1. Introduction
Inventory management faces many replenishment difficulties. If the supplier’s activities are affected during replenishment, customers may be unable to get vital supplies. As a result, we use two suppliers, referred to in this paper as “regular suppliers” and “temporary suppliers”. For instance, let’s consider textile shops. The system always gives the order to the regular supplier, but the temporary supplier supplies trending dresses, so the system decides to make a trial for a temporary supplier at the same time to ensure the system doesn’t lose the regular supplier. Good connections with servers and customers are among these objectives, and customers should avoid being sent back due to inventory shortage and helping to minimise the total cost. Following that, all arriving customers have the option of purchasing or not purchasing their items after the detailed inquiries in N different states of random environment (RE) in this paper.
Soujanyaa and Vijaya Laxmi [
1] investigated perishable inventory systems with dual supply chains and negative customers. They considered the lead time and perishable rate to be exponentially distributed, the arrival process followed by Poisson, and found a limiting time distribution. Finally, they demonstrate numerical examples for cost rate function and system performance parameters. Yang and Tseng [
2] considered a three-echelon inventory model with a backorder and permissible delay in payments under a controllable lead time. The queueing-inventory system literature has more ordering policies such as [
3,
4,
5,
6].
The system’s behavior is determined by the current state of the Markovian random process with a finite state space. The technique of this operation in a fixed state is called a “random environment”. The system makes an effort to avoid crowds while also being mindful of not losing customers. The system also has several modalities for receiving the demand. This effort helps to satisfy all the customers without any customer accumulation. Kim and Dudin [
7] considered a multi-server and impatient customer in the queueing system with a random environment. The service time is a phase-type distribution. They derived ergodicity conditions for this system with an infinite buffer. Some numerical results are presented. Kim et al. [
8] investigated a multi-server queueing system and two types of customers with infinite buffers. The system is operating in a random environment, and the arrival stream is marked as a Markovian arrival process. Non-preemptive priority is given to type 1 customers over type 2 customers. The service time for two types of customers has a phase-type distribution with different parameters. The correlation in the arrival process is depicted numerically in the section.
In this article, we consider the arrival process as MAP under the fixed state of a random environment. Whenever the states of a random environment are changed, the parameters of the arrival flow and retrying flow are changed as well. Here, we give some reviews about the random environment (see [
9,
10,
11,
12,
13,
14]). In the 1970s, Neuts introduced the Versatile Markovian Point Process (VMPP) as the Markovian arrival process. The VMPP had extensive notations when it was first developed; however, it was considerably simplified by Lucantoni [
15], and this process has been known as the Markovian arrival process ever since. The Markovian arrival process was greatly vetted in the literature [
16,
17,
18,
19,
20,
21]. See Krishnamurthy et al. [
22] for details on the batch Markovian arrival process in the queueing-inventory system.
In this paper, all the arriving customers go into orbit at inventory level zero, and the arriving customers from orbit may or may not buy the product. For example, customers wait for the trending dresses at a nearby textile shop. At the textile shop, after coming across the trending dresses, the customer went to the shop, but the dress colour or size did not fit the customer. At that point, the customer didn’t buy the product and left the system. The responses to the last poll inspired our investigation, and there is no research into random environments with two supply chains to the best of our knowledge.
1.1. Research Gap
The authors above work with RE in the queuing system. This article examines RE in the inventory. The authors above handle the two-echelon chain, and both orders are obtained. In this article, we place an order with a temporary supplier, in case we have not received the order from the temporary supplier when the inventory level has reached some reordering point, we cancel that order and place the order with our regular supplier.
1.2. The Perspective of This Work
The model representation of the stochastic inventory model with two classes of suppliers and impulse customers on RE is portrayed in
Section 2. In
Section 3 the model analysis is portayed. The joint probability distribution of the number of customers in the orbit of infinite size, and the inventory level is evaluated in
Section 4. We measure some important system peculiarities and construct the cost function in
Section 5. Some numerical examples are provided in
Section 6 and, finally, we describe a conclusion in
Section 7.
2. Mathematical Framework of the Model
In this section, we discuss the random environment in the stochastic inventory system with two classes of suppliers named “temporary suppliers” and “regular suppliers”. In this system, an infinite-sized orbit is added, and here there is no waiting space attached to the system. The system has a maximum S units of items. Whenever the inventory level approaches r, we place an order with the temporary supplier for items, where . Similarly, whenever the inventory level reaches , we will order items to the regular supplier and immediately cancel the order for items from the temporary supplier. Both lead times of temporary and regular suppliers are exponentially distributed with and respectively.
The customers arrive according to a Markovian arrival process (MAP) from N different states of RE. Customers who require a single item have the option to choose any arrival mode (state) of the RE. The behaviour of the model depends on the state of the RE. The RE is used on the stochastic process , which is an irreducible continuous time Markov chain with the state space and the infinitesimal generator H. The stationary row vector of the RE is to be obtained by using The underlying Markov chain of the MAP has a generator , which is a square matrix of dimension m with in a fixed state . In this case, denotes no arrival matrix of size m and denotes an arrival matrix of size m under fixed state . The arrival process of customer representation is under fixed state . The average arrival rate of a customer is defined by , where stationary row vector of size is to be obtained by using under fixed state .
Not all incoming customers buy the product, those customers are referred to as impulse customers. Assume that customers buy the product with probability and the complementary probability under fixed state . The service process of the system is assumed to be instantaneous when the inventory level is positive. When the arriving customers find the stock stage is empty, they should enter into orbit. Even though the mode of approach to entering the system is different for each customer, once the customer enters the orbit, they will be considered retrial customers. Retrial customers have the option of choosing another arriving mode of RE during the retry without considering the previous choice of different states of RE. The retrial customers will get into service only if the stock stage is positive. The time between two successive retrials is exponentially distributed with a transition rate .
3. Model Analysis
In this sector, we construct the transition rate matrix on the stochastic inventory system. The Markov process of the form
with state space
where
: The number of customers in the orbit of infinite size waiting place at time t.
: The state of random environment at time t.
: The number of items in the inventory at time t.
: Phase of the arrival process at time t.
The process’s infinitesimal generator P is generated by
Here,
Here,
For
and
:
where
and
.
It may be noted that the , , , and are all square matrices of dimension and the matrices are dimension , where
4. Joint Probability Distribution under Steady State
We give the condition for stability through a theorem on the stochastic inventory system. Let us consider . It is easy to visualize that G is given by
Here
where
and
Here, denotes the steady-state probability vector of G.
The below system of equations are derived from
:
The steady-state probability vector is obtained from the above system of equations and the normalizing condition ().
4.1. Stability Condition
We give the condition for stability through theorem on the stochastic inventory system.
Theorem 1. The stochastic inventory system under study is stable if and only if Proof. From the standard results of Neuts [
23] on the positive recurrence of
P we have
and by applying the structure of the matrices
and
and
the declared results follow. □
It can be seen from the structure of the rate matrix P and from the Theorem 1, that the Markov process with the state space E is regular.
4.2. Steady-State Probability Vector
In this section, we calculate steady-state probability. Let
be the steady-state probability vector of the matrix
P and it is obtained from
,
, where
where
The relation
leads to the following system of equations:
The Markov process
on the state space E and the limiting distribution
where
is the steady-state probability for the state
, exists and is independent of the initial state.
Theorem 2. When the stability condition holds well, the steady-state probability vector Θ is given bywhere the rate matrix , it is defined byand satisfies the matrix quadratic equation Proof. The proof follows from the well-known result on matrix geometric property given by Neuts [
23]. □
Theorem 3. The stationary probability vector is the unique solution of the systemand subject to the normalising condition Proof. The proof is based on Neuts’ [
23] well-known matrix geometric property. And see [
24] more details about the property. □
Once
is found, the boundary probability vector
is computed by using the Equations (
7) and (
8), and the probability vectors
can be obtained from Equation (
5).
5. A Few Significant System Peculiarities
In the segment, we examine a few significant peculiarities measures.
Mean inventory level, is given by
Mean reorder for temporary supplier, is given by
Mean reorder for regular supplier, is given by
Mean number of customers in the orbit
Mean loss rate of arrival of impulse customers
Expected number of time the replenishment is to be done from temporary supplier
Expected number of time the replenishment is to be done from regular supplier
Overall retrial rate
Success retrial rate
Fractional success retrial rate
Construction of the Cost Feature
To construct the expected total cost function per unit time is provided by
where : The inventory carrying cost per unit time.
: Waiting cost of a customer in the orbit per unit time.
: Setup cost per order for temporary supplier.
: Setup cost per order for regular supplier.
: Lost cost of a impulse customer per unit time.
6. Numerical Analysis
We give a few descriptive numerical examples that expose the convexity of the expected cost rate and consider that orders are received with 2 different states of RE; these are confined by the infinitesimal generator
The MAP for appearance of customers are
Negative Correlation (NC):
Positive Correlation (PC):
The demand process has negative (positive) correlated arrival with coefficient of variance and coefficient of correlation with an arrival rate .
The demand process has negative (positive) correlated arrival with coefficient of variance and coefficient of correlation with an arrival rate .
Some discussion about numerical examples of our perspective model and its obtainment with the parameters and cost values are are listed in the Results and Discussion section.
Results and Discussion
We discuss the behavior of the cost function of two variables,
, under hyper-exponential distribution. The values are divulged in
bold in each column to indicate the minimum cost rate, whereas the least cost rate is specified in each row by underlining the values. As a result, a value (bold and underlined) represents the local minimum of the function
. At
=
28 and
=
5, the optimal cost value
=
5.7025. The function
is convex, as shown in
Table 1 and
Figure 1.
Figure 2 depicts a contour plot of the total cost function, which also demonstrates that the function
is convex.
Figure 3 and
Figure 4 show that the mean reorder for temporary supplier
and regular supplier (
) compare with
s and
r respectively.
Figure 3 demonstrates that
decreases and
increases, whenever
s increases.
Figure 4 shows that
is increased and regular suppliers
is decreased whenever
r is increased.
Figure 5 and
Figure 6 show that the mean number of times the replenishment is to be done from a temporary supplier
and regular supplier
. Here,
Figure 5 shows that
decreases and
increases whenever
s increases.
Figure 6 shows that when
r increases,
increases and
decreases.
Figure 7 compares the lead time rate of temporary (
) and regular (
) suppliers with their total expected cost value. Here, the total expected cost value decreases whenever
and
rates are increased.
Table 2 shows that when increasing the lead time rate of the temporary supplier (
) increases the optimal cost value
, whereas when increasing the lead time rate of the regular supplier (
) decreases the
. The
for the lead time rates of two suppliers decreases as the maximum inventory level (
S) increases.
When the setup cost for temporary supplier (
) and regular supplier (
) increases, the value of
increases in
Table 3. When
S increases, the value of
decreases. When the values of
increase, so does the value of
as shown in
Table 4.
As shown in
Table 5, when s and s values rise, the mean inventory level (
) rises, but the mean number of customers in the orbit (
) and the mean loss rate of arrival of impulse customers (
) decreases.
7. Conclusions
In this paper, we discussed two classes of suppliers in the stochastic inventory system on the RE. These two classes of suppliers have a huge impact on the management of the inventory system and also reduce replenishment difficulties. We showed the minimised total expected cost rate (refer to
Table 1 and
Figure 1) and showed it via a contour plot (refer
Figure 2). When the lead time rate of both suppliers is smaller, the total expected cost value of the regular supplier is greater than the total expected cost value of the temporary supplier. Moreover, when the lead time rate of both suppliers goes up, the total expected cost value of the regular supplier is smaller than the total expected cost value of the temporary supplier (refer
Table 2). Similarly, when the setup cost value is small, the total expected cost value of the temporary supplier is smaller than the total expected cost value of the regular supplier. Furthermore, when the setup cost value goes up, the total expected cost value of the temporary supplier is greater than the total expected cost value of the regular supplier (refer
Table 3). Finally, we showed that both the value of the setup cost and the lead time rate are small, the best supplier is a temporary supplier, and both have higher value, making the best supplier a regular supplier.
7.1. Limitations
This study deals with impulse customers and the probability of customers who may buy an item, and the complementary probability . Assuming the probability value is zero, in this case, not all incoming customers will purchase the item, which does not always happen in real-life situations.
The sum of the fixed probability distribution values must be one.
7.2. Future Directions
We will discuss multi-server with phase-type distribution for service.
We plan to investigate multi-suppliers.
We plan to study RE use with payment mode.
Author Contributions
Conceptualization, N.A. and V.V.; data curation, N.A., V.V., and K.J.; formal analysis, N.A. and V.V.; funding acquisition, H.-K.S., G.P.J., and H.M.; methodology, S.A. and K.J.; project administration, H.-K.S. and H.M.; resources, B.S. and H.M.; software, S.A.; visualization, B.S. and G.P.J.; writing—original draft, N.A.; writing—review & editing, G.P.J. All authors have read and agreed to the published version of the manuscript.
Funding
This work was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2020R1A6A1A03038540) and National Research Foundation of Korea (NRF) grant funded by the Korean government, Ministry of Science and ICT (MSIT) (2021R1F1A1046339) and by a grant (20212020900150) from “Development and Demonstration of Technology for Customers Bigdata-based Energy Management in the Field of Heat Supply Chain” funded by Ministry of Trade, Industry and Energy of Korean government.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
Not applicable.
Acknowledgments
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), Government of India.
Conflicts of Interest
The authors declare no conflict of interest.
Notations and Abbreviations
The following Notations and Abbreviations are used in this manuscript:
| The element of submatrix at (i,j) the position of A. |
| Zero matrix. |
| Identity matrix of appropriate dimension. |
| Identity matrix of dimension m. |
| A column vector of 1’s appropriate dimension. |
| Kronecker product of matrices A and B. |
| Kronecker sum of matrices A and B. |
RE | Random Envirnonment. |
MAP | Markovian Arrival Process. |
| |
| |
| |
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