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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

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

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.)

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

Four of the spanning trees of the graph

A model for combination of set covering and network connectivity

Minimal Spanning TreeA 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.

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)}.

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

A model for combination of set covering and network connectivity MODEL

A model for combination of set covering and network connectivity MODEL

A model for combination of set covering and network connectivity Model

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.

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

A model for combination of set covering and network connectivity Sensitivity Analysis

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

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

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

A model for combination of set covering and network connectivity Future Work

- Developing heuristic
- Using Meta-heuristics for solving the problem in Large-size

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