Exploring Further Advantages in an Alternative Formulation for the Set Covering Problem

Table 1

Notation.

	Cardinal of a given set.
	Relative importance of pheromone trails, .
	The set of columns that row covers, .
	Binary matrix of m-rows and n-columns. The rows are the elements of the universe and the columns are the subsets .
	Percentage of columns to be added during the generation of a solution in ABC.
	Percentage of columns to be removed during the generation of a solution in ABC.
	Heuristic factor of column at step for the alternative formulation, .
	Value in the cell of . It equals 1 if the -th column covers the -th row and 0 otherwise, , .
	Point/points in which a function gets its maximum value/values.
	Point/points in which a function gets its minimum value/values.
	The sum of the gains of covering the noncovered rows which could be covered by column at .
	Relative importance of heuristic information, .
	Set of costs, .
	Heuristic factor of column at step for the classical formulation, .
	Cost associated to the -th column, .
	Cost of the cheapest set among the sets covering , , .
	Number of noncovered rows which could be covered by column at step .
	Solution generated by an ant at step , .
	A solution to the problem, .
	Ratio coefficient, .
	Heuristic factor of column at step , .
	Universe of elements, .
	-th element of , , .
	Average fitness evaluations needed for reaching the best solution found during the exploration.
	Fitness function of the classical SCP formulation, .
	Fitness function of the alternative SCP formulation, .
	Gain of covering an element , , .
	Row set, .
	Row set covered by column , .
	Percentage of improvement by considering the alternative approach of the algorithm instead of the default version.
	Column set, .
	The best solution found from the beginning of the algorithm, .
	Threshold value based on trials for deciding if a worker bee is transformed into a scout one in ABC, .
	Cardinal of or number of rows.
	Cardinal of or number of columns.
	Number of worker bees in ABC, .
	Amount of pheromone put on column , .
	Very small positive constant, .
	Population size in ACO.
	Population size in ABC.
	Landscape solution quality metric, .
	Random number uniformly distributed in .
	Parameter determining the relative importance of exploitation versus exploration, .
	Pheromone persistence, .
	Set of unselected columns in , , .
	Average RPD from a distribution, .
	Average RPD of algorithm , with .
	Average RPD of algorithm , with .
	Average RPD of algorithm , with .
	Average RPD of algorithm , with .
	Minimum RPD from a distribution, .
	Maximum RPD from a distribution, .
	Set of subsets, , .
	-th subset of , , , .
	Landscape rate of success, .
	Landscape speed of reaching a solution, .
	Pheromone trail of column , .
	Construction step in ACO, .
	Average execution time of algorithm , with .
	Average execution time of algorithm , with .
	Average execution time of algorithm , with .
	Average execution time of algorithm , with .
	Number of uncovered rows in a solution, .
	Set of uncovered rows in , , .
	Number of columns in a solution that cover the row , .
	An SCP solution expressed as binary vector, .
	-th element of . It equals 1 if is part of the solution and 0 otherwise, .
	Variable equaling 1 if the element is covered in and 0 otherwise, .
	Column provided by SROM at step , .
	Average cost obtained from a distribution, .
	Maximum cost obtained from a distribution, .
	Minimum cost obtained from a distribution, .
	Optimum solution of a given instance, .
	Column selected at step , .