what is network analysis in operations research

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Chapter 3: network analysis.

Many real-world problems in various fields can be formulated as optimization problems on networks. A network consists of nodes and edges that connect two nodes of the network to each other. A network is called directed if all edges can only be traversed in one direction. In an undirected network, every edge may be traversed in either direction. An undirected network can always be reduced to a directed network (why?). Network optimization problems can generally be solved with (continuous) linear programming. However, using the special structure of such problems, one can develop algorithms that reach the optimal solution much more quickly than linear programming algorithms. In this chapter, we discuss some of the most important models from network theory. In Section 3.1, we consider the shortest-path problem. We discuss Dijkstra’s algorithm for finding the shortest path between nodes in a network with nonnegative distances. Section 3.2 deals with the maximum-flow problem, in which the maximum amount of flow (for example, liquid or traffic) that can go from one node in the network to another node is sought, taking into account capacity restrictions on the edges between the nodes. In Section 3.3, we discuss the minimum spanning tree problem. We discuss Prim’s algorithm, which can be used to calculate how to connect a given number of nodes by a spanning tree at the lowest cost. We also briefly touch upon the related Steiner tree problem. Section 3.4 deals with the Euler circuit problem, where in an undirected graph, a path is sought that passes through every edge of the graph exactly once and ends at the starting point of the path. We then discuss the Chinese postman problem, in which costs are attached to the edges of the graph and a circuit is sought with the lowest total cost. Finally, in Section 3.5, we briefly discuss the general minimum-cost flow model that can be used for a large number of network optimization problems; in particular, we mention the assignment problem and the associated Hungarian solution method.

Recommended

What is Network Analysis?

A brief introduction with examples.

Mengsay Loem

Towards Data Science

If you are using any social media applications, you may have experienced the friend or follower suggestions functions. Have you ever wondered how these functions work? One common technology used in these cases is Network Analysis.

What is a Network?

A network refers to a structure representing a group of objects/people and relationships between them. It is also known as a graph in mathematics. A network structure consists of nodes and edges. Here, nodes represent objects we are going to analyze while edges represent the relationships between those objects.

For example, if we are studying a social relationship between Facebook users, nodes are target users and edges are relationships such as friendships between users or group memberships. In Twitter, edges can be following/follower relationships.

Why Network Analysis?

Network Analysis is useful in many living application tasks. It helps us in deep understanding the structure of a relationship in social networks, a structure or process of change in natural phenomenons, or even the analysis of biological systems of organisms.

Again, let’s use the network of social media users as an example. Analyzing this network helps in

• Identifying the most influent person/people in a group
• Defining characteristics of groups of users
• Prediction of suitable items for users
• Identifying CM targets ,etc.

Other easy-to-understand examples are the Friend Suggestion function in Facebook or Follow Suggestion function in Twitter.

Who is the Important Person?

A crucial application of network analysis is identifying the important node in a network. This task is called Measuring Network Centrality. In social network analysis, it can refer to the task of identifying the most influential member, or the representative of the group.

For example, which node do you think is the most important one in the above figure?

Of course, to define the most important node, we need a specific definition of the Important Node. There are several indicators used to measure the centrality of a node.

• Degree centrality : node with a higher degree has higher centrality
• Eigenvector centrality : adding to the degree of one node, the centralities of neighbor nodes are considered. As a result, the eigenvector corresponding to the highest eigenvalue of the adjacency matrix represents the centrality of nodes in the network
• Betweenness centrality : the number of paths between two nodes that go through the i-th node is considered as the i-th node’s betweenness centrality.
• Closeness centrality : the length of the path from the i-th node to other nodes in the network is considered as the i-th node’s closeness centrality. With this definition, for example, this centrality can be applied in the task of defining a suitable evacuation site in a city.

Who are we?

Another application of network analysis is the Community Detection task. This task purpose to divide a network into groups of nodes that are similar in any specific features. Examples of this task are a task of defining groups of users in SNS who share common interests/opinions, find groups of customers to advertise specific items, recommendation systems in online shopping systems, etc.

Many researchers are working on algorithms to effectively solve community detection problems. Some well-known algorithms/methods in this task are Kernighan-Lin algorithms, Spectral Clustering, Label propagation, Modularity Optimization, etc.

What is else?

Besides these applications, network analysis also plays important role in time series analysis, natural language processing, telecommunication network analysis, etc. Recently, the technology of Machine Learning (Deep Learning) is also used in network analysis. In this case, research on Graph Embedding and Graph Neural Networks are interesting topics.

For more detail, I recommend the following sites and textbooks.

• Network Science ( http://networksciencebook.com/ )
• Networks: A Very Short Introduction ( http://www.veryshortintroductions.com/view/10.1093/actrade/9780199588077.001.0001/actrade-9780199588077 )
• Networks, Crowds, and Markets ( https://www.cs.cornell.edu/home/kleinber/networks-book/ )

Written by Mengsay Loem

Researcher; Machine Learning; Natural Language Processing;

Text to speech

Operations Research OER

Network Model ¶

Introduction ¶.

Many problems in computer science can be presented in the form of graphs which consists of nodes & links between them. Network flow problems can be taken as an example, as it involves the transportation of goods and material across networks. The motivation for taking advantage of their structure usually has been the need to solve larger problems than otherwise would be possible to solve with existing computer technology. Historically, the first of these special structures to be analyzed was the transportation problem, which is a particular type of network problem. The development of an efficient solution procedure for this problem resulted in the first widespread application of linear programming to problems of industrial logistics. More recently, the development of algorithms to efficiently solve particular large-scale systems has become a major concern in applied mathematical programming.

History Of Network Model ¶

It was introduced by Charles Bachaman and developed into standard specifications by Conference on Data System Language (CODASYL) Consortium in 1969.Researchers sponsered by the U.S. Department of Defensen’s Advanced Researvh Projects Agency created the First Packet-switched network, called the ARPANET, in 1969.

Today’s globalization as the world has evolved to an advanced planet in Information Technology because of computer networks. One of the key contributing factors of the information technology rise in the world is network and data communication because of network and technology’s advancement as the system. The goal of network model has always been same, which is convergence and achieving service integration. From early 1960s to present, the development and progress that network model has gone through has always accommodated the rapidly increasing number of users and applications. It has made the global environment of information and technology transparent to applications.

Example 1 ¶

Shortest route/path problem ¶.

In Network Models, problems can be presented in many ways. Here, one of the common problem is the shortest route problems. Shortest route problem is a network model problem which has received a great deal of attention for both the practical and theoretical reasons. For instance, given a network with a distance associated with each arc, this network model helps us to find the shortest distance from origin (source) to the destination (which is called sink).

These problems can be formulated in real life problems like equipment replacement, capital, investment, project scheduling, and inventory planning. The theoretical interest of this problem is due to the efficient solution problems. Shortest route problems can be interpreted as a network problem very easily.

A graph’s shortest route issue asks for a path between vertices that has the least total weight of all its edges [5-7]. In which we must identify all the paths and then we must sum of all the distances of specific path. Then the path has less distance it will be considered as the shortest route.

Find the shortest path in the given graph from point A to E.

It’s a directed graph firstly we will see all possible path in it. So, the possible paths are:

After executing the whole path, we will calculate the distance of every node included in the selected path then the path has overall less distance will be the shortest path.

In the bracket [m, n] m is basically distance between two nodes and the n is parent node from which we are coming from. So, as above 3 is distance between A to C which is m and, C parent node is A which is n. So, the distance of path 1 is 10

The distance of path 2 is 12

The distance of path 3 is 15

The distance of path 4 is 17

The distance of path 5 is 7.

So, the results are in the given table:

So, the shortest path is path 5 which is A-B-D-E with distance 7.

Solving Example 1 to find shortest path using python ¶

Example 2 ¶, maximal overflow ¶.

The maximum quantity of flow that can flow from a source to a sink across a network is what this term refers to. The maximum flow problem may be solved using a variety of techniques [1]. Ford-Fulkerson method and Dinic’s Algorithm are two popular solutions methods to this type of problem. Here, we are going to be using Ford-Fulkerson method to solve the problems.

Heres how to implement the problems. ¶

Find the Max flow for the given graph G from the source S to destination T.

**Path 1:**Set the edge flow to 0 on all of them. Find a augmenting path in the residual network that has a higher capacity. In this case, I’m going to choose the S-a-b-d-c-T. A bottleneck capacity is the maximum flow that a route can handle in a certain time. You can see that the bottleneck capacity is 2 along this line. Update the edge flow values in the augmenting path now, but don’t go over the capacity limit. After that, you’ll have the flow network below and the residual network to deal with.

Flow= S-a-b-d-c-T =2

Path 2: Then I select the augmenting path S-a-c–T. Now the bottleneck capacity is 4.

Flow= S-a-c-T = 4

Path 3: Then I select the augmenting path S-a-d-c-T. Now the bottleneck capacity is 4.

Flow= S-a-d-c-T= 4

Path 4: Then I select the augmenting path S-b-a-d-T. Reverse flow in it so there is no bottleneck.

Reverse FLOW

Path 5: Then I select the augmenting path S-b-d-T. Now the bottleneck capacity is 9.

Flow=S-b-d-T= 9 The residual graph is now empty since there are no more routes connecting s and t. As a result, there’s no way to make it flow. The Ford Fulkerson technique is complete, and we can now proceed to finding the highest flow rate possible using this information. The maximum flow is equal to the source flow. And the max flow for this is given below Max Flow= 2+4+4+9=19

Solving the above Question in example 2 to find maximal flow using python ¶

Example 3 ¶, minimum spanning tree ¶.

Minimizing the lengths (or “weights”) of the edges of a tree is known as a minimal spanning tree [2],[3], [4]. A cable company, for example, could wish to build cable in several areas at the same time. By laying less cable, the cable business saves money. For any pair of vertices in a tree, there is only one path that connects them. A graph’s spanning tree is one that:

Step 1: Here we are writing the edges that connected with each other and their distance is also mentioned in the table so find the minimum spanning tree for this tree.

**Step 2:**Firstly, we are selecting the edge which has most less distance between the so D to E has less distance that’s why we choose it.

**Step 3:**Now, select the node that is connected to D or E with with the less distance as we see in the tree D to G is 3 which is less distance among all the nodes connected with the D and E. Also adding their entry in the table.

**Step 4:**Now here we are also searching for the node that has least distance among all the nodes connected to D, E, and G.

Step 5: Searching for the node that has least distance among all the nodes connected to D, E, C and G.

Step 6: Searching for the node that has least distance among all the nodes connected to D, E, C, H and G.

Step 7: Searching for the node that has least distance among all the nodes connected to D, E, C, H, F and G.

Step 8: Searching for the node that has least distance among all the nodes connected to D, E, C, H, F, B and G.

Step 9: So here we discover all the nodes and discovered all these through visiting the path with smallest distance and we can’t make a cycle in it and cant visit the node twice.

So, the total cost we got = 28

Solving the above Question in example 3 for Minimum spanning tree using python ¶

Question 1 ¶.

Find the shortest path from the given graph from point A to E.

Question 2 ¶

Find the shortest path from the given graph from point A to F.

Question 3 ¶

Question 4 ¶

Question 5 ¶.

Question 6 ¶

Question 7 ¶

Find the Minimum Spanning Tree of the following from point A to G.

Question 8 ¶

Find the Minimum Spanning Tree of the following from point A to H.

Question 9 ¶

Find the Minimum Spanning Tree of the following network from point 1 to 8.

Question 10 ¶

Find the Minimum Spanning Tree of the following from poinnt 1 to 9.

Project Ideas ¶

Network model is such a huge topic, it pretty much covers every concepts within the information and technology. One of the reasons why one must understand the concept of network model is because it helps to break down larger problems into smallers ones and split them into chunks so that they can be easilt managed. They represent a framework for dividing up the tasks needed to implement a network. The basic features of network model, such as optimization and modularity, has always been a boon for large scale industries to small businessess which as continuouly helped them enhance and modify systems for a common cause.

Overall, we were so fascinated by this topic and the idea of network science, we wanted to explore the diversity, history and culture of this model. This really got us going.

References ¶

Bradley, Hax, and Magnanti. Network Models. Applied Mathematical Programming. Addison – Wesley, 1977. http://web.mit.edu/15.053/www/AMP.htm . Accessed 16 Sep 2021.

Barua, Saurav. (2019). Operation Research Problems Solving in Python. 10.13140/RG.2.2.20719.38565. https://www.researchgate.net/publication/335572221_Operation_Research_Problems_Solving_in_Python . Accessed 18 Sep 2021.

Taha, H. A. (2017). Network Model). In Operations research: An introduction (10th ed.). pp. 220, 225, 235, Pearson. Accessed 19 Sept 2021.

L. R. Ford and D. R. Fulkerson, “Maximal Flow Through a Network,” Canadian Journal of Mathematics, vol. 8, pp. 399–404, 1956, doi: 10.4153/cjm-1956-045-5. Accessed 20 Oct 2021.

A. Kershenbaum, “Computing capacitated minimal spanning trees efficiently,” Networks, vol. 4, no. 4, pp. 299–310, 1974, doi: 10.1002/net.3230040403. Accessed 18 Oct 2021.

T. C. Hu, “Optimum Communication Spanning Trees,” SIAM Journal on Computing, vol. 3, no. 3, pp. 188–195, Sep. 1974, doi: 10.1137/0203015. Accessed 18 Sep 2021.

A. A. and N. Eckson, “Robust Water Distribution System using Minimal Spanning Tree Algorithm,” International Journal of Computer Applications, vol. 158, no. 1, pp. 1–4, Jan. 2017, doi: 10.5120/ijca2017912724. Accessed 18 Oct 2021.

A. Fitro, S. Suryono, and R. Kusumaningrum, “Shortest Route at Dynamic Location with Node Combination-Dijkstra Algorithm,” Proceeding of the Electrical Engineering Computer Science and Informatics, vol. 5, no. 1, Nov. 2018, doi: 10.11591/eecsi.v5.1604. Accessed 18 Oct 2021.

Bing Liu et al., “Finding the shortest route using cases, knowledge, and Djikstra’s algorithm,” IEEE Expert, vol. 9, no. 5, pp. 7–11, Oct. 1994, doi: 10.1109/64.331478. Accessed 18 Oct 2021.

J. Wang and T. Kaempke, “Shortest route computation in distributed systems,” Computers & Operations Research, vol. 31, no. 10, pp. 1621–1633, Sep. 2004, doi: 10.1016/s0305-0548(03)00111-4. Accessed 9 Oct 2021.

Principal author of this chapter were: Sachin Samal, Prakash Silwal, Watan Chaudhary

Contributions were also made by: Dr. Nicholas Jacob

Goal Programming

Transportation Theory

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OPERATIONS RESEARCH NETWORK OPTIMIZATION MODELS

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In this paper, we are concerned with the maximum flow problem in the distribution network, a new kind of network recently introduced by Fang and Qi. It differs from the traditional network by the presence of the D-node through which the commodities are to be distributed proportionally. Adding D-nodes complicates the network structure. Particularly, flows in the distribution network are frequently increased through multiple cycles. To this end, we develop a type of depth-first-search algorithm which counts and finds all unsaturated subgraphs. The unsaturated subgraphs, however, could be invalid either topologically or numerically. The validity are then judged by computing the °ow increment with a method we call the multi-labeling method. Finally, we also provide a phase-one procedure for finding an initial flow.

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Numerous networks, such as transportation, distribution and delivery networks optimize their designs in order to increase efficiency and lower costs, improving the stability of its intended functions, etc. Networks that distribute goods, such as electricity, water, gas, telephone and data (Internet), or services as mail, railways and roads are examples of transportation networks. The optimal design fixes network architecture, including clustering, degree distribution, hierarchy, community structures and other structural metrics. These networks are specifically designed for efficient transportation, minimizing transit times and costs. All sorts of transportation networks face the same problem: traffic congestion among their channels. In this work we considered a transportation network model in which we optimize/minimize a cost function for the flux/current at each channel/link of the network. We performed simulations and an analytical study of this problem, focusing on the fraction o...

European Journal of Operational Research

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operations research (OR)

• Sarah Lewis

Operations research (OR) is an analytical method of problem-solving and decision-making that is useful in the management of organizations. In operations research, problems are broken down into basic components and then solved in defined steps by mathematical analysis.

The process of operations research can be broadly broken down into the following steps:

• Identifying a problem that needs to be solved.
• Constructing a model around the problem that resembles the real world and variables.
• Using the model to derive solutions to the problem.
• Testing each solution on the model and analyzing its success.
• Implementing the solution to the actual problem.

Disciplines that are similar to, or overlap with, operations research include statistical analysis , management science, game theory, optimization theory, artificial intelligence and network analysis. All of these techniques have the goal of solving complex problems and improving quantitative decisions.

The concept of operations research arose during World War II by military planners. After the war, the techniques used in their operations research were applied to addressing problems in business, the government and society.

Characteristics of operations research

There are three primary characteristics of all operations research efforts:

• Optimization- The purpose of operations research is to achieve the best performance under the given circumstances. Optimization also involves comparing and narrowing down potential options.
• Simulation- This involves building models or replications in order to try out and test solutions before applying them.
• Probability and statistics- This includes using mathematical algorithms and data to uncover helpful insights and risks, make reliable predictions and test possible solutions.

Importance of operations research

The field of operations research provides a more powerful approach to decision making than ordinary software and data analytics tools. Employing operations research professionals can help companies achieve more complete datasets, consider all available options, predict all possible outcomes and estimate risk. Additionally, operations research can be tailored to specific business processes or use cases to determine which techniques are most appropriate to solve the problem.

Uses of operations research

Operations research can be applied to a variety of use cases, including:

• Scheduling and time management.
• Urban and agricultural planning.
• Enterprise resource planning ( ERP ) and supply chain management ( SCM ).
• Inventory management .
• Network optimization and engineering.
• Packet routing optimization.
• Risk management .

Continue Reading About operations research (OR)

• The big picture of Operations Research

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Operations Research

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operations research

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operations research , application of scientific methods to the management and administration of organized military, governmental, commercial, and industrial processes.

Operations research attempts to provide those who manage organized systems with an objective and quantitative basis for decision; it is normally carried out by teams of scientists and engineers drawn from a variety of disciplines . Thus, operations research is not a science itself but rather the application of science to the solution of managerial and administrative problems, and it focuses on the performance of organized systems taken as a whole rather than on their parts taken separately. Usually concerned with systems in which human behaviour plays an important part, operations research differs in this respect from systems engineering , which, using a similar approach, tends to concentrate on systems in which human behaviour is not important. Operations research was originally concerned with improving the operations of existing systems rather than developing new ones; the converse was true of systems engineering. This difference, however, has been disappearing as both fields have matured.

The subject matter of operations research consists of decisions that control the operations of systems. Hence, it is concerned with how managerial decisions are and should be made , how to acquire and process data and information required to make decisions effectively, how to monitor decisions once they are implemented , and how to organize the decision-making and decision-implementation process. Extensive use is made of older disciplines such as logic, mathematics, and statistics, as well as more recent scientific developments such as communications theory, decision theory , cybernetics, organization theory , the behavioral sciences, and general systems theory .

In the 19th century the Industrial Revolution involved mechanization or replacement of human by machine as a source of physical work. Study and improvement of such work formed the basis of the field of industrial engineering . Many contemporary issues are concerned with automation or mechanization of mental work. The primary technologies involved are mechanization of symbol generation (observation by machines such as radar and sonar), mechanization of symbol transmission (communication by telephone, radio, and television), and mechanization of logical manipulation of symbols (data processing and decision making by computer). Operations research applies the scientific method to the study of mental work and provides the knowledge and understanding required to make effective use of personnel and machines to carry it out.