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A model for combination of set covering and network connectivity in facility location

A model for combination of set covering and network connectivity in facility location. Rana Afzali and Shaghayegh Parhizi. Introduction Set Covering Network Connectivity Model Formulation Case Study Conclusion Future Works.

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A model for combination of set covering and network connectivity in facility location

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  1. A model for combination of set covering and network connectivity in facility location RanaAfzali and ShaghayeghParhizi

  2. A model for combination of set covering and network connectivity • Introduction • Set Covering • Network Connectivity • Model Formulation • Case Study • Conclusion • Future Works

  3. A model for combination of set covering and network connectivity • One of the classical objectives in location modeling is “coverage”. • In many optimization problems in networking ,connectivity is a main requirement.

  4. A model for combination of set covering and network connectivity • Both of these two models have been studied a lot separately, but the studies which consider these two together are rare. • Goal: Minimizing the total Cost ,subject to two main constraints • covering and connectivity.

  5. A model for combination of set covering and network connectivity • The problem of locating sensors to minimize the total cost with covering demands points by using sensors while all sensors are connected to each other is considered. • where to put sensors • Each demand point is covered by which sensor • How sensors are connected to each other

  6. A model for combination of set covering and network connectivity Problem Description

  7. A model for combination of set covering and network connectivity Problem Description

  8. A model for combination of set covering and network connectivity Problem Description

  9. A model for combination of set covering and network connectivity Problem Description

  10. A model for combination of set covering and network connectivity SET COVERING • Ensure that each customer considered to be “served” by a set of facilities has a facility within reasonable travel distance. • Introduced by Church and ReVelle(1974) • Many applications such as location of emergency services, the location of retail facilities and signal-transmission facilities (cell-phone towers, light standards, etc.)

  11. A model for combination of set covering and network connectivity NETWOK CONNECTIVITY • several optimization problems with many applications, in which the network connectivity is a requirement. • One of those problems is the minimum cost spanning tree problem. The goal is to find a minimum cost connected subgraph of a network • spanning tree of the graph is a connected subgraph in which there are no cycles

  12. Four of the spanning trees of the graph A model for combination of set covering and network connectivity Minimal Spanning Tree

  13. A model for combination of set covering and network connectivity CHANGING CONTINUOUS REGION TO DISCRETE • feasible region for sitting sensors is continuous • We define the potential nodes as nodes belonging to the network intersect point set .Any point on the network that is r distance away from demand point i∈ N is a NIP. The NIPS is the set of all NIPs plus all demand points.

  14. A model for combination of set covering and network connectivity CHANGING CONTINUOUS REGION TO DISCRETE Define (a, x, b) a non-nodal point at a distance of x from node a on link (a, b) When r =4, the NIPS is {1, 2, 3, (1, 2, 2), (1, 4, 2), (2, 4, 3), (2, 6, 3), (1, 2, 3), (1, 4, 3)}.

  15. A A B B D D C C A model for combination of set covering and network connectivity MODEL FORMULATION • The goal :minimizing the total cost • cost of locating facilities • cost of connecting the facilities

  16. A model for combination of set covering and network connectivity MODEL

  17. A model for combination of set covering and network connectivity MODEL

  18. A model for combination of set covering and network connectivity Model

  19. A model for combination of set covering and network connectivity Problem Size • This model can solve a problem in size of 300 potential points and 500 demand points.

  20. A model for combination of set covering and network connectivity Numerical Example • Locating sensors in 20 potential capitals of states to cover all states in USA

  21. A model for combination of set covering and network connectivity Result (Location of Sensors)

  22. A model for combination of set covering and network connectivity Result(covering)

  23. A model for combination of set covering and network connectivity Sensitivity Analysis • parameters :radius coverage and the cost of locating and connecting the facilities.

  24. A model for combination of set covering and network connectivity Conclusion • Solving a problem of a combination of set covering and network connectivity problems. • Developing a model • Applying the model for a real case

  25. A model for combination of set covering and network connectivity Future Work • A more reasonable model would have a gradual decline in the coverage frequency as a function of distance from the sensor. • Difference if demand points cover by one sensor or more. • Consider coverage radius as a decision variable

  26. A model for combination of set covering and network connectivity Future Work • Developing heuristic • Using Meta-heuristics for solving the problem in Large-size

  27. A model for combination of set covering and network connectivity Thank you for your attention

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