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

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

RanaAfzali and ShaghayeghParhizi


A model for combination of set covering and network connectivity

  • Introduction

  • Set Covering

  • Network Connectivity

  • Model Formulation

  • Case Study

  • Conclusion

  • Future Works


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.


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.


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


Problem description

A model for combination of set covering and network connectivity

Problem Description


Problem description1

A model for combination of set covering and network connectivity

Problem Description


Problem description2

A model for combination of set covering and network connectivity

Problem Description


Problem description3

A model for combination of set covering and network connectivity

Problem Description


Set covering

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.)


Netwok connectivity

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


Minimal spanning tree

Four of the spanning trees of the graph connectivity

A model for combination of set covering and network connectivity

Minimal Spanning Tree


Changing continuous region to discrete

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.


Changing continuous region to discrete1

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)}.


Model formulation

A connectivity

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





Problem size

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.


Numerical example

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


Result location of sensors

A model for combination of set covering and network connectivity

Result (Location of Sensors)



Sensitivity analysis

A model for combination of set covering and network connectivity

Sensitivity Analysis

  • parameters :radius coverage and the cost of locating and connecting the facilities.


Conclusion

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


Future work

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


Future work1

A model for combination of set covering and network connectivity

Future Work

  • Developing heuristic

  • Using Meta-heuristics for solving the problem in Large-size


Thank you for your attention

A model for combination of set covering and network connectivity

Thank you for your attention


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