 6.8785 6.8788
0.10 54 6 54 6 54 6 54 6 54 6 54 6
5.8771 5.8777 5.8780 6.9767 6.9790 6.9793
0.11 54 6 54 6 54 6 54 6 54 6 54 6
5.8784 5.8787 5.8790 6.9777 6.9870 6.9873
0.20 0.09 54 6 54 6 54 6 54 6 54 6 54 6
6.1761 6.1784 6.1787 7.2202 7.2565 7.2568
0.10 53 6 53 6 53 6 53 6 53 6 53 6
6.1786 6.1789 6.1792 7.2566 7.2569 7.2572
0.11 53 6 53 6 53 6 53 6 53 6 53 6
6.1791 6.1793 6.1796 7.2671 7.2674 7.2677
0.02 0.10 0.09 53 6 53 6 53 6 52 6 52 6 52 6
6.5627 6.5670 6.5823 7.6797 7.6894 7.6898
0.10 53 6 53 6 53 6 52 6 52 6 52 6
6.5792 6.5802 6.5907 7.6857 7.6920 7.6923
0.11 53 5 53 5 53 5 52 5 52 5 52 5
6.5906 6.5929 6.5942 7.6919 7.6942 7.6958
0.20 0.09 53 5 53 5 53 5 52 5 52 5 52 5
6.7281 6.7284 6.7287 7.7312 7.7315 7.7318
0.10 52 5 52 5 52 5 52 5 52 5 52 5
6.7296 6.7299 6.7309 7.7426 7.7429 7.7512
0.11 52 5 52 5 52 5 52 5 52 5 52 5
6.8371 6.8373 6.8376 7.7516 7.7534 7.7547
https://doi.org/10.17993/3cemp.2022.110250.33-48
45
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
Figure 1 – TC versus γ1
γ2=0.8,S
1= 52,s1=5,S
2= 20,s
2=3,N
1=3,N
2=3,c
h1=0.01,c
h2=0.01,c
r=0.55,c
r1=0.45,c
r2=0.5,c
b1=0.1,c
b2=
0.1,c
p1=0.1,c
p2=0.4..
Figure 2 – TC versus γ2
γ1=0.01,S
1= 52,s
1=5,S
2= 20,s
2=3,N
1=3,N
2=3,c
h1=0.01,c
h2=0.01,c
r=0.55,c
r1=0.45,c
r2=0.5,c
b1=0.1,c
b2=
0.1,c
p1=0.1,c
p2=0.4.
7 CONCLUSION
In this article, we examined the substitutable perishable inventory system. Specifically, we analyzed the
structure of the system performance that takes place when a local purchase is made to clear the backlog
instantaneously if both commodities have reached zero and demand is backlogged up to predetermined
levels. Arriving customers follow a Markovian arrival process. The commodities are assumed to be
substitutable. If both commodities have reached zero, demand is backlogged up to predetermined levels.
Graphical results of perishable rates and replenishment rates had been presented. This shows that if
the perishable and replenished rate increases then the total cost would increases. The results of the
contribution were illustrated using numerical patterns to estimate the convexity of the overall cost rate
of this system. The impact of cost values on total expected cost rate were shown. In the future, our
proposed model can be expanded by various reordering policies and described by real data values.
ACKNOWLEDGMENT
Anbazhagan and Amutha would like to thank RUSA Phase 2.0 (F 24-51/2014-U), DST-FIST (SR/FIST/MS-
I/2018/17), DST-PURSE 2nd Phase programme (SR/PURSE Phase 2/38), Govt. of India.
REFERENCES
[1]
Adak Sudip, and Mahapatra, G.S.,(2020). Effect of reliability on multi-item inventory system
with shortages and partial backlog incorporating time dependent demand and deterioration. Ann.
https://doi.org/10.17993/3cemp.2022.110250.33-48
Oper. Res. https://doi.org/10.1007/s10479-020-03694-6.
[2]
Anbazhagan, N., and Arivarignan, G.(2001). Analysis of two commodity Markovian iventory
system with lead time. Korean J. Comput. Appl. Math., 8(2),427-438 .
[3]
Anbazhagan, N., Arivarignan, G., and Irle, A. (2012). A Two-commodity continuous review
inventory system with substitutable items. Stoch. Anal. Appl., 30, 1-19.
[4]
Anbazhagan, N., Goh, M., and Vigneshwaran, B.(2015). Substitutable inventory systems
with coordinated reorder levels. Stat. Appl. Prob., 2(3), 221-234.
[5]
Balintfy, J.L.(1964). On a basic class of multi-item inventory problems. Manag. Sci., 10(2),
287-297.
[6]
Benny, B., Chakravarthy, S.R., and Krishnamoorthy, A. (2018). Queueing-Inventory System
with Two Commodities. J. Indian Soc. Probab. Stat., 19(2), 437-454.
[7]
C
´a
rdenas-Barr
´o
n Leopoldo, Chung Kun-Jen, Kazemi Nima and Shekarian Eh-
san.(2018).Optimal inventory system with two backlog costs in response to a discount offer: corrections
and complements. Oper. Res. Int. J., https://doi.org/18. 10.1007/s12351-016-0255-8.
[8]
Chen, K., Xiao, T., Wang, S., and Lei, D. (2021). Inventory strategies for perishable products
with two-period shelf-life and lost sales. International Journal of Production Research, 59(17),
5301-5320.
[9]
Helal Md Abu, Bensoussan Alain, Ramakrishna Viswanath, and Sethi Suresh, P. (2021).
A mathematical method for optimal inventory policies With backlog sales. Int. J. Transp. Eng.,
11(2), 323 - 340 .
[10]
Karthikeyan, K., and Sudhesh, R. (2016). Recent review article on queueing inventory systems.
Res. J. Pharm. Technol., 9(11), 1451-1461 .
[11]
Khan, M.A.A., Shaikh, A.A., C
´a
rdenas-Barr
´o
n, L.E., Mashud, A.H.M., Trevi
˜n
o-Garza,
G., and C
´e
spedes-Mota, A. (2022). An Inventory Model for Non-Instantaneously Deteriorating
Items with Nonlinear Stock-Dependent Demand, Hybrid Payment Scheme and Partially Backlogged
Shortages. Mathematics 10, 434.
[12]
Krishnamoorthy, A., Iqbal Basha, R., and Lakshmy, B. (1994). Analysis of a two commodity
inventory problem. Inf. Manag. Sci., 5(1), 127-136 .
[13]
Kurt, H.S, and James, H.S. (1962). A System for Control of Order Backlog. Manage. Sci.,2(1),
1-6
[14]
Nithya, N., Anbazhagan, N., Amutha, S., Jeganathan, K., and Koushick, B. (2021).
Working Vacation in Queueing-Stock System with Delusive Server. Global and Stochastic Analysis.,
8(1), 53-71.
[15] Nahmias, S. (2011). Perishable Inventory Systems. Springer.,New York, USA.
[16]
Ozkar, S., and Uzunoglu Kocer, U. (2020). Two-commodity queueing-inventory system with
two classes of customers. OPSEARCH., 58(1), 234-256.
[17]
San Jos
´e
, L.A., Sicilia, J., Manuel Gonz
´a
lez de la Rosa, and Jaime Febles-Acosta. (2021).
Profit maximization in an inventory system with time-varying demand, partial backordering and
discrete inventory cycle. Ann. Oper. Res., https://doi.org/10.1007/s10479-021-04161-6.
[18]
Senthil Kumar, P. (2021). A finite source two commodity inventory system with retrial demands
and multiple server vacation. J.Phys.:Conf.Ser. 1850, 1-12.
[19]
Silver, E.A. (1974). A control system for coordinated inventory replenishment. Int. J. Prod. Res.,
12(6), 647-671 .
https://doi.org/10.17993/3cemp.2022.110250.33-48
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
46
Figure 1 – TC versus γ1
γ2=0.8,S
1= 52,s1=5,S
2= 20,s
2=3,N
1=3,N
2=3,c
h1=0.01,c
h2=0.01,c
r=0.55,c
r1=0.45,c
r2=0.5,c
b1=0.1,c
b2=
0.1,c
p1=0.1,c
p2=0.4..
Figure 2 – TC versus γ2
γ1=0.01,S
1= 52,s
1=5,S
2= 20,s
2=3,N
1=3,N
2=3,c
h1=0.01,c
h2=0.01,c
r=0.55,c
r1=0.45,c
r2=0.5,c
b1=0.1,c
b2=
0.1,c
p1=0.1,c
p2=0.4.
7 CONCLUSION
In this article, we examined the substitutable perishable inventory system. Specifically, we analyzed the
structure of the system performance that takes place when a local purchase is made to clear the backlog
instantaneously if both commodities have reached zero and demand is backlogged up to predetermined
levels. Arriving customers follow a Markovian arrival process. The commodities are assumed to be
substitutable. If both commodities have reached zero, demand is backlogged up to predetermined levels.
Graphical results of perishable rates and replenishment rates had been presented. This shows that if
the perishable and replenished rate increases then the total cost would increases. The results of the
contribution were illustrated using numerical patterns to estimate the convexity of the overall cost rate
of this system. The impact of cost values on total expected cost rate were shown. In the future, our
proposed model can be expanded by various reordering policies and described by real data values.
ACKNOWLEDGMENT
Anbazhagan and Amutha would like to thank RUSA Phase 2.0 (F 24-51/2014-U), DST-FIST (SR/FIST/MS-
I/2018/17), DST-PURSE 2nd Phase programme (SR/PURSE Phase 2/38), Govt. of India.
REFERENCES
[1]
Adak Sudip, and Mahapatra, G.S.,(2020). Effect of reliability on multi-item inventory system
with shortages and partial backlog incorporating time dependent demand and deterioration. Ann.
https://doi.org/10.17993/3cemp.2022.110250.33-48
Oper. Res. https://doi.org/10.1007/s10479-020-03694-6.
[2]
Anbazhagan, N., and Arivarignan, G.(2001). Analysis of two commodity Markovian iventory
system with lead time. Korean J. Comput. Appl. Math., 8(2),427-438 .
[3]
Anbazhagan, N., Arivarignan, G., and Irle, A. (2012). A Two-commodity continuous review
inventory system with substitutable items. Stoch. Anal. Appl., 30, 1-19.
[4]
Anbazhagan, N., Goh, M., and Vigneshwaran, B.(2015). Substitutable inventory systems
with coordinated reorder levels. Stat. Appl. Prob., 2(3), 221-234.
[5]
Balintfy, J.L.(1964). On a basic class of multi-item inventory problems. Manag. Sci., 10(2),
287-297.
[6]
Benny, B., Chakravarthy, S.R., and Krishnamoorthy, A. (2018). Queueing-Inventory System
with Two Commodities. J. Indian Soc. Probab. Stat., 19(2), 437-454.
[7]
C
´a
rdenas-Barr
´o
n Leopoldo, Chung Kun-Jen, Kazemi Nima and Shekarian Eh-
san.(2018).Optimal inventory system with two backlog costs in response to a discount offer: corrections
and complements. Oper. Res. Int. J., https://doi.org/18. 10.1007/s12351-016-0255-8.
[8]
Chen, K., Xiao, T., Wang, S., and Lei, D. (2021). Inventory strategies for perishable products
with two-period shelf-life and lost sales. International Journal of Production Research, 59(17),
5301-5320.
[9]
Helal Md Abu, Bensoussan Alain, Ramakrishna Viswanath, and Sethi Suresh, P. (2021).
A mathematical method for optimal inventory policies With backlog sales. Int. J. Transp. Eng.,
11(2), 323 - 340 .
[10]
Karthikeyan, K., and Sudhesh, R. (2016). Recent review article on queueing inventory systems.
Res. J. Pharm. Technol., 9(11), 1451-1461 .
[11]
Khan, M.A.A., Shaikh, A.A., C
´a
rdenas-Barr
´o
n, L.E., Mashud, A.H.M., Trevi
˜n
o-Garza,
G., and C
´e
spedes-Mota, A. (2022). An Inventory Model for Non-Instantaneously Deteriorating
Items with Nonlinear Stock-Dependent Demand, Hybrid Payment Scheme and Partially Backlogged
Shortages. Mathematics 10, 434.
[12]
Krishnamoorthy, A., Iqbal Basha, R., and Lakshmy, B. (1994). Analysis of a two commodity
inventory problem. Inf. Manag. Sci., 5(1), 127-136 .
[13]
Kurt, H.S, and James, H.S. (1962). A System for Control of Order Backlog. Manage. Sci.,2(1),
1-6
[14]
Nithya, N., Anbazhagan, N., Amutha, S., Jeganathan, K., and Koushick, B. (2021).
Working Vacation in Queueing-Stock System with Delusive Server. Global and Stochastic Analysis.,
8(1), 53-71.
[15] Nahmias, S. (2011). Perishable Inventory Systems. Springer.,New York, USA.
[16]
Ozkar, S., and Uzunoglu Kocer, U. (2020). Two-commodity queueing-inventory system with
two classes of customers. OPSEARCH., 58(1), 234-256.
[17]
San Jos
´e
, L.A., Sicilia, J., Manuel Gonz
´a
lez de la Rosa, and Jaime Febles-Acosta. (2021).
Profit maximization in an inventory system with time-varying demand, partial backordering and
discrete inventory cycle. Ann. Oper. Res., https://doi.org/10.1007/s10479-021-04161-6.
[18]
Senthil Kumar, P. (2021). A finite source two commodity inventory system with retrial demands
and multiple server vacation. J.Phys.:Conf.Ser. 1850, 1-12.
[19]
Silver, E.A. (1974). A control system for coordinated inventory replenishment. Int. J. Prod. Res.,
12(6), 647-671 .
https://doi.org/10.17993/3cemp.2022.110250.33-48
47
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
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Sivakumar, K., Vasanthi, C., and Nagarajan, A. (2020). Optimization Of Substitutable
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Smrutirekha, D., Milu, A., and Samanta, G.C. (2015). An inventory model for perishable
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Tai, P., Huyen, P., and Buddhakulsomsiri, J. (2021). A novel modeling approach for a
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s, Q
)inventory system
with random lifetime and two demand classes. OPSEARCH., 57, 104-118.
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Yadavalli, V.S.S., Adetunji, O., Sivakumar, B., and Arivarignan, G. (2010). Two-
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Yadavalli, V.S.S., Sundar, D.K., and Udayabaskaran, S. (2015). Two substitutable
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https://doi.org/10.17993/3cemp.2022.110250.33-48
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
48