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 - 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
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

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

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
• 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
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