a model for combination of set covering and network connectivity in facility location rana afzali...

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

Rana Afzali and Shaghayegh Parhizi

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•Introduction•Set Covering•Network Connectivity•Model Formulation•Case Study•Conclusion•Future Works

A model for combination of set covering and network connectivity

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•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

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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

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• 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 sensorsEach demand point is covered by which

sensor How sensors are connected to each other

A model for combination of set covering and network connectivity

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

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Problem DescriptionA model for combination of set covering and network

connectivity

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Problem DescriptionA model for combination of set covering and network

connectivity

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Problem DescriptionA model for combination of set covering and network

connectivity

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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.)

A model for combination of set covering and network connectivity

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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

A model for combination of set covering and network connectivity

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Minimal Spanning Tree

Four of the spanning trees of the graph

A model for combination of set covering and network connectivity

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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.

A model for combination of set covering and network connectivity

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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 model for combination of set covering and network connectivity

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MODEL FORMULATION

•The goal :minimizing the total cost

cost of locating facilities cost of connecting the facilities

A model for combination of set covering and network connectivity

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D C

B A

D C

B

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MODELA model for combination of set covering and network connectivity

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MODEL A model for combination of set covering and network connectivity

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ModelA model for combination of set covering and network connectivity

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Problem Size

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

A model for combination of set covering and network connectivity

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Numerical Example• Locating sensors in 20 potential capitals of states

to cover all states in USA

A model for combination of set covering and network connectivity

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Result (Location of Sensors)

A model for combination of set covering and network connectivity

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Result(covering)

A model for combination of set covering and network connectivity

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Sensitivity Analysis

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

A model for combination of set covering and network connectivity

Radius Number of facilities (Sensors)

450 8

550 4

1050 2

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Conclusion

•Solving a problem of a combination of set covering and network connectivity problems.

•Developing a model•Applying the model for a real case

A model for combination of set covering and network connectivity

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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

A model for combination of set covering and network connectivity

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Future Work

•Developing heuristic•Using Meta-heuristics for solving the

problem in Large-size

A model for combination of set covering and network connectivity

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Thank you for your attention

A model for combination of set covering and network connectivity