# An Analysis on the Markovian Model for Production Management and Establishment of New Pharmacies Using Hyper Graph

Authors: Anitha Florence F. Vinola; G. Padma; Prabu Velusamy
Journal: Journal of Mathematics
 (2022)

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## Abstract

Stochastic models are mainly used to rectify the real-life problems in different areas of the busy world such as marketing, trading, networking, communicating media, and challenging biomedical inventions, for the survival of mankind. Any business which runs on the basis of getting more profit with less investment includes all the resources for production and the man power too, during the current century. To bring the equality in business gain and man power, many stochastic models have been introduced since the 1940s. The different states and the steady state probabilities are discussed by the Markov models to overcome the insufficiency of man power. An Iterative Hidden Markov Model has been introduced for effective production by considering the observable states depend on hidden states under the circumstances of real-world probabilities are known and unknown. This paper mainly deals with the various machines, pharmacies, and different methods, namely, Nonadjacency Search Method and Adjacency Search Method applied through the Hyper graph. Comparison made between Nonadjacency Search Method and Adjacency Search Method through hyper assignments using cliques helps us to obtain tremendous profit through effective production and to start new pharmacies, instead of predictions through computational methods. Results were discussed based on the information obtained from Apollo pharmacies.


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## Body

## 1. Introduction

Markov models are always act as an effective medium among the inventions of various areas such as inventory control, Marketing, customer classification, Economics, Finance, Risk analysis, and Bio-Informatics. Probabilistic Markov models are widely applied in inventories to burnish control over the business. Customer’s demand, available product, production management, and the manpower are the resources involved in the business. The Iterative Hidden Markov Model has been applied in man power and machine power for the effective production management involves the real-world parameters which are estimated by using the minimization problem [1]. Hidden Markov Models (HMMs) are used to identify the threats involved in national security, diseases involved in the human body, and the data sequence analysis in human cell movement. Though the diseases can be arrested and treated by the variety of medicines given by the doctors, paying the bills raised by the hospitals are very big challenges to the patients. The economic evaluation and also the effective cost management are discussed in[2, 3]. During the COVID-19 pandemic situation, people are seeking for the medicines to be delivered to their home and searching for the better services provided by pharmacies. The nonadjacency search method has been applied in bipartite hyper graph to find the route map for purchasing the better medicines in better medicals in minimum cost and to maximize the profit in the production field too [4, 5]. The trend has changed to the new modern society which includes the levels of people and the different methodologies used for the better survival. As the population increases, the production of manufacturing also increases with greater competitions [6]. These healthy competitions make the business world to fight for the survival of their concerns. The workers are always shifted from one place to another place based on the atmospheric conditions, salary, and other benefits. Because of frequent transformations, only concerns remain but not the workers [7, 8].To avoid the shortage of workers, Markov models and many algorithms, namely, the Viterbi algorithm, Maximum likelihood estimators have been applied [9, 10]. Applications of man power in various stages of business are discussed and solution were obtained by using eight point scale and ten point scale of stochastic models. A group of people lying under the same initial point of different diseases are categorized and treated them with an appropriate therapy and the patients of each state are also estimated by simulated Markov models [11, 12]. Monte Carlo methods have been used for data sequence analysis [13, 14]. Beyond the mathematical models and calculations, graphical representations are always the supreme power for validating the Markov models and giving the quick and exact judgments of the models [15]. Hyper graphs are constructed in different methods on the basis of (1) Visual features and (2) View clustering. Samples selected for an experiment are denoted by vertices and the relationships between the samples are represented by the edges of a graph in general. Preferably graphs might be determined by directed or undirected in which the relationships between samples are commensurate or not. The hyper graph has been explored innumerable data mining and information recuperation assignments, as it has the potential for assimilating an interrelationship between samples in higher order [16, 17]. The survey has been conducted among the different pharmacies, whether the proposed models through hypergraph are effective to open more pharmacies of the respective type. Outcomes were discussed in later sections and the solutions arrived at using cliques were shown graphically.


## 2. Basic Definitions

The following are the definitions which are related to the hyper graph that exists in [18–20].Definition 1.
A graph in which each edge is associated with any number of vertices is called hyper graph [21].
(i.e.) A hyper graphχ contains (X, C), where X denotes the vertex set and C denotes the set of hyper edges. Each hyper edge can accommodate any number of vertices from the vertex set X and the set is not empty. Therefore, each hyper edge is a nonempty subset of the vertex set X.Definition 2.
A hyper graphχ (X, C) is bipartite if X is disassociated into two subsets A and B of X whose intersection is null and X=A∪B.
The bipartite hyper graph is represented asχ=A,B,C, A and B are the set of vertices whose intersection is null and C contains all the hyper edges of a given graph. The disassociated vertex sets are equal in number as assumptions made and all hyper edge c ∈ C has the same number c∩A=c∩A>0 of vertices in A and B.Definition 3.
χ=A,B,C is a bipartite hyper graph. The subset C1 of C in the hyper graph χ is called an assignment if any two hyper edges in C1 are nonadjacent.Definition 4.
χ=A,B,C is a bipartite hyper graph. An assignment H in χ is said to be a hyper assignment in χ if every vertex belongs to its union (i.e) v ∈ A∪B is contained in exactly one hyper edge c ∈ H.Definition 5.
Letχ=A,B,C be bipartite hyper graph. A clique of a hyper graph is defined as for every two different vertices of vertex set, there is an edge of hyper graph which includes both the vertices.


## 3. Algorithm for Nonadjacency Search Method

The hyper assignments of a bipartite graph are identified using a Nonadjacency Search Method (NASM).Stage 1: group all the edges which are not adjacent to each edge in the given graph and are denoted byCi (set of all edges which are not adjacent to each of ci, i = 1, 2, …, n).Stage 2: erect the edge setC∗=m∈C1,C2,…Cn:∪c∈mc=A∪B.Stage 3: ifC∗ is empty, then there is no hyper assignment. If C∗ is nonempty then go to stage 4.Stage 4: takeC∗=C1∗,C2∗,…Cm∗,1≤m≤n. Consider C1∗.Stage 5: enumerate the complete edge set which are adjacent to each edge ofC1∗ with respect to C1∗.Stage 6: enumerate the complete edge set whose intersection is null w.r.tC1∗ by applying the set obtained from stage 5. Consider this set as D∗=D1,D2,…Dp:1≤p≤n.Stage 7: ifD∗ is empty then the graph has no hyper assignment related to C1∗ and proceed to stage 9. If D∗ is nonempty then proceed to stage 8.Step 8: for each1≤i≤p,∪c∈Dic=A∪B, then Di is a hyper assignment in χ, or else it is an assignment in χ.Stage 9: apply stage 5 to stage 8 forC2∗,C3∗,…,Cm∗ and eject the enumeration.Stage 10: get the hyper assignments from Stage 8.


## 4. Hyper Graphical Structure for Effective Production

Let us consider the man power with High, Moderate, and Low levels are denoted as set of verticesa1, a2, and a3, and the machines used to manufacture the products with High, Moderate, and Low levels are denoted as a set of vertices b1, b2, and b3. The production companies are denoted as a set of edges c1, c2, c3, c4, c5, and c6. The production companies which are related to the different levels of man power and machines are defined as follows: the production companies c1 and c2 avail high man power a1and highly production rate machine b1. The production companies’ c2, c3, and c4 avail moderate man power a2 and moderate production rate machine b2. The production companies’ c2 and c5 avail high man power a1and moderate production rate machine b2. The production companies’ c5 and c6 avail low man power a3 and low production rate machine b3. The edge set which contains the collection of vertices are represented as follows: c1=a1,b1,c2=a1,a2,b1,b2,c3=a2,b2,c4=a2,b2,c5=a1,a3,b2,b3,c6=a3,b3. The relation among the vertices and edges are shown in Figure 1.Figure 1
Hyper graph (manpower, machines, production companies).LetC1, C2, C3, C4, C5, and C6 be the set of all edges which are not adjacent to each of ci, i = 1, 2, 3, 4, 5, 6.(1)C1=c1,c3,c6,C2=c2,c6,C3=c3,c1,c6,C4=c4,c1,c6,C5=c5,C6=c6,c1,c3,and ∪c∈Cic=A∪B,i=1,2,3,4,6.To apply stage 5 to stage 8 inC1, c1,c3,c6 is the hyper assignment. To apply stage 5 to stage 8 in C2, c2,c6 is the hyper assignment. To apply stage 5 to stage 8 in C4, c4,c1,c6 is the hyper assignment.Therefore,H1=c1,c3,c6,H2=c4,c1,c6,H3=c2,c6 are the hyper assignments which shows that the manpower is more whenever the machines are more, manpower is moderate when the machines are moderate and the manpower is low whenever the machines are low.The production companies can follow any one of these hyper assignments for effective production of products to get maximum profit.


## 5. Algorithm for Adjacency Search Method Using Cliques

The hyper assignments of a bipartite graph are identified using a new model called Adjacency Search Method (ASM) using maximal cliques.Stage 1: collect the set of all cliques of each edgesci, i = 1, 2, …, n of a hypergraphStage 2: identify the edge which contains the clique of maximum order is called the maximal clique represented byci∗,i=1,2,…nStage 3: group all the edges which are adjacent to each edge in the given hypergraph. Let it beCi, i = 1, 2, …, nStage 4: identify theCi, i = 1 in which the maximal clique exists is called the hyper assignment otherwise called an assignment. If the hyper assignment does not exist then go to stage 5Stage 5: proceed stage 4 for eachCi, i = 2, 3, …, n. to get the hyper assignments


## 6. Structure of Hyper Graph for Opening Maximum of New Pharmacies

Reasons for opening a new pharmacy are listed as follows:(1)
More sales⇒ More discount(2)
Less rent for shops⇒ Flexibility to pay medicine bill(3)
More residential area⇒ Expect home delivery(4)
More parking area in shops⇒ Customer’s convenient to park their vehicle(5)
Common transport facility for the customers to reach the branch⇒ Customers do not have own vehicle(6)
Located in visible region⇒ An advertisement to the people passing by(7)
Availability of medicines for customers’ satisfaction⇒ Prescribed by different doctors for various diseasesThe production and sales of a particular product depends on the demand and best service from customers and shop keepers. The manufacturers are always working for customers’ satisfaction. The following are the key points for opening a greater number of Apollo pharmacies in Tamil Nadu. For example, different types of medicines are prepared for different diseases by different manufacturers. Customers avail the medicine from the manufacturers is quite difficult. The numbers of medical shops in different names have been opened to serve the customers. Some shops are getting developed by opening more branches and some shops are tending to close. The following model analyzes the reason for opening/closing medical shops using hyper graph.Letai, i = 1 to 7 be the vertices which represents the strength of manufacturers in view of shop keepers of a hyper graph. Let bi, i = 1 to 7 be the vertices which represents the strength of manufacturers in view of customers of a hyper graph. Let ci, i = 1 to 6 be the edges which represents the medical shops on different names of a hyper graph. The vertices of a hyper graph in view of shop keepers and customers are defined as follows:a1- More salesa2- Less rent for shopsa3- More residential areaa4- More parking area in shopsa5- Common transport facility for the customers to reach the brancha6- Located in visible regiona7- Availability of medicines for customers’ satisfactionb1- More discountb2- Flexibility to pay medicine billb3- Expect home deliveryb4- Customer’s convenient to park their vehicleb5- Customers do not have own vehicleb6- An advertisement to the people passing byb7- Prescribed by different doctors for various diseasesThe edges of a hyper graph are the famous group of pharmacies which are located in Tamil Nadu are defined as follows:c1- Apollo Pharmacyc2- Med Plusc3- Green Pharmacyc4- Muthu Medicalsc5- Bawa Medicalsc6- Thulasi MedicalsThe edge set which contains the collection of vertices are represented as follows:c1=a1,a2,a4,a5,a6,a7,b1,b2,b4,b5,b6,b7,c2=a1,a3,a6,a7,b1,b3,b6,b7,c3=a3,a4,a6,b3,b4,b6,c4=a2,a4,b2,b4,c5=a3,a7,b3,b7,c6=a1,a2,a4,b1,b2,b4. The relation among the vertices and edges are shown in Figure 2.Figure 2
Hyper graph (strength of manufacturers in view of shop keepers, customers, pharmacies).From stage 1, let the cliques of each edgeci, i = 1, 2, …, 6 be.c1 contains the clique of order 12c2 contains the clique of order 8c3 contains the clique of order 6c4 contains the clique of order 4c5 contains the clique of order 4c6 contains the clique of order 6From stage 2, sincec1 contains the clique of maximum order 12, c1 has a maximal clique and is denoted by c1∗.From stage 3, let the set of all adjacent edges ofci, i = 1, 2, …, 6 be.(2)C1=c1∗,c2,c3,c4,c5,c6,C2=c2,c1∗,c3,c5,c6,C3=c3,c1∗c2,c4,c5,c6,C4=c4,c1∗,c3,c6,C5=c5,c1∗,c2,c3,C6=c6,c1∗,c2,c3,c4.From stage 4, allCi’s, i = 1, 2, …, 6 contains a maximal clique c1∗. Hence Ci’s, i = 1, 2, …, 6 are the hyper assignments.Since the maximal cliquec1∗ exists in all hyper assignments and the corresponding edge c1 is adjacent to all other edges of a given hypergraph, every clique of a hyper assignment is connected to each other. Therefore, the hypergraph corresponding to the given model representing the Markov chain is irreducible [22]. Hyper assignments are identified using cliques rather than the nonadjacency search method is reliable and adapted to different circumstances.Hence, it is concluded that the edgec1 which represents the Apollo pharmacy meets the maximum number of conditions on customers view and the shopkeepers view and it is being opened in large numbers in recent years compared to other pharmacies.


## 7. Results and Discussion

Based on the survey conducted among the production machines and the pharmacies around 850 numbers, the following queries were put forth whether the proposed modified model ASM in terms of cliques would be (1) Applicable, (2) Implemented, (3) Agreed, (4) Easy, and (5) Effective? It has been answered by the people that the model is applicable by 800 numbers, implemented by 780 numbers, agreed by 830 numbers, easy by 830 numbers, and effective by 780 numbers. Figure3 and Table 1 show the data sheet of the sample analyzed in the study.Figure 3
Bar chart representation of number of pharmacies accepting different criteria for starting new pharmacies.Table 1
Data for the pharmacies to adopt ASM.
S. noCriteriaNo. of pharmacies1Applicable8002Implemented7803Agreed8304Easy8305Effective780From the bar chart, it is very clear that the production machines and the pharmacies which agreed and easy to manipulate the ASM have the high production rate, and also the number of Apollo pharmacies has been started in more numbers than the other pharmacies discussed in earlier sections 4, 6. Though the Apollo hospitals originated in Andhra Pradesh, their branches have started deliberately in Chennai in the year 2000 with less than 5 hospitals. But thereafter the Apollo pharmacies were started gradually in and around Chennai along with the tough competitors like Med plus, Green pharmacy, Muthu Medicals, Bawa medicals, and Thulasi medicals. The growth of these pharmacies is analyzed in the years 2000, 2005, 2010, 2015, 2020, and 2021 and shown in the following Table2.Table 2
Data for the growth of different pharmacies in Chennai from 2000 to 2021.
S. noYearsApollo pharmacyMed plusGreen pharmacyMuthu medicalsBawa medicalsThulasi medicals12000165152014022005200210023132010265270244242015280290168325202030032518104562021322325191247The growth of different pharmacies in the years 2000, 2005, 2010, 2015, 2020, and 2021 are represented as numbers 1 to 6 in theX-axis of Figure 4. From the trend lines, it is very clear that the growth of Apollo Pharmacy increased gradually than the other pharmacies. Though the Apollo pharmacies and Med plus have equally good in the growth till 2020, and the year 2021 shows more advancement of Apollo pharmacies. The growth of Apollo pharmacies is highly effective and started new pharmacies in different corners of cities and out of cities. The Markovian model applied in Hyper graph is discussed in terms of cliques. Hyper assignments were identified using cliques and the pharmacy which has been started in larger numbers is highlighted. The ASM is adequate to the Apollo pharmacies and the reasons for opening new pharmacies in view of customers and shopkeepers are interrelated. Therefore, the relation between the customers and the shopkeepers is very strong and shows the ASM is suitable for Apollo pharmacies. The pandemic situation due to COVID-19 is also being organized by the Apollo pharmacies in 24 × 7 and the medicines are delivered to the door step of customers [23]. The result and discussion conclude that the pharmacies in different names which adopt ASM have started more new branches.Figure 4
Trend line for the growth of pharmacies in Chennai.


## 8. Conclusion

In this paper, the nonadjacency search method (NASM) has been applied in the bipartite hyper graph and the hyper assignments are identified to analyze the better combinations of machines and manpower for effective production management. The modified method called adjacency search method (ASM) has been applied in the bipartite hyper graph and the hyper assignments are identified to analyze the reasons for opening new pharmacies using maximal clique. It is concluded that the hyper assignments of the given hypergraph preserve an irreducible Markov chain which gives strengths to the model construction and a greater number of Apollo Pharmacies are opened in recent years compared to the other pharmacies. In comparison with the NASM and ASM, it is observed that the adjacency search method plays an important role to determine the Apollo pharmacy has strong reasons to open in more numbers than the other pharmacies and is easily adaptable. ASM emphasize the role of graphical representation of HMMs in terms of hyper graph. Hyper assignments obtained from ASM using maximal clique is more efficient and time limited than the NASM.



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