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# A model for combination of set covering and network connectivity in facility location - PowerPoint PPT Presentation

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

RanaAfzali and ShaghayeghParhizi

• Introduction

• Set Covering

• Network Connectivity

• Model Formulation

• Case Study

• Conclusion

• Future Works

• One of the classical objectives in location modeling is “coverage”.

• In many optimization problems in networking ,connectivity is a main requirement.

• 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.

• 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

Problem Description

Problem Description

Problem Description

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

• 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

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

• 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 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)}.

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

MODEL

MODEL

Model

Problem Size

• This model can solve a problem in size of 300 potential points and 500 demand points.

Numerical Example

• Locating sensors in 20 potential capitals of states to cover all states in USA

Result (Location of Sensors)

Result(covering)

Sensitivity Analysis

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

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 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 Work

• Developing heuristic

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