A Fuzzy Graph Theory Approach to the Facility Location Problem: A Case Study in the Indian Banking System
A fuzzy graph <i>G</i> is stated to have a set of trees as its tree cover if all the vertices of <i>G</i> are in their union. The maximum weight tree in the tree cover is assumed to be the cost of a tree cover for a fuzzy graph. For an integer <inline-formula><math x...
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MDPI AG
2023-07-01
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| author | Anushree Bhattacharya Madhumangal Pal |
| author_facet | Anushree Bhattacharya Madhumangal Pal |
| author_sort | Anushree Bhattacharya |
| collection | DOAJ |
| description | A fuzzy graph <i>G</i> is stated to have a set of trees as its tree cover if all the vertices of <i>G</i> are in their union. The maximum weight tree in the tree cover is assumed to be the cost of a tree cover for a fuzzy graph. For an integer <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, finding a set of trees to cover all the vertices of a graph with minimum cost and at most <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> number of spanning trees is known as the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>-tree cover problem. Combining the tree-covering concept and facility location problem in a fuzzy environment for solving critical real-life problems in the recent era is a more fruitful approach. This issue strongly inspires us to develop a model with a practical algorithm. This paper provides an algorithm and complexity analysis to determine the number of rooted trees <i>s</i> covering the given fuzzy graph. In addition, a model is constructed with three optimization programming problems in the facility location problem and a tree covering fuzzy graphs. The model includes two types of the facility location problem, simultaneously addressing a variable covering radius and a fixed covering radius. A numerical example is provided to further describe the model, then, in the application part of the paper, the proposed model is applied to solve the real-life problem of maximizing demand saturation by minimizing the number of small denominations in the Indian banking system. This problem involves the data input of different indicators in the banking system along with details of the denominations of banknotes. |
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| series | Mathematics |
| spelling | doaj.art-cb1884473d124eeeaa03f0d66adf6fe52023-11-18T17:04:15ZengMDPI AGMathematics2227-73902023-07-011113299210.3390/math11132992A Fuzzy Graph Theory Approach to the Facility Location Problem: A Case Study in the Indian Banking SystemAnushree Bhattacharya0Madhumangal Pal1Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore 721102, IndiaDepartment of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore 721102, IndiaA fuzzy graph <i>G</i> is stated to have a set of trees as its tree cover if all the vertices of <i>G</i> are in their union. The maximum weight tree in the tree cover is assumed to be the cost of a tree cover for a fuzzy graph. For an integer <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, finding a set of trees to cover all the vertices of a graph with minimum cost and at most <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> number of spanning trees is known as the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>-tree cover problem. Combining the tree-covering concept and facility location problem in a fuzzy environment for solving critical real-life problems in the recent era is a more fruitful approach. This issue strongly inspires us to develop a model with a practical algorithm. This paper provides an algorithm and complexity analysis to determine the number of rooted trees <i>s</i> covering the given fuzzy graph. In addition, a model is constructed with three optimization programming problems in the facility location problem and a tree covering fuzzy graphs. The model includes two types of the facility location problem, simultaneously addressing a variable covering radius and a fixed covering radius. A numerical example is provided to further describe the model, then, in the application part of the paper, the proposed model is applied to solve the real-life problem of maximizing demand saturation by minimizing the number of small denominations in the Indian banking system. This problem involves the data input of different indicators in the banking system along with details of the denominations of banknotes.https://www.mdpi.com/2227-7390/11/13/2992fuzzy graphtree coverscovering problemtree covering numberfuzzy optimization |
| spellingShingle | Anushree Bhattacharya Madhumangal Pal A Fuzzy Graph Theory Approach to the Facility Location Problem: A Case Study in the Indian Banking System Mathematics fuzzy graph tree covers covering problem tree covering number fuzzy optimization |
| title | A Fuzzy Graph Theory Approach to the Facility Location Problem: A Case Study in the Indian Banking System |
| title_full | A Fuzzy Graph Theory Approach to the Facility Location Problem: A Case Study in the Indian Banking System |
| title_fullStr | A Fuzzy Graph Theory Approach to the Facility Location Problem: A Case Study in the Indian Banking System |
| title_full_unstemmed | A Fuzzy Graph Theory Approach to the Facility Location Problem: A Case Study in the Indian Banking System |
| title_short | A Fuzzy Graph Theory Approach to the Facility Location Problem: A Case Study in the Indian Banking System |
| title_sort | fuzzy graph theory approach to the facility location problem a case study in the indian banking system |
| topic | fuzzy graph tree covers covering problem tree covering number fuzzy optimization |
| url | https://www.mdpi.com/2227-7390/11/13/2992 |
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