1. B. Jayalakshmi and Alok Singh 2015
A hybrid artificial bee colony algorithm for the cooperative maximum covering location problem
B. Jayalakshmi and Alok Singh 2015

2. The Problem
-The Cooperative Cover Location Problems: the Planar Case comes about in the paper by Berman, Drezner, and Krass (2009).
-This is a Special Case of the Maximum Covering problem
-In this case, facilities emit a “signal” that dissipates as distance increases and if the “signal” strength exceeds a certain threshold when it reaches a demand then the demand is covered.
-The cooperative problem assumes that facilities cooperate to provide coverage to nodes.

3. Formulation of the Cooperative Maximum Covering Location (CMCLP) Problem
-Let G be an undirected graph G= (V,E) where V = (1, …, n) is the set of vertices and E = (e1, … en) is the set of edges
- p = number of facilities and T is the coverage threshold
-All nodes have non-negative weight 𝑤 𝑖 and all edges have a positive length 𝑙 𝑘
-Objective is to maximize the sum total of the weights of the covered demands
𝑓 𝑋 𝑝 , 𝑇 = 𝑖: Φ i 𝑥 𝑝 ≥𝑇 𝑤 𝑖
-The signal strength is defined by a function of the total signals received by a point
Φ i 𝑥 𝑝 = 𝑘=1 Φ(𝑑 i 𝑥 𝑘
-We can locate facilities on edges or nodes

4. Example Cooperative Maximum Covering Problem
Consider the graph in figure 1 with p = 3 and a threshold T = .5 and a signal strength function
of Φ 𝑑 =max⁡{0, 1 − 𝑑 10 }
Solving the problem produces the following solution illustrated on the graph
X = {(4), ([7,8], .6, ([1,2]), .4) where p1 is on node 4, p2 on edge 7,8 with t = .6 and p3 is on edge 1,2 with t = .4.
Here examining node 9 shows how the cooperative coverage problem works. As location 9 is not covered by either point 2 or point 3 on its own. But that the signal strength from Φ(p3) = 1− = .34
And from p2 to node 9 Φ(p2) = 1− = .22 which together produces a signal strength of .56 at 9 and leads to an objective value of 29 as all demands are covered.
1.6 P1
1.8 P2

5. Artificial Bee Covering (ABC) Algorithm
First proposed by Karaboga in 2005
-This algorithm creates three groups of “bees”: scouts, employed, and onlooker
-Initialize the algorithm by sending scouts to find initial food sources
-Repeat:
Send Employed Bees onto food sources and determine nectar amounts
Calculate the probability value of the sources with which they are preferred by the onlooker bees
Send onlooker bees onto food sources and determine their nectar amounts
Stop the exploitation process of the sources exhausted by the bees
Randomly, send the scouts to find new food sources
Memorize the best food source found so far
Stop

6. Bee Employment
In the employed bee phase, the employed bees generate food sources in the proximity of its associated food source and evaluate food quality
If the new food source is better than the employed bee moves to that food sources
The bee employment phase ends when all employed bees finish this process than the onlooker phase begins

7. Bee Onlookers Employed bees share their information with onlookers
Onlookers select food based on quality based on the fitness of the solution
After onlookers select food sources they then determine food sources in the proximity and amongst all food sources in all neighborhoods the best quality food source is determined.

8. How and Why this Approach
In employed bee phase all solutions are equally likely to improve
While in the observer phase good quality solutions are more likely to improve than poor quality solutions
The inclination toward selecting good quality solutions while searching in the proximity of good solutions for better solutions lends strength to the algorithm’s search for an optimal solution
The scout bees randomly searching for new food sources aims to avoid the trap of local optima where our employed and observer bees might get honey trapped.

9. ABC Algorithm and the CMCLP
In this approach, a solution X’ is considered better than X if it has a better objective value or if it provides a larger total coverage
𝑖𝜖V Φ 𝑖 𝑋′ > 𝑖𝜖V Φ 𝑖 𝑋
Food source selection for onlookers selects two food sources from by uniform random distribution and then the better of the two is selected with probability 𝑝 𝑜𝑛𝑙 and the worse 𝑝 𝑜𝑛𝑙 in a binary tournament.

10. Determining an Initial Solution
Start from an empty solution and add one facility at a time
In each iteration to add a facility to the current partial solution S compute the set of points Y that provide exact or greater coverage for yet uncovered nodes V’. x points considered on edges at relative intervals of .1. Evaluate points based on how much coverage they provide and choose R best points from the set Y and select one of the points at random
𝑌=𝑉 𝑈 {𝑥 𝜖 𝐺 Φ 𝑖 𝑥 ≥ 𝑇 𝑖 (𝑋) for some 𝑖 𝜖 𝑉’
After locating a facility the threshold is updated as
𝑇 𝑖 𝑆 = max 0, 𝑇− Φ 𝑖 𝑆 for 𝑖 𝜖 𝑉

11. Neighborhood Solution Generation
To generate a solution 𝑋 ′ in the neighborhood of X, the author deletes F facilities from X randomly
Then the algorithm adds F facilities from a set 𝐹 𝑛𝑒𝑤 that contains all points x in the graph that provide exact coverage for at least one yet uncovered node in 𝑉′.
𝐹 𝑛𝑒𝑤 =𝑍 −𝑋
𝑍=𝑉 𝑈 𝑥 𝜖 𝐺 Φ 𝑖 𝑥 = 𝑇 𝑖 (𝑋)}
These facilities to add are chosen from a set of R best points randomly and 𝐹 𝑛𝑒𝑤 is updated after each facility is added.
This manner is similar to greedy search with the exception of the random aspect.

12. Other Features of the Algorithm
If an employed bees solution does not improve for a specified number of iterations then the associated employed bee becomes a scout
There are no restrictions on the number of scout bees in an iteration
The scout bee is reemployed by assigning it to a new solution generated in the same way as our initial solution

13. Local Search as a Means of Improvement
Local search utilized to improve the algorithm
Each facility x in a solution S is considered one by one and the best point to relocate 𝑏 𝑥 is determined
Then the set 𝐹 𝑛𝑒𝑤 for the solution 𝑆\{𝑥} is computed to find the location of the next facility
If the solution 𝑆\{𝑥} 𝑈 𝑏 𝑥 is better than S then we replace S with the new solution

14. Pseudocode

15. Computational Results Setup
The authors compare their results on the same instance done in the paper: Cooperative Covering Problems on Networks by Averbakh, Berman, Krass, Kalcsics, and Nickel (2014)
They generate five instances of nodes {40, 60, 80, 100, 120, 140, 160, 180, 200} with average degrees {5,6,7}. They solve the problem with p = 3, 4, 5 for n = 40, 60, 80: p =4,5,6 for n = 100, 120, 140: and p =5,6,7 for n = 160, 180, 200
They use three T for signal threshold [.6, .8, 1]. These formulations result in 3625 instances.
They assume 10 employed bees and 20 observer bees. The limit on employed search is set at 𝑝 𝑜𝑛𝑙 = .9 for U (the fraction of diameter of the network) = .15, .25, .35 and T = .6, while 𝑝 𝑜𝑛𝑙 is set to .8 for all other combinations
R= 20 and F = 2 for n = 40; 60; 80 and p = 3; 4; 5, F = 3 for n = 100; 120; 140 and p = 4; 5; 6, F = 4 for n = 160; 180; 200 and p = 5; 6; 7
The algorithm terminates after 500 iterations.
All computations done in C on a Linux based Intel Core i5 2400

16. Computational Results
The authors compare their results with the interchange algorithms proposed in the paper on Cooperative Covering Problems on Networks (2014)
The improvement is measured by 100 ∗ Φ 𝑖 𝑆 𝐴 − Φ 𝑖 𝑆 𝐵 Φ 𝑖 𝑆 𝐵 where 𝑆 𝐴 is the solution from Algorithm A and 𝑆 𝐵 from algorithm B
R is the average execution
On the largest instance
N = 200

17. Conclusion
The ABC algorithm outperforms the existing interchange algorithms in solution quality but are slower than the existing methods.
This is the first metaheuristic presented for the CMCLP problem as tabu search and neighborhood that were experimented with only resulted in slight improvement over interchange and did not represent full metaheuristics.
This papers shows how population based metaheuristics offer an appropriate tool in solving the CMCLP problem.

18. Questions ?